The associated long exact sequence of perverse cohomology on $V$ splits, due to the weights, into short exact sequences

$$0 \to \mathcal{P}H^0(\pi_*j'_*\mathbf{K}) \xrightarrow{u_0} \mathcal{P}H^0(\pi_*j'_*\mathbf{L}) \xrightarrow{v_0} \mathcal{P}H^0(\pi_*j'_*\mathbf{I}) \to 0$$

In particular, we have induced sub-exact sequence of perverse sheaves on $V$:

$$0 \to j_* \mathbf{K} \xrightarrow{j_* u} j_* \mathbf{L} \xrightarrow{j_* v} j_* \mathbf{I} \to 0$$

Hence $(j_* \mathbf{K}, j_* u)$ is a kernel, and $j_* \mathbf{I}$ is the image of $j_* g : (j_* \mathbf{L}, F) \to (j_* \mathbf{L}', F)$. The proof of the existence of co-image and co-kernel is similar.

**Remark 7.1.** The morphism $j_* g : j_* (\mathbf{L}, F) \to (j_* \mathbf{L}', F)$ is a canonical extension of $g$ and all morphisms are obtained in this way in the category.

For each irreducible closed subset $S \subset V$, we have a natural functor $\mathcal{I}\mathcal{H}_S \to \mathcal{P}(V)$ to the category $\mathcal{P}(V)$ of perverse sheaves on $V$.

**Definition 7.1 (IH).** The category of Hodge intermediate extensions $\mathcal{I}\mathcal{H}$ attached to $V$ is the direct sum for all closed irreducible subsets $S$ of $V$: $\mathcal{I}\mathcal{H} := \bigoplus_{S \subset V} \mathcal{I}\mathcal{H}_S$.

Since a morphism on intermediate extensions of polarized VHS with support $S$ is determined by its restriction to the big stratum $S^*$, a morphism $i_{S_*} \mathbf{L} \to i_{S'} \mathbf{L}'$ between two of intermediate extensions of polarized VHS with support $S$ and $S'$ is necessarily 0 unless $S = S'$.

**Proposition 7.1.** Let $f : X \to V$ be a projective morphism and $j_* \mathbf{L}$ a polarized VHS on $X$. For all integers $i$, the perverse cohomology sheaves $(\mathcal{P}\mathcal{H}^i(f_*j_*\mathbf{L}), F)$ are in $\mathcal{I}\mathcal{H}$ on $V$.

We prove that any component $i_{S_*} \mathbf{L}_S^i$ of $\mathcal{P}\mathcal{H}^i(f_*j_*\mathbf{L})$ with support on the closure $\bar{S}$ of a stratum $S$ of $V$ is in $\mathcal{I}\mathcal{H}$ on $V$.
Let $\pi : S' \to \bar{S}$ be a desingularization of $\bar{S}$ and $k : S \hookrightarrow S'$.

We have to prove that $(i_{S_*} \mathbf{L}_S^i, F)$ is a component of $(\mathcal{P}\mathcal{H}^0(\pi_* k_* \mathbf{L}_S^i), F)$, We construct a variety $\tilde{V}$ in the diagram

<figure><img src="image_10.png" /></figure>

by blowings up above the singularities $S_s$ of $\bar{S}$ (including the singularities of $\mathbf{L}_S^i$) such that there exists a desingularization $\tilde{S} \subset \tilde{V}$ of $S$, strict transform of $S$ in $\tilde{V}$ and an embedding $i_{\tilde{S}} : S \hookrightarrow \tilde{S}$ such that $V \setminus S_s \simeq \tilde{V} \setminus \pi^{-1}S_s$ (without modification of $V$ outside the singularities $S_s \subset S$).
Set $\tilde{X} := X \times_V \tilde{V}$, $X_s := f^{-1}(S_s)$ such that $X \setminus X_s \simeq \tilde{X} \setminus \pi_1^{-1}X_s$ ($\tilde{X}$ is a modification of $X$). Let $j'_s \mathcal{L}'$ denote the intermediate extension of $\mathcal{L}$ on $\tilde{X}$ ($\tilde{X}$ may be singular, we recall that $j'_s \mathcal{L}$ on $X$ and $j'_s \mathcal{L}'$ on $\tilde{X}$ are constructed out of a desingularization of $\tilde{X}$).

The component $i_{S_*} \mathcal{L}^i$ may be constructed out of the decomposition of $h$. In the decomposition of $\mathcal{H}^i(f'_s j'_s \mathcal{L}')$ the component $i_{\tilde{S}!*} \mathcal{L}^i$ is exactly the extension of $\mathcal{L}^i$ on the desingularization $\tilde{S}$ of $S$ and by the decomposition theorem

$$ (i_{S_*} \mathcal{L}^i, F) \subset (\mathcal{H}^0(\pi_* i_{\tilde{S}!*} \mathcal{L}^i), F) \subset (\mathcal{H}^i(h_* j'_s \mathcal{L}'), F). $$

**Corollary 7.1.** Let $V$ be an algebraic variety. The category of Hodge intermediate extensions $\mathcal{I}\mathcal{H}$ is well defined on $V$.

We can suppose $V$ irreducible, since the category is the direct sum over irreducible components. In the previous definitions, the fact that $V$ is projective was convenient for the proof but not essential.

By uniqueness of the extension we can remove the condition $V$ is projective. Let $\mathcal{L}$ be defined on an open subset $U$. For any affine open subset $A \subset V$, the extension of the restriction $\mathcal{L}|_{A \cap U}$ to a projective closure is well defined, hence it is well defined on $A$. All extensions over an affine covering of $V$ glue into a unique extension on $V$.

**Definition 7.2 (Perverse variation of MHS (PVMHS)).** A bifiltered complex $(K, W, F)$ is a perverse variation of mixed Hodge structures (PVMHS) on a singular $X$ if:

1. $K$ is a $\mathbb{Q}$-perverse sheaf on $X$,

2. $W$ is an increasing filtration by $\mathbb{Q}$-perverse sub-sheaves,

3. $F$ is a decreasing filtration by $\mathbb{C}$-sub-sheaves of $K \otimes_{\mathbb{Q}} \mathbb{C}$ such that:

$$ \forall r \in \mathbb{Z}, (\mathrm{Gr}_r^W K, F) \in \mathcal{I}\mathcal{H} \text{ is a direct sum of intermediate extensions of polarized VHS on } V. $$

In the text all PVMHS are graded polarized. The following result may be deduced as in ([Sa 90], 2.d):

**Corollary 7.2 (M. Saito).** Let $f: X \to V$ be a projective morphism and $(K, W, F)$ a PVMHS on $X$. The perverse cohomology $(\mathcal{H}^i(f_*K), W, F)$ with induced filtrations $W$ and $F$ is a PVMHS on $V$ and we have:

$$ {}^pE_1^{p,q} := \mathcal{H}^{p+q}(f_*Gr_{-p}^W K) \implies Gr_q^W \mathcal{H}^{p+q}(f_*K) \quad (7.2) $$

degenerating at rank 2: ${}^pE_2^{p,q}(f_*K, W) \simeq Gr_q^W \mathcal{H}^{p+q}(f_*K)$.

By the decomposition theorem, already proved (see ch.6, Corollary 6.3) the terms of the perverse weight spectral sequence ${}^pE_r^{p,q}(f_*K', W)$ are in the abelian category $\mathcal{I}\mathcal{H}$ on $V$ and the spectral sequence degenerates at rank 2 by an argument similar to Deligne's results on spectral sequences with HS ([De 72] revisited in [De 75]).

The main interest in the abelian category $\mathcal{I}\mathcal{H}$ is to construct the terms ${}^pE_2^{p,q}$ as cohomology of the complex of terms ${}^pE_1^{p,q}$.
### 7.1.1 Hodge theory along locally principal divisors

Let $j_*^p \mathcal{L}$ be an intermediate extension of a VHS $p\mathcal{L}$ of weight $p_w$ on a non-singular dense algebraic open subset $U$ of a variety $V$ and $j: U \to V$.

Let $H \subset V$ be a principal divisor defined by an equation $g$. There exists a perverse variation of mixed Hodge structure (PVMHS) on $p\psi_g j_*^p \mathcal{L}$ (resp. $p\varphi j_*^p \mathcal{L}$) (see Proposition 7.3 below).

Let $i: H \to V$. There exists a PVMHS on $i^*j_*^p \mathcal{L}[-1]$ (resp. $i^!j_*^p \mathcal{L}[1]$)

Let $v \in V$ be a closed point, $B_v$ a small ball with center $v$ and $B_v^* := B_v \setminus v$. There exists a MHS on $\mathbb{H}^j(B_v^*, j_*^p \mathcal{L})$ (Proposition 7.4 below).

#### 7.1.1.1 Factoriality of the monodromy filtration

Let $f: X \to V$ be a projective morphism of algebraic complex varieties, $U' \subset V$ an open subset (algebraic or analytic) and $g: U' \to \mathbb{C}$ a morphism. Let $H := g^{-1}(0)$. We suppose $X$ non singular and $f^{-1}H \subset X$ a NCD.

Let $j_*^p \mathcal{L}$ be the intermediate extension of a shifted polarized VHS on $X$, $h := g \circ f$ and $(K, F) := f_*(j_*^p \mathcal{L}, F)$, then:

$$ p\psi_g p\mathcal{H}^i(f_*K) = p\mathcal{H}^i(f_* p\psi_h j_*^p \mathcal{L}), \quad p\varphi_g p\mathcal{H}^i(f_*K) = p\mathcal{H}^i(f_* p\varphi_h j_*^p \mathcal{L}) \quad (7.3) $$

The weight filtration $W$ is related to the monodromy filtration $W(N_h)$ on $p\psi_h j_*^p \mathcal{L}$ (resp. on $\varphi_h j_*^p \mathcal{L}$) (see Theorem 5.1). The image of the perverse cohomology $p\mathcal{H}^i(f_*W)$ defines a filtration on $p\psi_g p\mathcal{H}^i(f_*K)$ (resp. $p\varphi_g p\mathcal{H}^i(f_*K)$). We have a perverse spectral sequence

$$ \begin{aligned} pE_1^{p,q} &:= p\mathcal{H}^{p+q}(f_*Gr_{-p}^W p\psi_h j_*^p \mathcal{L}) &\implies & Gr_q^W p\psi_g p\mathcal{H}^{p+q}(f_*j_*^p \mathcal{L}) \\ pE_1^{p,q} &:= p\mathcal{H}^{p+q}(f_*Gr_{-p}^W p\varphi_h j_*^p \mathcal{L}) &\implies & Gr_q^W p\varphi_g p\mathcal{H}^{p+q}(f_*j_*^p \mathcal{L}) \end{aligned} \quad (7.4) $$

**Proposition 7.2 (Saito [Sa 88] Proposition 5.3.5).** i) The perverse weight spectral sequences (7.4) degenerate at rank 2 in the abelian category $\mathcal{IH}$ on $H$ (Definition 7.1).

ii) The filtration induced by $\mathrm{Im} \, p\mathcal{H}^i(f_*W(N_h)) \to p\psi_g p\mathcal{H}^i(f_*j_*^p \mathcal{L})$ (resp. $p\varphi_g p\mathcal{H}^{p+q}(f_*j_*^p \mathcal{L})$) coincides with the monodromy filtration $W(N_g)$.

iii) The terms $pE_2^{p,q}$ are in $\mathcal{IH}$ and we have a structure of PVMHS on $p\psi_g p\mathcal{H}^{p+q}(f_*K)$ (resp. $p\varphi_g p\mathcal{H}^{p+q}(f_*K)$).

The proof is based on the bigraded objects of intermediate extensions of polarized VHS on $H$ in the abelian category $\mathcal{IH}$ by comparison with the bigraded polarized HS in chapter 5, §5.2.1.1.

We use the perverse Hard Lefshetz theorem (resp. the morphism $N$) on the terms $pE_1^{p,q}$.
### 7.1.2 Nearby and vanishing cycles on a variety V

Let $j_*{}^p\mathcal{L}$ be the intermediate extension of a polarized VHS on a non singular Zariski open subset $U$ dense in a projective variety $V$ possibly singular and $g: U \to \mathbb{C}$ a projective analytic morphism.

**Proposition 7.3.** Let $j_*{}^p\mathcal{L}$ on $V$ and $N := \log T^u$. There exists on the perverse nearby cycle complex $\psi_g j_*{}^p\mathcal{L}[−1]$ (resp. $\varphi_g j_*{}^p\mathcal{L}[−1]$) a filtration $F$ inducing with the filtration $W := W(N)[p^w − 1]$ (resp. $W := W(N)[p^w]$) a structure of PVMHS where the weight $W$ is a shifted monodromy weight filtration $W(N)$.

Let $W$ be an increasing filtration, recall: $W[n]_{i+n} = W_i$. The proof is by reduction to the non singular case. Let $H := g^{-1}(0)$ and $Y$ the singularities of ${}^p\mathcal{L}$ in $V$ including the singularities of $V$. Let $\pi: \tilde{V} \to V$ be a desingularization such that $Y' := \pi^{-1}(Y)$ and $H' := \pi^{-1}(H)$ are NCD. We suppose that $H'$ is a sub-divisor of $Y'$ as we can enlarge $Y$ if necessary. Let $j': U \to \tilde{V}$ ($U \simeq (\tilde{V} \setminus Y') \simeq V \setminus Y$. We apply the proposition 7.2 to $j'_*{}^p\mathcal{L}$ on $\tilde{V}$. The proof follows, since $j_*{}^p\mathcal{L}$ is a component of ${}^p\mathcal{H}^0(\pi_*j'_*{}^p\mathcal{L})$.

The uniqueness follows by considering a diagram $\tilde{V}_1 \leftarrow \tilde{V} \to \tilde{V}_2$ of modifications of $V$ in the case of two choices $\tilde{V}_1$ and $\tilde{V}_2$.

**Definition 7.3.** We define $\psi_g j_*{}^p\mathcal{L}$ as a component of $\psi_g {}^p\mathcal{H}^0(\pi_*j_*{}^p\mathcal{L}) := {}^p\mathcal{H}^0(\pi_*\psi_{g\circ\pi}(j'_*{}^p\mathcal{L}))$. A similar definition of $\varphi_g j_*{}^p\mathcal{L}$ apply.

**Corollary 7.3.** Let $V$ be projective and $i: H \hookrightarrow V$ the embedding of a locally principal divisor.

i) There exists on $i^*j_*{}^p\mathcal{L}[−1]$ (resp. on $i^!j_*{}^p\mathcal{L}[1]$) a natural PVMH inducing on $\mathbb{H}^j(H, j_*{}^p\mathcal{L})$ (resp. on $\mathbb{H}_H^j(V, j_*{}^p\mathcal{L}[1])$) a filtration by MHS of weights $w \le {}^p w + j$ (resp. $w > {}^p w + j$).

ii) Let $k: V \to H \hookrightarrow V$. There exists on $k_*k^*j_*{}^p\mathcal{L}$ a natural PVMH of weights $w \ge {}^p w$ and a MHS on $\mathbb{H}^j(V \setminus H, j_*{}^p\mathcal{L})$ of weights $w \ge {}^p w + j$.

i) Let $g$ be a local equation of $H$. The corollary follows since $i^*j_*{}^p\mathcal{L}[−1]$ may be constructed locally as kernel of the morphism $N: \psi_g^u\psi_g j_*{}^p\mathcal{L}[−1] \to \psi_g^u\psi_g j_*{}^p\mathcal{L}[−1]$ (see ch.5, Equation 5.50) with induced weight filtration. The local constructions glue globally on $H$ (the case $i^!j_*{}^p\mathcal{L}[1]$ is dual).

It is equivalent to construct the filtrations $W$ and $F$ by push-forward from a desingularization of $V$.

ii) The quotient $k_*k^*j_*{}^p\mathcal{L}/j_*{}^p\mathcal{L}$ is isomorphic to $i^!j_*{}^p\mathcal{L}[1]$.

**Proposition 7.4 (MHS on $\mathbb{H}^j(B_v^*, j_*{}^p\mathcal{L}))$.** Let $v \in V$, $k_v: (V \setminus v) \to V$. There exists a natural MHS on $\mathbb{H}^j(v, k_v k_v^* j_*{}^p\mathcal{L}) \simeq \mathbb{H}^j(B_v^*, j_*{}^p\mathcal{L})$.

*Proof.* Since the proof is local at $v$, we change $V$ into a small ball $B_v$ with center $v \in V$. By blowing up $v \in B_v$, we have a diagram
$$ \begin{gathered} \tilde{Z} \stackrel{\tilde{i}}{\hookrightarrow} \tilde{B}_v \stackrel{\tilde{k}}{\hookleftarrow} \tilde{B}_v^* \\ \pi_v \downarrow \quad \downarrow \pi \quad \simeq \downarrow \pi' \\ v \stackrel{i_v}{\hookrightarrow} B_v \stackrel{k}{\hookleftarrow} B_v^* \end{gathered} \qquad (7.5) $$

where $B_v^* := B_v \setminus \{v\}$ and $\pi: \tilde{B}_v \to B_v$ is the blow up of $v \in B_v$.

Let $P$ be a complex on $\tilde{B}_v^*$ such that $\pi_*P = (j_{\uparrow*} p_L)_{B_v^*}$. Since $\pi'$ is an isomorphism, we have:

$$ i_v^* k_* \pi_* P \simeq \pi_{v*} \tilde{i}^* \tilde{k}_* P $$

where $\tilde{i}^* \tilde{k}_* P$ has two non vanishing perverse cohomology on the locally principal divisor $\tilde{Z}$:

$$ {}^p\mathcal{H}^{-1}(\tilde{i}^* \tilde{k}_* P) = \tilde{i}^* \tilde{k}_{\uparrow*} P[-1], \quad {}^p\mathcal{H}^0(\tilde{i}^* \tilde{k}_* P) = \tilde{i}^* \tilde{k}_{\uparrow*} P[1] \qquad (7.6) $$

The complex $\tilde{i}^* \tilde{k}_* P$ is isomorphic to a mixed cone over the intersection complex $I: \tilde{i}^* \tilde{k}_{\uparrow*} P \to \tilde{i}^* \tilde{k}_{\uparrow*} P$ with filtrations $W$ and $F$ such that

$$ (H^i(\pi_{v*} \tilde{i}^* \tilde{k}_* P), W, F) \simeq \mathbb{H}^i(\tilde{Z}, \tilde{i}^* \tilde{k}_* P, W, F) \qquad (7.7) $$

$H^i(v, k_{v*} k_v^* j_{\uparrow*} p_L) \simeq H^i(\pi_{v*} \tilde{i}^* \tilde{k}_* P)$ has a natural MHS.

**Remark 7.2.** 1) Let $i: H \hookrightarrow V$ is a closed subvariety of $V$, then the results of Corollary 7.3) can be generalized to this case:

i) There exists on $i^* j_{\uparrow*} p_L[-1]$ (resp. on $i^! j_{\uparrow*} p_L[1]$) a natural PVMH inducing on $\mathbb{H}^j(H, j_{\uparrow*} p_L)$ (resp. on $\mathbb{H}_H^j(V, j_{\uparrow*} p_L[1])$) a filtration by MHS of weights $w \le pw + j$ (resp. $w > pw + j$).

ii) Let $k: V \setminus H \hookrightarrow V$. There exists on $k_* k^* j_{\uparrow*} p_L$ a natural PVMH of weights $w \ge pw$ and a MHS on $\mathbb{H}^j(V \setminus H, j_{\uparrow*} p_L)$ of weights $w \ge pw + j$.

*Proof.* We blow-up $H$ to construct $H' \subset V'$ and $f: V' \to V$ such that $f^{-1}(H) = H'$, then apply (Corollary 7.3) to the locally locally principal divisor $H'$ to define a PVMHS, then apply (Proposition 7.2).

2) Then Proposition 7.4 follows since $i_v^* k_{v*} k_v^* j_{\uparrow*} p_L) \simeq \mathbb{H}^j(B_v^*, j_{\uparrow*} p_L)$ is a mixed cone over the intersection morphism $I: i_v^! \to i_v^*$ (Equation 2.1.3.1, §5.2.2, Lemma 5.3).

**Remark 7.3 (Compatibility with the perverse filtration).** (See a general method to prove the compatibility with the perverse filtration by (Equation 10.7)).

i) Let $f: X \to V$ be a projective morphism (X may be singular), $H \subset V$ a locally principal divisor on $V$ and $(j_{\uparrow*} p_L, W, F)$ on $X$. Set $H' := f^{-1}(H) \subset X$, $i_H: H \to V$, $i_{H'}: H' \to X$.

Let $(K', W, F) := (i_{H'}^* j_{\uparrow*} p_L, W, F)$ (resp. $(K', W, F) := (i_{H'}^! j_{\uparrow*} p_L, W, F)$) and denote again $f_{|H_*} W$ by $W$, $f_{|H_*} F$ by $F$ on $f_* K'$.

We deduce from the perverse filtration $p_\tau$ on $f_* j_{\uparrow*} p_L$ a filtration $i_H^* p_\tau$ on $i_H^* f_* j_{\uparrow*} p_L$ (resp. $i_H^! p_\tau$ on $i_H^! f_* j_{\uparrow*} p_L$) and by transport of structure a filtration
$i_H^* p_\tau$ on $f_{|H_*} K'$ by the isomorphism $f_{|H_*} K' \simeq i_H^* f_* j_*^p \mathbf{L}$ (resp. $i_H^! p_\tau$ on $i_H^* f_* j_*^p \mathbf{L}$). Thus we have three filtrations

$$ (f_{|H_*} K', W, F, i_H^* p_\tau) \text{ (resp. } (f_{|H_*} K', W, F, i_H^! p_\tau) \text{).} $$

We deduce from (Corollary 7.3 and Proposition 7.4) an isomorphism of MHS of weights $w \le pw + j$ (resp. of weights $w \ge pw + j$).

$$ (\mathbb{H}^j(H, p\mathcal{H}^i(f_*j_*^p\mathbf{L})), W, F) \simeq (Gr_i^{i_H^* p_\tau} \mathbb{H}^{i+j}(X_H, j_*^p\mathbf{L}), W, F) \quad (7.8) $$

$$ (\mathbb{H}_H^j(V, p\mathcal{H}^i(f_*j_*^p\mathbf{L})), W, F) \simeq (Gr_i^{i_H^* p_\tau} \mathbb{H}_{X_H}^{i+j}(X, j_*^p\mathbf{L}), W, F) $$

ii) We may even consider the induced filtration in the category of PVMHS on the desingularization to deduce PVMHS on a singular variety.

iii) MHS on $H^j(B_v^*, p\mathcal{H}^i(f_*j_*^p\mathbf{L}))$: Let $v \in V$ be a closed point such that $Z = X_v := f^{-1}(v)$ is a locally principal divisor in $X$. Set $k : V \setminus \{v\} \to V$, $i_v : v \to V$, $i : Z \to X$ and $j_Z : X \setminus Z \to X$ and $j_*^p \mathbf{L}$ on $X$.

We deduce from the bifiltered complex $(K', W, F) := (i^*j_Z*(j_*^p\mathbf{L})|_{(X\setminus Z)}, W, F)$ a structure of MHC on the derived global sections complex $\Gamma(Z, K', W, F)$. We denote again by $W$ and $F$ the filtrations $(f_{|Z})_*W, (f_{|Z})_*F$ on $(f_{|Z})_*K'$.

We use the isomorphism $(f_{|Z})_*K' \simeq i_{v_*} k_* ((f_*j_*^p\mathbf{L})|_{(V\setminus v)})$ to deduce a filtration $p_\tau^*$ on $(f_{|Z})_*K'$ from the perverse filtration $p_\tau$ on $(f_*j_*^p\mathbf{L})|_{(V\setminus v)}$

$$ p_{\tau^*}(f_{|Z})_*K' \simeq p_{\tau^*}(i_{v_*} k_*(f_*j_*^p\mathbf{L})|_{(V\setminus v)}) := i_{v_*} k_*(p_{\tau}(f_*j_*^p\mathbf{L})|_{(V\setminus v)}) $$

Thus we have three filtrations $W, F, p_\tau^*$ on the complex $(f_{|Z})_*K'$ inducing filtrations $W, F$ and $p_\tau^*$ on the hypercohomology space $\mathbb{H}^j(Z, j_*^p\mathbf{L})$:

1) The induced filtration $p_\tau^*$ on $\mathbb{H}^j(X_v, K')$ is a filtration by MHS.

2) There exists a natural MHS on $H^j(v, k_*k^* p\mathcal{H}^i(f_*j_*^p\mathbf{L}))$ compatible with the isomorphism of MHS:

$$ H^j(v, k_*k^* p\mathcal{H}^i(f_*j_*^p\mathbf{L})) \simeq H^j(v, \mathrm{Gr}_i^{p_\tau^*} (f_{|X_v})_*K', W, F) \quad (7.9) $$

If $Z$ is a NCD in a non singular $X$, the bifiltered complex $K' := i^*j_Z^*j_Z^*j_*^p\mathbf{L}$ with induced $W$ and $F$ is defined by (chapter 5, §5.2.2, Definition 5.6).

**Corollary 7.4 (Perverse filtration by sub-HS).** With the notations of (Corollary 6.3), the filtration $p_\tau$ on $f_*j_*^p\mathbf{L}$ induces a filtration by Hodge substructures of $\mathbb{H}^{i+j}(X, j_*^p\mathbf{L})$ such that

$$ (Gr_i^{p_\tau} \mathbb{H}^{i+j}(X, j_*^p \mathbf{L}), F) \simeq (\mathbb{H}^j(V, p\mathcal{H}^i(f_*j_*^p \mathbf{L})), F). $$

We have a decomposition into a direct sum of polarized HS of weight $pw+i+j$:

$$ \mathbb{H}^j(V, p\mathcal{H}^i(f_*j_*^p\mathbf{L})) \xrightarrow{\sim} \bigoplus_{S \in \mathcal{S}} \mathbb{H}^j(X, i_{S_*}^*(j_*^p\mathbf{L})_S^i) $$
## 7.2 Local and global invariant cycle theorems

We prove the local and global invariant cycle theorems on eventually singular varieties as announced in the introduction (ch. 4, Theorems 4.1 and 4.2). The results on non singular varieties extend to singular varieties with coefficients in intermediate extensions of VHS by the previous remark 7.3.

Also, Deligne's proof on non singular varieties extends to singular varieties.

The octahedron diagram Diagram (7.13) and Equation (7.15) are applications of the decomposition of $\varphi_g$ (see ch.6, Equation 6.17).

### 7.2.0.1 Local invariant cycle theorem on singular varieties

Let $V$ be an algebraic variety, $\mathcal{P}\mathbf{L}$ a shifted polarized VHS of weight $p_w$ on a non singular Zariski open subset $U \subset V$ and $j_{1*}\mathcal{P}\mathbf{L}$ the intermediate extension of $\mathcal{P}\mathbf{L}$ endowed with a Hodge filtration $F$ (see ch.6, Definition 6.2).

Let $g: V_D \to D$ be a projective morphism from a complex open subset $V_D$ to the complex disc $D \in \mathbb{C}$ with central fiber $H := V_0$.

**Proposition 7.5 (Local invariant cycle).** *The sequence*

$$ \mathbb{H}^q(H, j_{1*}\mathcal{P}\mathbf{L}[-1]) \xrightarrow{sp} \mathbb{H}^q(H, \psi_g^u j_{1*}\mathcal{P}\mathbf{L}[-1]) \xrightarrow{N} \mathbb{H}^q(H, \psi_g^u j_{1*}\mathcal{P}\mathbf{L}[-1])(-1) \quad (7.10) $$

*is an exact sequence of MHS.*

The proof is similar to Deligne's proof for constant coefficients [De 80]. For $D$ small enough, $V_D$ is a retract by deformation of $H$ as $g$ is proper. Hence the restriction map $H^i(V_D, j_{1*}\mathcal{P}\mathbf{L}) \to H^i(H, j_{1*}\mathcal{P}\mathbf{L})$ is an isomorphism. Let $p_w$ be the weight of $\mathcal{P}\mathbf{L}$. We consider the diagram

<figure><img src="image_11.png" /></figure>

Since the topological Wang exact sequence

$$ \mathbb{H}^r(V_D \setminus H, j_{1*}\mathcal{P}\mathbf{L}) \to \mathbb{H}^r(H, \psi_g^u j_{1*}\mathcal{P}\mathbf{L}) \xrightarrow{N} \mathbb{H}^r(H, \psi_g^u j_{1*}\mathcal{P}\mathbf{L}) \to \quad (7.11) $$

is compatible with MHS (see Remark 7.3 ii), the morphism $\beta$ is surjective on the invariant subspace by $N$ and compatible with MHS.
Let $j_H : V \backslash H \to V$. The whole diagram can be constructed in the category of MHS, since the following hypercohomology spaces carry MHS:

1) $\mathbb{H}^i(V_D \backslash Z, j_{!*} p_L) \simeq \mathbb{H}^i(H, i_H^* j_{H*} j_H^* j_{!*} p_L)$ ($V_D$ retracts on $H$),

2) $H^i(H, \psi_f^u j_{!*} p_L)$, (Proposition 7.3)

3) $\mathbb{H}^i(V_D, j_{!*} p_L) \simeq \mathbb{H}^i(H, j_{!*} p_L)$ of weight $\le p_w + i$ (Corollary 7.3) and

4) $\mathbb{H}_Z^{i+1}(V_D, j_{!*} p_L)$ of weight $\ge p_w + i + 1$ (Corollary 7.3).

Since $H_H^{i+1}(V_D, j_{!*} p_L)$ has weights $\ge p_w + i + 1$, and the weight of $\mathrm{Im} \alpha$ is $\le p_w + i$, we have: $W_{p_w+i} \mathbb{H}^i(V_D \backslash H, j_{!*} p_L) \subset \mathrm{Im} \alpha$.

Since $\beta$ is surjective by Wang sequence, we deduce that $\mathrm{sp} := \beta \circ \alpha$ is surjective as the weight of $\mathbb{H}^i(H, \psi_f^u j_{!*} p_L)^N$ is $\le p_w + i$.

### 7.2.0.2 The octahedron diagram

Let $g : V \to \mathbb{P}$ be a morphism to the complex projective line $\mathbb{P} := \mathbb{P}_\mathbb{C}^1$, $H := g^{-1}(0)$ and $f : X \to V$ such that $Z := (g \circ f)^{-1}(0) = f^{-1}(H)$ is a NCD in $X$

$$ \begin{array}{ccc} Z & \stackrel{i'}{\hookrightarrow} & X \hookrightarrow X \backslash Z \\ \downarrow & & \downarrow f \\ H & \stackrel{i}{\hookrightarrow} & V \stackrel{k'}{\hookrightarrow} V \backslash H \\ \downarrow & & \downarrow g \\ 0 & \stackrel{i''}{\hookrightarrow} & \mathbb{P} \hookrightarrow \mathbb{P} \backslash 0 \end{array} \qquad (7.12) $$

Let $K := f_* j_{!*} p_L$. We attach to the Diagram 7.12, the following upper and lower 'caps' of the octahedron diagram ([BBD 83], 1.1.6):

<figure><img src="image_12.png" /></figure>

where the cones $C(N) \simeq C(I)$ are isomorphic. The middle term of the sequence:

$$ (p\mathcal{H}^i(p\psi_g^u K) \xrightarrow{\mathrm{can}} p\mathcal{H}^i(p\varphi_g^u K)) \xrightarrow{\mathrm{var}} p\mathcal{H}^i(p\psi_g^u K)) $$

where $p\mathcal{H}^i(p\psi_g K) = p\psi_g p\mathcal{H}^i(K)$ and $p\mathcal{H}^i(p\varphi_g K) = p\varphi_g p\mathcal{H}^i(K)$, splits into a direct sum by (Lemma 6.4)

$$ p\mathcal{H}^i(p\varphi_g^u K) \xrightarrow{\sim} \mathrm{Im} \mathrm{can} \oplus \mathrm{Ker} \mathrm{var} \qquad (7.14) $$
Typically for an octahedron diagram, this decomposition is equivalent to the long exact sequence with four terms defined by the common boundary of the diagrams:

$$ p\mathcal{H}^{i-1}(i^*K) \to p\mathcal{H}^i(p\psi_g^u K) \xrightarrow{N} p\mathcal{H}^i(p\psi_g^u K)(-1) \to p\mathcal{H}^{i+1}(i!K) \xrightarrow{I} p\mathcal{H}^{i+1}(i^*K) \to \\ \text{or the exact sequence} $$

$$ i^* p\mathcal{H}^i(K)[-1] \to p\psi_g^u p\mathcal{H}^i(K) \xrightarrow{N} p\psi_g^u p\mathcal{H}^i(K)(-1) \to i^! p\mathcal{H}^i(K)[1] \to i^* p\mathcal{H}^{i+2}(K)[-1] \to \\ \text{since } p\mathcal{H}^{i-1}(i^*K) = i^* p\mathcal{H}^i(K)[-1], \ p\mathcal{H}^{i+1}(i!K) = i^! p\mathcal{H}^i(K)[1]. $$

Since $H$ is locally principal and $p\mathcal{H}^i(K)$ decomposes into a sum of intermediate extensions, this exact sequence splits into various perverse exact sequences

$$ 0 \to i^* p\mathcal{H}^i(K)[-1] \to p\psi_g^u p\mathcal{H}^i(K) \xrightarrow{N} p\psi_g^u p\mathcal{H}^i(K)(-1) \to i^! p\mathcal{H}^i(K)[1] \to 0 \quad (7.15) $$

This exact sequence appears in ([Sa 88] 5.2.2.2), see also [BBD 83]. The local invariant theorem follows again:

**Corollary 7.5.** i) We have an exact sequence of MHS:

$$ \mathbb{H}^q(H, p\mathcal{H}^0(K)[-1]) \xrightarrow{sp} \mathbb{H}^q(H, \psi_g^u p\mathcal{H}^0(K)[-1]) \xrightarrow{N} \mathbb{H}^q(H, \psi_g^u p\mathcal{H}^0(K)[-1])(-1) $$

ii) Let $p\mathcal{L}$ be on $V$. We have an exact sequence of MHS:

$$ \mathbb{H}^{q-1}(H, j_{i_*} p\mathcal{L}) \xrightarrow{sp} \mathbb{H}^{q-1}(H, \psi_g^u j_{i_*} p\mathcal{L}) \xrightarrow{N} \mathbb{H}^{q-1}(H, \psi_g^u j_{i_*} p\mathcal{L})(-1) $$

i) The weights $w(\mathbb{H}^q(H, p\mathcal{H}^0(K)[-1]) \le p_w+q-1$ and $w(\mathrm{Ker} N) \le p_w+q-1$. The exact sequence (Equation 7.15) is reduced to two short exact sequences:

$$ 0 \to i^* p\mathcal{H}^0(K)[-1] \to p\mathcal{H}^0(p\psi_g^u K) \to \mathrm{Im} N \to 0, \\ 0 \to \mathrm{Im} N \to p\mathcal{H}^0(p\psi_g^u K)(-1) \to i^! p\mathcal{H}^0(K)[1] \to 0 $$

We deduce the associated long exact sequences

$$ \mathbb{H}^q(H, p\mathcal{H}^0(K)[-1]) \xrightarrow{sp} \mathbb{H}^q(H, \psi_g^u p\mathcal{H}^0(K)[-1]) \xrightarrow{\alpha} \mathbb{H}^q(H, \mathrm{Im} N) \to \\ \mathbb{H}^{q-1}(H, i^! p\mathcal{H}^0(K)[1]) \xrightarrow{\delta} \mathbb{H}^q(H, \mathrm{Im} N) \xrightarrow{\beta} \mathbb{H}^q(H, \psi_g^u p\mathcal{H}^0(K)[-1])(-1) \to $$

where $N = \beta \circ \alpha$. Since the weights $w$ of $\mathbb{H}^q(H, p\mathcal{H}^0(i^*K)[-1])$ are $\le p_w+q-1$, we prove that any element of $a \in \mathrm{Ker} N \subset \mathbb{H}^q(H, \psi_g^u j_{i_*} p\mathcal{L}[-1])$ of weight $w(a) \le p_w+q-1$ satisfy necessarily $\alpha(a) = 0$, hence $a$ is in the image of sp.

We have $Na = \beta \circ \alpha(a) = 0$ and $w(\alpha(a)) = w(a) \le p_w+q-1$.

Since the weights $w$ of $\mathbb{H}^{q-1}(H, i^! p\mathcal{H}^0(K)[1])$ satisfy $w \ge p_w+q$ we have: $w(\mathrm{Im}, \delta) = w(\mathrm{Ker} \beta) \ge p_w+q$ and $\alpha(a) \subset \mathrm{Ker} \beta$, hence $\alpha(a) = 0$.
ii) We apply i) to the diagram defined by a desingularization $\tilde{V}$ of $V$ and to the intermediate extension of $p\mathbf{L}$ on $\tilde{V}$. Since $j_{1*}p\mathbf{L} \subset p\mathcal{H}^0(K)$ in this case, we deduce the statement ii) from i).

**Remark 7.4.** i) We deduce from the exact sequence of the boundary, the decomposition in the lower cap of the octahedron on its turn:

$$ p\mathcal{H}^i(p\varphi_g^u K) \xrightarrow{\sim} (\mathrm{Im} : p\mathcal{H}^i(i!K) \to p\mathcal{H}^i(p\varphi_g^u K)) \oplus (\mathrm{Ker} : p\mathcal{H}^i(p\varphi_g^u K) \to p\mathcal{H}^i(i^*K)) $$

ii) The exact sequence in the corollary may be extended by the same method, to a long exact sequence of four terms known after Clemens-Schmid in the case of constant coefficients:

$$ \begin{align*} & \to \mathbb{H}^{q-1}(H, j_{1*}p\mathbf{L}) \xrightarrow{sp} \mathbb{H}^{q-1}(H, \psi_g^u j_{1*}p\mathbf{L}) \xrightarrow{N} \mathbb{H}^{q-1}(H, \psi_g^u j_{1*}p\mathbf{L})(-1) \\ & \to \mathbb{H}_H^{q+1}(V, j_{1*}p\mathbf{L}) \xrightarrow{I} \mathbb{H}^{q+1}(H, j_{1*}p\mathbf{L}) \end{align*} $$

**Remark 7.5.** i) Let $V_D$ be a tubular analytic neighbourhood of $H$ defined over a small disc $D \subset \mathbb{C}$. We have a long exact Wang sequence

$$ \mathbb{H}^r(V_D \setminus H, j_{1*}p\mathbf{L}) \to \mathbb{H}^r(H, \psi_g^u j_{1*}p\mathbf{L}) \xrightarrow{N} \mathbb{H}^r(H, \psi_g^u j_{1*}p\mathbf{L})(-1) \to $$

where the weight is deduced from the weight on $\Psi_g^u j_{1*}p\mathbf{L}$. From this we extract the short exact sequence of MHS

$$ 0 \to \mathrm{Coker} N |_{\mathbb{H}^{r-1}(H, \psi_g^u j_{1*}p\mathbf{L})}(-1) \to \mathbb{H}^r(V_D \setminus H, j_{1*}p\mathbf{L}) \to \mathbb{H}^r(H, \psi_g^u j_{1*}p\mathbf{L})^N \to 0 $$

ii) In Corollary 7.5 ii), we may change $\mathbb{H}^q(H, j_{1*}p\mathbf{L})$ into $\mathbb{H}^q(V_D, j_{1*}p\mathbf{L})$ since $V_D$ is a tubular neighborhood of $Z$.

iii) Let $H$ be transverse to the strata of $V$ except at a stratum $v \in V$ of dimension 0. A weaker version of the theorem where invariant cycles are lifted on $V \setminus \{v\}$ is used in [DeG 81].

### 7.2.1 Global invariant cycle theorem

Let $h : X \to T$ be a morphism of complex projective varieties, $p\mathbf{L}$ a shifted polarized VHS of weight $pw$ on a nonsingular Zariski open subset $U \subset X$ and $j_{1*}p\mathbf{L}$ the intermediate extension of the perverse sheaf $p\mathbf{L}$.

**Lemma 7.2 (Global invariant cycle).** Let $T' \subset T$ denote the Zariski open dense subset of parameters $t \in T$ over which $R^q h_{1*}p\mathbf{L}$ is a polarized VHS for all $q$. Let $t_0 \in T'$ (note that $T' \neq \emptyset$). The image of the restriction map

$$ \mathbb{H}^q(X, j_{1*}p\mathbf{L}) \to \mathbb{H}^q(X_{t_0}, j_{1*}p\mathbf{L}) \qquad (7.16) $$

is the invariant subset $\mathbb{H}^q(X_{t_0}, j_{1*}p\mathbf{L})^\pi$ of fixed elements under the monodromy action $\pi := \pi_1(T', t_0) \to \mathrm{Aut} \, \mathbb{H}^q(X_{t_0}, j_{1*}p\mathbf{L})$.
*Proof.* (Similar to the non singular case ([De 80] Corollary 6.2.12)). By the invariant theory of local systems, the restriction morphism:

$\mathbb{H}^0(T', R^q h_* j_!^* p_L) \stackrel{c}{\hookrightarrow} (R^q h_* j_!^* p_L)_{t_0} = \mathbb{H}^q(X_{t_0}, j_!^* p_L)$ is injective with image the invariant subspace $(R^q h_* j_!^* p_L)_{t_0}^\pi$. In the diagram

$$ \mathbb{H}^q(X, j_*^p \mathbf{L}) \xrightarrow{a} \mathbb{H}^q(X_{T'}, j_*^p \mathbf{L}) \xrightarrow{b} \mathbb{H}^0(T', R^q h_* j_*^p \mathbf{L}) \xrightarrow{c} \mathbb{H}^q(X_t, j_*^p \mathbf{L}) $$

the morphism $b$ is surjective by the Decomposition Theorem. The restriction morphisms $c \circ b$ and $c \circ b \circ a$ have the same image since $\mathbb{H}^q(X, j_*^p \mathbf{L})$ and $\mathbb{H}^q(X_{t_0}, j_*^p \mathbf{L})$ are pure of weight $pw + q$ while $\mathbb{H}^q(X_{T'}, j_*^p \mathbf{L})$ is mixed of weights $w \ge pw + q$.

Moreover, the morphism $a$ is surjective on $W_{pw+q} \mathbb{H}^q(X_{T'}, j_*^p \mathbf{L})$, since we may apply an extension of (ch. 5, Corollary 5.7 i) to this case. The lemma follows by strictness of morphisms of MHS (we may suppose $W := X \setminus X_{T'}$ a NCD by blowing up subvarieties above $W$ without modification of $X_{T'}$).
<figure><img src="image_13.png" /></figure>
Chapter 8
Deligne Gabber local purity

We illustrate, on complex algebraic varieties, the proof of the local purity
theorem in positive characteristic which preceded the decomposition theorem
(see [DeG 81] and Theorem 1.1).

In some sense, we have already expanded the proof in the previous chap-
ters, including the local and global invariant cycle theorems in chapter 7.
Hence, we adapt to complex algebraic varieties the original dense text in
[DeG 81] without further comments.

In chapter 6 the method has been applied to give a simple proof of the
decomposition once we have at our disposal the preliminaries on perverse
sheaves in chapter 2.

A simplified proof combined with the decomposition theorem is given in
chapter 10.

8.0.0.1 Induction

Let $V$ be a complex algebraic variety. Since the statement of the local purity theorem is local at a point $v \in V$, we can suppose $V$ projective by considering an affine neighborhood that can be embedded into a projective space. Let $j_{1*}P_L$ be the intermediate extension of a shifted polarized VHS on $V$ of weight $Pw$ and $B_v \subset V$ a small ball with center $v$.

The inequalities on the weights $w$ on $\mathbb{H}^i(B_v \setminus \{v\}, j_{1*}P_L)$, except the
crucial case $i=0$, $w = Pw$ are easily deduced by induction from the inequali-
ties on the intersection with a general hyperplane section $H$ of $V$ through $v$ by
results on the Stein inclusion $B_v \setminus (H \cap B_v) \hookrightarrow B_v$ similar to Artin-Lefschetz
theorems on affine morphisms ([BBD 83] 4.1). The proof will concentrate
essentially on the crucial case $i=0$, $w = Pw$ in §8.1,

**Lemma 8.1 (Induction).** If we assume the local purity theorem (Theorem 1.1) in the case of dimension $\dim V = m-1$, the inequalities on the weights $w$ of the MHS on $\mathbb{H}^i(B_v \setminus \{v\}, j_{1*}P_L)$ (Proposition 7.4) are satisfied in the
case dim $V = m$:

$$w \leq {}^{p}w + i \text{ if } i \leq -1 \text{ and } w > {}^{p}w + i \text{ if } i \geq 0.$$

except: $\mathrm{Gr}_{p_w}^W \mathbb{H}^0(B_v \setminus \{v\}, j_{1*}^p \mathbf{L}) = 0$ and $\mathrm{Gr}_{p_w}^W \mathbb{H}^{-1}(B_v \setminus \{v\}, j_{1*}^p \mathbf{L}) = 0$.

The intersection cohomology $\mathbb{H}^*(\partial B_v, j_{1*}^p \mathbf{L})$ satisfy Poincaré's selfduality. The boundary $\partial B_v$ intersects transversely a stratification of $V$ when $B_v$ is small enough. Then $B_v^*$ retracts by deformation to its boundary and the duality by transport of structure transforms one set of inequalities on $w$ into the other set. Hence, the proof is reduced to either one of the set of inequalities.

If $\dim V = 0$, the assertion is trivially satisfied. If $\dim V = 1$ and $V$ is nonsingular at $v$, then $B_v$ is isomorphic to a disc in $\mathbb{C}$. Let $L := \psi_z(p\mathbf{L})(v)$ be a nilpotent orbit of weight $pw - 1$; as in (Example 5.4, 2) in §5.2.1.4, the weights $w$ satisfy:

$$\mathrm{Gr}_w^W \mathbb{H}^{-1}(B_v \setminus \{v\}, j_{1*}^p \mathbf{L}) = \mathrm{Gr}_w^W (\mathrm{Ker} \, N : L \to L) \text{ for } w \le {}^p w - 1 \\ \mathrm{Gr}_w^W \mathbb{H}^0(B_v \setminus \{v\}, j_{1*}^p \mathbf{L}) = \mathrm{Gr}_{w-1}^W (\mathrm{Coker} \, N : L \to L) = \mathrm{Gr}_{w-1}^W \mathrm{Prim} \, L \text{ for } w > {}^p w.$$

If $V$ is singular at $v$, the proof is reduced to the previous case by a desingularization $\pi : \tilde{V} \to V$, since at each point $\tilde{v} \in \pi^{-1}(v)$, the map $\pi$ induces an isomorphism from $B_v^*$ to a connected component of $B_v^*$.

If $\dim V > 1$, let $p\mathbf{L}$ be defined on the complement of a divisor $Y \subset V$, $S$ a stratification of $V$ compatible with $v$ and $Y$ and $H$ a general hyperplane section of $V$ through $v$, transverse to all strata $S \subset V \setminus v$.

*Proof (of the lemma by induction).* Let $K = j_{1*}^p \mathbf{L}$, $V^* := V \setminus v$, $H^* := H \setminus v$, $j : X \setminus Y \to Y$ and $i_H : H \to V$. Let $i \neq 0$, $B_v^* := B_v \setminus v$, $H_v^* := H \cap B_v^*$ and $K_H = i_{H^*} i_H^* K[-1]$. By transversality, Thom isomorphisms apply on $V^*$ with a twist of MHS:

$$\mathbb{H}^{i-1}(H_v^*, K_H) = \mathbb{H}^{i-2}(H_v^*, K) \simeq \mathbb{H}_{H_v^*}^i(X_v^*, K)(1)$$

Hence, the exact sequence $\mathbb{H}_{H_v^*}^i(B_v^*, K) \to \mathbb{H}^i(B_v^*, K) \to \mathbb{H}^i(B_v^* \setminus H_v^*, K) \xrightarrow{1} $ may be rewritten as

$$\mathbb{H}^{i-1}(B_v^* \setminus H_v^*, K) \to \mathbb{H}^{i-1}(H_v^*, K_H) \to \mathbb{H}^i(B_v^*, K) \to$$

Since $B_v^* \setminus H_v^*$ is Stein, a version of the Artin-Lefschetz vanishing theorem applies in this case to prove that $\mathbb{H}^i(B_v^* \setminus H_v^*, K) = 0$ for $i > 0$. Hence we have: $\mathbb{H}^{i-1}(H_v^*, K_H)(-1) \simeq \mathbb{H}^i(B_v^*, K)$ if $i > 1$ and $\mathbb{H}^0(H_v^*, K_H)(-1) \to \mathbb{H}^1(B_v^*, K) \to 0$ is surjective.

Let $j' : H \setminus (H \cap Y) \to H$. Since, by transversality, the shifted restriction $K_H \simeq j_{1*}^p \mathbf{L}|_H|_{V^*}[-1]$ is perverse of weight $pw - 1$, the inequalities $w > {}^p w - 1 + i - 1 = {}^p w + i - 2$ apply to $\mathbb{H}^{i-1}(H_v^*, K_H)$ by induction and necessarily $w > {}^p w + i$ holds for $\mathbb{H}^i(B_v^*, K)$ for $i \geq 1$.

We prove that $\mathbb{H}^0(B_v^*, K)$ has weights $w \geq {}^p w$ (weaker than $w > {}^p w$): Let $H_1 \not\supseteq v$ be a general hyperplane section not containing $v$ and let $k : V^* \to V$. From the distinguished triangle $k_1k^*K \to k_*k^*K \to i_{v_*}i_v^*k_*k^*K,$
we deduce the exact sequence of hypercohomology on the affine open subset $U_1 := V \setminus H_1$

$$ \mathbb{H}^0(U_1, k_*k^*K) \xrightarrow{\gamma_0} \mathbb{H}^0(i_v^*k_*k^*K) \to \mathbb{H}^1(U_1, k_*k^*K) = 0 $$

where the last term vanishes as $k_*k^*K \in \mathbb{P}^{pD_c^0}$ ([BBD 83] Théorème 4.1.1).

This proves the lemma since the weights $w$ of $\mathbb{H}^0(U_1, k_*k^*K)$ satisfy $w \ge p_w$ on the open set $U_1 \setminus v$, and $\gamma_0$ is surjective.

### 8.0.1 The crucial case: $i = 0, -1, w = p_w$

The proof by induction on $\dim V$ is based on a fibration of $V$ defined by a Lefschetz pencil of hyperplane sections. This will occupy the rest of this section.

**Proposition 8.1.** Let $j_*{}^p\mathbf{L}$ be the intermediate extension of a shifted polarized VHS on a complex algebraic variety $V$ and $v \in V$, then:

$$ \mathrm{Gr}_{p_w}^W \mathbb{H}^{-1}(B_v^*, j_*{}^p\mathbf{L}) = 0 \quad \text{and dually} \quad \mathrm{Gr}_{p_w}^W \mathbb{H}^0(B_v^*, j_*{}^p\mathbf{L}) = 0. $$

#### 8.0.1.1 Extension to hyperplane sections

Let $H$ be a general hyperplane section of $V$ through $v$. By induction, we suppose that the local Purity theorem holds on $H$.

*Recall the notations:* $V^* := V \setminus \{v\}$, $H^* := H \setminus \{v\}$, $k: V^* \to V$, $h: H^* \to H$, $i_v: \{v\} \to H$, $i_H: H \to V$, $H_v := B_v \cap H$, $H_v^* := H_v \setminus \{v\}$ and $B_v^* := B_v \setminus \{v\}$.

Two distinguished triangles are attached to any complex $L \in D_c^b(H)$:

$$ h_!h^*L \to L \to i_{v_*}i_v^*L \xrightarrow{+1} \text{ and } h_!h^*L \to h_*h^*L \to i_{v_*}i_{h_*}h^*L \xrightarrow{+1} \quad (8.1) $$

from which two exact sequences are deduced:

$$ \begin{array}{ccc} \mathbb{H}^{-1}(H, L) & \xrightarrow{\partial} & \mathbb{H}_c^0(H^*, L) \xrightarrow{\gamma} \mathbb{H}^0(H, L) \\ \downarrow R & & \downarrow R_v \\ \mathbb{H}^{-1}(H^*, L) & \xrightarrow{\rho} & \mathbb{H}^{-1}(H_v^*, L) \xrightarrow{\partial} \mathbb{H}_c^0(H^*, L) \to \mathbb{H}^0(H^*, L) \end{array} \quad (8.2) $$

Let $L := i_H^*j_*{}^p\mathbf{L}$. Since $B_v - H_v$ is Stein and the fact that the cohomology of $i_v^*j_*{}^p\mathbf{L}$ is concentrated in degree $< 0$ we have the first isomorphism:

$$ \mathbb{H}^{-1}(H_v, L) \simeq \mathbb{H}^{-1}(B_v, j_*{}^p\mathbf{L}) \simeq \mathbb{H}^{-1}(B_v^*, j_*{}^p\mathbf{L}) $$

Sine $H^r(i_v^*j_*{}^p\mathbf{L}) = 0$ if $r \le 0$, we have the second isomorphism.
An element $\alpha \in \mathbb{H}^{-1}(B_v, j_1^*\mathbb{P}\mathbf{L})$ will be identified with its restrictions to $H_v$ and $B_v^*$ by the above isomorphisms. Let $R_v(\alpha) := \alpha|_{H_v^*}$ be the restriction to $H_v^*$, then we identify $\partial(\alpha|_{H_v^*}) = \partial(\alpha) \in \mathbb{H}_c^0(H^*, L)$.

The space $\mathbb{H}^{-1}(B_v^*, j_1^*\mathbb{P}\mathbf{L})$ being endowed with a MHS (Proposition 7.4), all these spaces are endowed with MHS. In particular the MHS on $\mathbb{H}^*(H, L)$ is defined by (Corollary 7.3).

**Lemma 8.2 (Extension).** i) Given an element $\alpha \in \text{Gr}_{p_w}^W \mathbb{H}^*(B_v^*, j_1^*\mathbb{P}\mathbf{L})$, its restriction $\alpha|_{H_v^*}$ to $H_v^*$ extends to an element

$$ \alpha_{H^*} \in \text{Gr}_{p_w}^W \mathbb{H}^{-1}(H^*, j_1^*\mathbb{P}\mathbf{L}) \text{ with } \rho(\alpha_{H^*}) = \alpha|_{H_v^*}. $$

ii) Equivalently, $\alpha \in \text{Gr}_{p_w}^W \mathbb{H}^{-1}(H_v, j_1^*\mathbb{P}\mathbf{L})$ extends to a section

$$ \alpha_H \in \text{Gr}_{p_w}^W \mathbb{H}^{-1}(H, j_1^*\mathbb{P}\mathbf{L}). $$

The obstruction to extend $\alpha|_{H_v^*}$ is the element $\partial(\alpha|_{H_v^*}) = \partial(\alpha)$ in $\mathbb{H}_c^0(H^*, L)$ (Equation 8.2). We prove:

$$ \partial(\alpha) = 0 \in \mathbb{H}_c^0(H^*, L), \text{ which is equivalent to } \partial(\alpha|_{H_v^*}) = 0. $$

We remark that the image of the obstruction $\gamma(\partial\alpha) = 0 \in \mathbb{H}^0(H, L)$.

$$ \text{Let } L' = h_{1*}(j_{1*}^*\mathbb{P}\mathbf{L})|_{H^*} := \tau_{\le -2}h_*(j_{1*}^*\mathbb{P}\mathbf{L})|_{H^*} = [1] \circ h_{1*}((j_{1*}^*\mathbb{P}\mathbf{L})|_{H^*}[-1]). $$

It differs from $L$ only at $v$ since $L|_{H^*} \simeq L'_{|_{H^*}}$. As $L'[-1]$ is perverse, $H^r(i_v^*L') = 0$ if $r \le -1$ and $H^r(i_v^*L') = H^{r+1}(i_v^*L'[-1]) = 0$ if $r \ge -1$. Hence

* If $i \le 0$: $\mathbb{H}^i(H, L') \simeq \mathbb{H}^i(H^*, L')$ is pure of weight $a + i$, and $\mathbb{H}^i(H_v, L') \simeq \mathbb{H}^i(H_v^*, L')$ satisfy local purity inequalities by induction on $H$.

* If $i \ge 0$: $\mathbb{H}_c^i(H^*, L') \simeq \mathbb{H}^i(H, L')$ is pure of weight $p_w + i$.

Let $H'$ be a general hyperplane section of $V$. Since $H \setminus H'$ is affine and $L|_{H^*}[-1]$ perverse, $\mathbb{H}^0(H \setminus H', h_1(L|_{H^*})) = 0$,

We apply the global section functor $\Gamma(H \setminus H', *)$ to the second triangle on the left of (8.1), to deduce the exact sequence of hypercohomology:

$$ \mathbb{H}^{-1}(H \setminus H', L) \xrightarrow{\partial} \mathbb{H}^{-1}(H_v, L)) \xrightarrow{\partial} \mathbb{H}^0(H \setminus H', h_1(L|_{H^*})) = 0. $$

It follows that the obstruction to extend $\alpha$ to $\beta \in \text{Gr}_{p_w}^W \mathbb{H}^{-1}(H \setminus H', L)$ is zero. Let $u: H \setminus H' \to H$. We apply $\Gamma(H, *)$ to the distinguished triangle:

$$ i_{H \cap H'} i_{H \cap H'}^! L \to L \to u_* u^* L \to $$

to deduce the exact sequence:

$$ \mathbb{H}^{-1}(H, L) \to \mathbb{H}^{-1}(H \setminus (H \cap H'), L) \xrightarrow{\partial} \mathbb{H}_{H \cap H'}^0(H, L). $$

The obstruction to extend $\beta$ to an element in $\text{Gr}_{p_w}^W \mathbb{H}^{-1}(H, L)$ is the element

$$ \xi = \partial\beta \in \text{Gr}_{p_w}^W \mathbb{H}_{H \cap H'}^0(H, L) = \mathbb{H}_{H \cap H'}^0(H, L') \text{ as } v \notin H \cap H'. $$
Remark 8.1. The image $\xi'$ of $\xi$ by the morphism $\mathbb{H}_{H \cap H'}^0(H, L') \to \mathbb{H}^0(H, L')$ coincides with the image $(\partial\alpha)'$ of $\partial(\alpha)$ by the isomorphism: $\mathbb{H}^0(H, h_1h^*L) \simeq \mathbb{H}^0(H, L')$.

To apply the global invariant cycle theorem on $H$ (Lemma 7.2), we fix $H$ and let $H'$ vary in a Lefschetz pencil $P$ of hyperplane sections of $V$. The hypercohomology spaces $\mathbb{H}^*(H \cap H', L')$ form a local system on a Zariski open subset $U$ of $P$.

We choose an embedding of $H \hookrightarrow V \hookrightarrow \mathbb{P}$ into a projective space $\mathbb{P}$ and construct the diagram

$$ \begin{array}{ccc} H & \stackrel{q}{\hookrightarrow} & \tilde{H}_{\mathbb{P}} \\ & g \downarrow & \downarrow g \\ \mathbb{P} & \hookrightarrow & P \end{array} $$

where $\tilde{H}_{\mathbb{P}}$ is the disjoint union of all hyperplane sections of $H$, and $\tilde{\mathbb{P}}$ the paramater space of all hyperplane sections. Moreover $P$ is a general line in the the dual projective space $\tilde{\mathbb{P}}$ and $\tilde{H}$ is the restriction of $\tilde{H}_{\mathbb{P}}$ over $P$. The morphism $q \circ i$ is defined by the blow-up of the axis of the pencil.

By the transversality of a general hyperplane section, $R^i g_* q^* L'$ is a local system over a dense Zariski open subset $W$ of $\tilde{\mathbb{P}}$.

Let $U := W \cap P$ and $t \in U$. The fundamental group $\pi := \pi_1(U, t)$ acts as a subgroup of $\mathrm{GL}(\mathbb{H}^*(\tilde{H}_t, L'))$ (In this case the action of $\pi_1(W, t)$ as a subgroup of $\mathrm{GL}(\mathbb{H}^*(\tilde{H}_t, L'))$ coincides with the action of $\pi$). We remark that $\tilde{H}_t := H \cap H'$ where $H'$ is the hyperplane section in $V$.

We apply the global invariant cycle theorem to the morphism $g$ over the Zariski-dense open subset $U$ of $P$. For $t \in U$, the fundamental group $\pi_1(U, t)$ acts linearly on $\mathbb{H}^*(\tilde{H}_t, L')$.

For a general $H'$ with $v \notin H'$, we can view $\xi \in \mathbb{H}_{H \cap H'}^0(H, L')$ as an element in $\mathbb{H}^{-2}(H \cap H', L')$ via the Thom isomorphism.

The image of $\mathbb{H}^{-2}(H, L')$ in $\mathbb{H}^{-2}(H \cap H', L')$ under the restriction to $H \cap H'$ coincides with the image of $\mathbb{H}^0(P, R^{-2}g_*(i^*q^*L'))$ and is equal to the subspace of fixed elements $\mathbb{H}^{-2}(H \cap H', L')^\pi$ under the monodromy action of the fundamental group $\pi$; this subspace contains $\xi$ ([De 80] 6.2.12, Corollary 4.1 ii).

By duality, the Gysin morphism

$$ G : \mathbb{H}^{-2}(H, L')^{\pi}(-1) \hookrightarrow \mathbb{H}^{0}(H, L') $$

is injective. With the notation of Remark 8.1, if $\partial(\alpha) \neq 0$, we have $\xi \neq 0$.

Moreover, by transversality at a general point $t \in U$, the restriction $L'_{|_{H \cap H'}}$ is a shifted intermediate extension on $H \cap H'$.

We introduce the following diagram of Gysin morphisms with coefficients in the derived category of sheaves of vector spaces:
$$
\begin{equation} \tag{8.3}
\begin{aligned}
\operatorname{Gr}_{p_{w-4}}^{W} \mathbb{H}^{-2}(H \cap H', j_{l_*} p_{L|_{V^*}}) &\xrightarrow{G_1} \operatorname{Gr}_{p_{w-2}}^{W} \mathbb{H}^{0}(H', j_{l_*} p_{L|_{V^*}}) \\
\downarrow G_2 &\downarrow G_3 \\
\operatorname{Gr}_{p_{w-2}}^{W} \mathbb{H}^{0}(H, k_{l_*}(j_{l_*} p_{L|_{V^*}})) &\xrightarrow{G_4} \operatorname{Gr}_{p_w}^{W} \mathbb{H}^{2}(V, k_{l_*}(j_{l_*} p_{L|_{V^*}}))
\end{aligned}
\end{equation}
$$

where $j_{l_*} p_{L} := k_{l_*}(j_{l_*} p_{L|_{V^*}})$. Recall that for any subvariety $S'$ of codimension $s$ in $V$ which intersects transversely the strata of $S$ (i.e. $S'$ is normally embedded in the sense of [GMacP 88] [Max 19] Definition 3.4.5), one has the definition of an extension element $\eta_{V}^{2s} \in \operatorname{Ext}^{2s}(\mathbb{Q}_{S'}, \mathbb{Q}_{V})$ called the fundamental class [Ve 76].

It is based on the Thom isomorphism $i_{S'}^! \mathbb{Q} \simeq i_{S'}^* \mathbb{Q}[-2s]$ and extends by tensor product to coefficients in $j_{l_*} p_{L}$ since we have: $i_{S',j_{l_*} p_{L}}^! \simeq i_{S',j_{l_*} p_{L}}^*[-2s]$. This construction applies to $G_1$ and $G_2$ (resp. $G_3$) in the case of $H \cap H'$ (resp. $H'$). Special attention is needed in the case of the embedding $i_H : H \hookrightarrow V$, since $H$ contains the stratum $v$. In this case we have a morphism in the derived category $\eta_{V^*}^2 : \mathbb{Q}_{H^*}[-2] \to \mathbb{Q}_{V^*}$ defined by the fundamental class.

To construct the morphism $G_4$ after $\eta_{V^*}^2$, let $L''$ denote $j_{l_*} p_{L} = k_{l_*}(j_{l_*} p_{L|_{V^*}})$ and introduce the long exact sequences

$$
\begin{align*}
H^1(i_v^* L'') &\xrightarrow{\delta_1} \mathbb{H}_c^2(V^*, L'') \xrightarrow{\gamma} \mathbb{H}^2(V, L'') \to H^2(i_v^* L'') \\
\uparrow \eta_{V^*}^2 &\quad \uparrow \eta_{cV^*}^2 \\
H^{-1}(i_v^* L'') &\xrightarrow{\delta_{-1}} \mathbb{H}_c^0(H^*, L'') \xrightarrow{\gamma'} \mathbb{H}^0(H, L'') \to H^0(i_v^* L'')
\end{align*}
$$

defined by the distinguished triangle $k_1 k^* L'' \to L'' \to i_{v_*} i_v^* L'' \xrightarrow{[1]}$.

Here $\gamma$ is an isomorphism and $\gamma'$ is surjective, since $H^i(i_v^* L'') = 0$ for $i = 0, 1$.

We have $\delta_1 \circ \eta_{V^*}^2 = \eta_{cV^*}^2 \circ \delta_{-1}$, hence $\gamma \circ \eta_{cV^*}^2 \circ \delta_{-1} = 0$.

Indeed, $\eta_{V^*}^2$ induces two morphisms $\eta_{B_v^*}^2 : \mathbb{H}^{-1}(B_v \cap H^*, L'') \to \mathbb{H}^1(B_v^*, L'')$ (equivalently $\mathbb{H}^{-1}(i_v^* L'') \to \mathbb{H}^1(i_v^* L'')$) and $\eta_{cV^*}^2 : \mathbb{H}_c^0(H^*, L'') \to \mathbb{H}_c^2(V^*, L'')$.

We deduce the morphism $G_4 : \mathbb{H}^0(H, L'') \to \mathbb{H}^2(V, L'')$.

Moreover, it is clear by this construction that $G_4$ is compatible with MHS.

We have $G_2(\xi) = \gamma(\partial\alpha) = 0$, hence $G_3 \circ G_1(\xi) = 0$. The morphism $G_3$ is injective since $\mathbb{H}^1(X - H', k_{l_*}(p\mathcal{H}^i)) = 0$, hence $G_1(\xi) = 0$.

Finally, we fix $H'$ and let $H$ vary in a Lefschetz pencil of hyperplane sections $H$ through $v$ and we apply the global invariant cycle theorem again to prove that the Gysin morphism $G_1$ is injective, hence $\xi = 0$.

**Remark 8.2.** The original proof is based on a diagram defined by cohomology classes with support

$$
\begin{tikzcd}
\mathbb{Q}_{H \cap H'}(-1)[-2] \arrow[r] & \mathbb{Q}'_H \\
\mathbb{Q}_H \arrow[d] & \mathbb{Q}_V(1)[2]
\end{tikzcd}
$$

which generates by tensor product with coefficients in $L'' := j_{l_*} p_{L}$ the commutative diagram 8.3 of topological morphisms. However, the reference [Ve 76] applies only for nonsingular $V$; moreover $H$ is not normally embedded since
it contains the stratum $v$. We modified the proof to apply directly to the
Diagram 8.3.

8.0.1.2 Reduction to a fibration

To reduce the crucial case to the case where a fibration $g : V \to \mathbb{P}^1$ exists, we use a general Lefschetz pencil of hyperplanes as already mentioned in chapter 6, §6.2.0.3. Then, the local and global invariant cycle theorems for perverse cohomology sheaves (Proposition 7.5, Corollary 7.5) and the extension property (Lemma 8.2) apply.

Let $g(v) = 0$, $D$ a complex disc with center 0 in $\mathbb{P}^1$ and $V_D := g^{-1}(D)$.
We have the diagram

$$
\begin{tikzcd}
H \setminus \{v\} \arrow[r, "u"] \arrow[d] & V_D \setminus \{v\} \arrow[r, "\phi"] \arrow[d] & V_D \setminus H \arrow[d] \\
0 \arrow[r, "u'"] & D \arrow[r, "\phi'"] & D^*
\end{tikzcd}
\qquad (8.4)
$$

**Lemma 8.3.** Let $H$ be a general hyperplane section and $P$ a perverse sheaf on $V_D \setminus \{v\}$. With the notations of Diagram 8.4, there exists a decomposition in the derived category of complexes of sheaves of vector spaces on $H \setminus \{v\}$:

$$
u^*P \oplus u^*P[-1] \simeq u^*\phi_*\phi^*P.
$$

The splitting of complexes of $\mathbb{Q}$-vector spaces at $0 \in D$

$$
(u')^* \phi'_* \mathbb{Q} \simeq \mathbb{Q} \oplus \mathbb{Q}[-1](-1)
$$

corresponds to the isomorphism $H^*(D^*, \mathbb{Q}) \simeq H^0(D^*, \mathbb{Q}) \oplus H^1(D^*, \mathbb{Q})$. In terms of the de Rham resolution $\mathcal{E}_{D^*}$ of $\mathbb{C}$, the generators are $1 \in \mathbb{Q}$ and $\frac{dz}{z}$ respectively.

In terms of a Whitney stratification $\mathcal{S}$ of $V$ compatible with the perverse sheaf $P$, the restriction of the cohomology sheaves of $P$ to a stratum $S$ near a point $a \in H \setminus v$ is constant (there is no monodromy near $a$), hence we deduce that the natural morphism $P \otimes g^*\phi_*\mathcal{E}_{D^*}^* \to \phi_*\phi^*P$ induces an isomorphism in the derived category of complexes of sheaves of vector spaces

$$
u^*P \oplus u^*P[-1] \xrightarrow{\gamma} u^*\phi_*\phi^*P.
$$

In view of this description we denote the decomposition as:

$$
\forall i \in \mathbb{Z}, \quad \mathbb{H}^i(H \setminus v, u^*\phi_*\phi^*P) \simeq \mathbb{H}^i(H \setminus v, P) \oplus \eta \mathbb{H}^{i-1}(H \setminus v, P) \quad (8.5)
$$
### 8.0.1.3 Lifting of $\alpha_H$

Let $P := j_*^*P_L$ on $V$, Coker$^{-2}N := \text{Coker}(N : \mathbb{H}^{-2}(H, \psi_g^u P) \to \mathbb{H}^{-2}(H, \psi_g^u P))$.

By (Remark 7.5), we have an exact sequence:

$$0 \to \text{Coker}^{-2} N(-1) \to \mathbb{H}^{-1}(V_D \setminus H, P) \to \mathbb{H}^{-1}(H, \psi_g^u P)^N \to 0$$

Since $V_D$ retracts by deformation onto $H$, the element $\alpha_H \in \text{Gr}_{p_w}^W \mathbb{H}^{-1}(H, P)$ in the Extension Lemma 8.2 corresponds to an element $\alpha_D \in \text{Gr}_{p_w}^W \mathbb{H}^{-1}(V_D, P)$, with restriction

$$\alpha_D^* \in \text{Gr}_{p_w}^W \mathbb{H}^{-1}(V_D \setminus H, P) \text{ on } V_D \setminus H.$$

The specialization (Equation 7.10)

$$\text{sp}(\alpha_H) := \tilde{\alpha}_D^* \in \text{Gr}_{p_w}^W \mathbb{H}^{-1}(H, \psi_g^u P)^N = 0$$

must vanish since the space $\mathbb{H}^{-1}(H, \psi_g^u P)^N$ is of weight $\le p_w - 1$, while $w(\alpha_H) = p_w$.

Hence the section $\alpha_D^*$ lifts to a section

$$\tilde{\beta} \in \text{Gr}_{p_w}^W (\mathbb{H}^{-2}(H, \psi_g^u P)(-1)) = \text{Gr}_{p_w-2}^W (\mathbb{H}^{-2}(H, \psi_g^u P) : (\tilde{\beta} \mapsto \alpha_D^*))$$

$\tilde{\beta}$ is unique in Coker $N := \text{Coker}(N : \text{Gr}_{p_w}^W \mathbb{H}^{-2}(H, \psi_g^u P) \to \text{Gr}_{p_w-2}^W \mathbb{H}^{-2}(H, \psi_g^u P))$.

Since $\text{Ker } N := \text{Ker}(N : \text{Gr}_{p_w-2}^W \mathbb{H}^{-2}(H, \psi_g^u P) \to \text{Gr}_{p_w-4}^W \mathbb{H}^{-2}(H, \psi_g^u P))$ projects isomorphically onto Coker $N$ as both are isomorphic to the space

$\text{Prim} \text{Gr}_{p_w-2}^W \mathbb{H}^{-2}(H, \psi_g^u P)$, we can choose the element $\tilde{\beta} \in \text{Gr}_{p_w-2}^W \mathbb{H}^{-2}(H, \psi_g^u P)^N$.

By the local invariant cycle theorem (Corollary 7.5), we lift this canonical choice of $\tilde{\beta}$ to an element $\beta_D \in \text{Gr}_{p_w-2}^W \mathbb{H}^{-2}(V_D, P)$.

Let $\beta_D^* \in \text{Gr}_{p_w-2}^W \mathbb{H}^{-2}(V_D \setminus H, P)$ denote the restriction of $\beta_D$; we have

$$\eta \wedge \beta_D^* = \alpha_D^* \in \text{Gr}_{p_w}^W \mathbb{H}^{-1}(V_D \setminus H, P) \quad (8.6)$$

### 8.0.1.4 First reduction to prove $\alpha = 0$

Let $B_v^* = B_v \setminus \{v\}$, $V_D^* = V_D \setminus \{v\}$, $H^* = H \setminus \{v\}$, $H_v = H \cap B_v$, $H_v^* = H_v \setminus \{v\}$, and consider the diagram

$$\alpha_*^* \in \text{Gr}_{p_w}^W \mathbb{H}^{-1}(B_v \setminus H_v, j_*^* P_L) \leftarrow \alpha \in \text{Gr}_{p_w}^W \mathbb{H}^{-1}(B_v^*, j_*^* P_L)$$

$$\uparrow \quad \leftarrow \quad \uparrow$$

$$\alpha_D^* \in \text{Gr}_{p_w}^W \mathbb{H}^{-1}(V_D \setminus H, j_*^* P_L) \leftarrow \alpha_D \in \text{Gr}_{p_w}^W \mathbb{H}^{-1}(V_D^*, j_*^* P_L)$$

$$\sigma \downarrow \quad \downarrow$$

$$A \quad \xleftarrow{\sigma_1} \quad \alpha_H \in \text{Gr}_{p_w}^W \mathbb{H}^{-1}(H^*, j_*^* P_L)$$

where $A := \text{Gr}_{p_w}^W (\mathbb{H}^{-1}(H^*, P) \oplus \eta \text{Gr}_{p_w-2}^W \mathbb{H}^{-2}(H^*, P))$ (by Equation 8.5)

The morphism $\sigma := (\sigma_1, \sigma_2)$ is defined by Equation 8.5. The element $\alpha_D$ restricts to $\alpha_D^* \in \text{Gr}_{p_w}^W \mathbb{H}^{-1}(V_D \setminus H, P)$ and to $\alpha_H \in \text{Gr}_a^W \mathbb{H}^{-1}(H^*, P)$ satisfying
the relation:

$$ \sigma(\alpha_D^*) = \sigma_1(\alpha_H) \in \mathrm{Gr}_{\mathrm{pw}}^W(\mathbb{H}^{-1}(H^*, P)) \quad (8.7) $$

The diagram above is commutative and by Lemma 8.2 the restriction of $\alpha_D$ coincides with $\alpha \in \mathrm{Gr}_{\mathrm{pw}}^W\mathbb{H}^{-1}(B_v^*, P)$.

**Lemma 8.4.** An element $\alpha \in \mathrm{Gr}_{\mathrm{pw}}^W\mathbb{H}^{-1}(B_v^*, j_*^p\mathbf{L})$ vanishes if and only if its restriction $\alpha_*^*$ to $B_v \setminus H_v$ is zero.

*Proof.* Since we have the exact sequence

$$ \mathrm{Gr}_{\mathrm{pw}-2}^W \mathbb{H}^{-3}(H_v^*, j_*^p \mathbf{L}) \xrightarrow{G} \mathrm{Gr}_{\mathrm{pw}}^W \mathbb{H}^{-1}(B_v^*, j_*^p \mathbf{L}) \to \mathrm{Gr}_{\mathrm{pw}}^W \mathbb{H}^{-1}(B_v \setminus H_v, j_*^p \mathbf{L}) $$

it is enough to prove $\mathrm{Gr}_{\mathrm{pw}-2}^W \mathbb{H}^{-3}(H_v^*, j_*^p \mathbf{L}) = 0$.

This follows by the inductive hypothesis on the small neighborhood $H_v$ of $v$ in $H$ since $\mathbb{H}^{-3}(H_v^*, j_*^p \mathbf{L}) = \mathbb{H}^{-2}(H_v^*, j_*^p \mathbf{L}[-1])$ satisfy $w \le p w - 3$ by induction.

### 8.0.1.5 Proof of $\alpha_*^* = 0$ in $\mathrm{Gr}_{\mathrm{pw}}^W \mathbb{H}^{-1}(B_v \setminus H_v, j_*^p \mathbf{L})$

We consider the diagram

$$ \begin{array}{ccc} \beta^* \in \mathrm{Gr}_{\mathrm{pw}-2}^W \mathbb{H}^{-2}(B_v^*, P) & \xrightarrow{\eta \circ \rho} & \alpha_*^* \in \mathrm{Gr}_{\mathrm{pw}}^W \mathbb{H}^{-1}(B_v \setminus H_v, P) \\ \uparrow & & \uparrow \\ \beta_D \in \mathrm{Gr}_{\mathrm{pw}-2}^W \mathbb{H}^{-2}(V_D^*, P) & \xrightarrow{\eta \circ \rho} & \alpha_D^* \in \mathrm{Gr}_{\mathrm{pw}}^W \mathbb{H}^{-1}(V_D \setminus H, P) \quad (8.8) \\ \downarrow & & \downarrow \sigma \\ \beta_H \in \mathrm{Gr}_{\mathrm{pw}-2}^W \mathbb{H}^{-2}(H^*, P) & \xrightarrow{\eta \cup} & A \end{array} $$

where $\rho$ denotes the restriction of $\beta^*$ to $B_v \setminus H_v$, resp. $\beta_D$ to $V_D \setminus H$.

**Lemma 8.5.** The restriction $\beta_H \in \mathrm{Gr}_{\mathrm{pw}-2}^W \mathbb{H}^{-2}(H^*, P)$ of $\beta_D$ vanishes.

*Proof.* The element $\beta_D \in \mathrm{Gr}_{\mathrm{pw}-2}^W \mathbb{H}^{-2}(V_D^*, j_*^p \mathbf{L})$ restricts to an element $\rho(\beta_D)$ on $V_D \setminus H$ such that $\eta \cup \rho(\beta_D) = \alpha_D^* \in \mathrm{Gr}_{\mathrm{pw}}^W \mathbb{H}^{-1}(V_D \setminus H, j_*^p \mathbf{L})$ by Equation (8.6). By restricting to $H^*$ we deduce:

$$ \eta \cup \beta_H = \sigma(\eta \cup \rho(\beta_D)) = \sigma(\alpha_D^*) = \sigma_2(\alpha_D^*) \in \eta \mathrm{Gr}_{\mathrm{pw}-2}^W(\mathbb{H}^{-2}(H^*, j_*^p \mathbf{L})) \subset A $$

By (8.7) we have the equality $\sigma(\alpha_D^*) = \sigma_1(\alpha_H)$, $\sigma_2(\alpha_D^*) = \sigma(\alpha_D^*) = \sigma_1(\alpha_H) = 0$ by the decomposition property of $A$; then $\beta_H = 0$ since the cup product $\eta \cup$ is injective on $\mathrm{Gr}_{\mathrm{a}-2}^W \mathbb{H}^{-2}(H^*, j_*^p \mathbf{L})$.

We denote by $\beta^*$ the restriction of $\beta_D$ to $B_v^*$ in Diagram (8.8), and by $\beta_{H_v^*}^*$ the restriction of $\beta_H$ to $H_v^*$.

**Lemma 8.6.** i) The restriction $\mathrm{Gr}_{\mathrm{pw}-2}^W \mathbb{H}^{-2}(B_v^*, j_*^p \mathbf{L}) \to \mathrm{Gr}_{\mathrm{pw}-2}^W \mathbb{H}^{-2}(H_v^*, j_*^p \mathbf{L})$ : $\beta^* \mapsto \beta_{H_v^*}^*$ is injective.

ii) We have: $\beta^* = \beta_{H_v^*}^* = 0$ and $\alpha_*^* = \eta \cup \rho(\beta^*) = 0$.
Proof. i) Let $j_H : B_v \setminus H_v \to B_v$ denote the open embedding, since the open subset $B_v \setminus H_v$ is Stein, we have

$$ \mathbb{H}^{-2}(B_v \setminus H_v, j_H, j_H^* P) = 0 = \mathbb{H}^{-1}(B_v \setminus H_v, j_H, j_H^* P). $$

ii) We have $\beta_{H_v}^* = 0$ as the restriction of $\beta_H = 0$, hence $\beta^* = 0$ and

$$ \alpha_* = \eta \cup \rho(\beta^*) = 0 \in \mathrm{Gr}_{p_{w-2}}^W \mathbb{H}^{-2}(B_v \setminus H_v, j_*^P L). $$

The local invariant cycle theorem is needed here as the lemma applies exactly to $\beta_D$ and not to $\alpha_D$. The final conclusion $\alpha = 0 \in \mathrm{Gr}_{p_w}^W \mathbb{H}^{-1}(B_v^*, j_*^P L)$ follows by (Lemma 8.4).

Remark 8.3. Let $f : X \to V$ be a projective morphism of complex algebraic varieties and $v \in V$, with fiber $X_v := f^{-1}(v) \subset X$. The local purity theorem may be interpreted as a relative local purity (ch. 9, Theorem 9.2) and in terms

of the intersection morphism $I: \mathrm{Gr}_j^{p_\tau} \mathbb{H}_{X_v}^i(X, j_*^P L) \xrightarrow{p_\tau I_*^j} \mathrm{Gr}_j^{p_\tau} \mathbb{H}^i(X_v, j_*^P L)$ (§10.4).
Chapter 9

Weight filtration, decomposition and
local purity

Fouad El Zein, Dũng Tráng Lê, Xuanming Ye

We start a new section of this volume consisting of chapters 9 – 12, to give a
new combined direct geometric proof of the local purity (ch. 1, Theorem 1.1)
and the decomposition theorem (ch. 1, Theorem 1.2).

In this section, we do not use the nearby or vanishing cycles complex to
construct the weight filtration on the logarithmic complex $\Omega^*\mathcal{L} \simeq j_*\mathbf{L}$.

The proof of local purity differs from the proof in [DeG 81]. An interpre-
tation in terms of Intersection morphisms is given at the end.

We refer to the necessary preliminaries in chapters 2 and 3, namely §2.3 on the degeneration of VMHS, §2.3.2 on infinitesimal mixed Hodge structure (IMHS), Remark 2.10 on IMHS defined by an admissible VMHS, chapter 2.4.0.1 on the embedding of the perverse filtration, §3 on perverse sub-sheaves of the logarithmic complex $\Omega^*\mathcal{L} \simeq j_*\mathbf{L}$ and Remark 3.6 vii on the decomposition of the Hodge filtration following the decomposition of spaces with underlying MHS.

9.1 Introduction to chapters 9 – 11 (section 3)

Given a graded polarized admissible variation of MHS ($\mathbf{L}, W^0, F$) on the
complement of a NCD $Y \subset X$, we develop in §9.1.0.2 a direct definition of the
weight filtration $W$ on the logarithmic complex $\Omega^*\mathcal{L}$ (§3.1) by perverse sub-
sheaves, extending naturally the weight filtration $W^0$ on $\mathbf{L}$, as an application
of the local study by Kashiwara [Ka 86] (see ch. 2, 2.3 and ch. 3, §3.2.6).

The weight filtration $W$ is defined in this chapter in terms of the local
monodromy defined by the coordinates (§3.1.1) at points of the NCD $Y$.

<footnote>IMJ-PRG, Université de Paris, Sophie Germain, F-75013 Paris, France</footnote>

<footnote>UMR-CNRS 7353, Université d'Aix-Marseille, F-13453 Marseille, France</footnote>

<footnote>School of Mathematics and Information Science, Guangzhou University, P.R.China.</footnote>
Since $Y$ is defined locally by an equation $f = z_{i_1} \cdots z_{i_n}$ product of local co-ordinates, the weight filtration is related to the monodromy weight filtration on the nearby cycle complex $\psi_f L$ by (Equation 5.54, §5.1.2 Remark 5.4).

Since $Gr_*^W \Omega^*\mathcal{L}$ satisfy a decompositon property (§9.2.2) into a direct sum of (shifted) intermediate extensions with Hodge filtration induced by $(\Omega^*\mathcal{L}, F)$, we deduce from the purity theorem on intermediate extensions with induced Hodge filtration (Theorem 3.1), the existence of a structure of MHC on $(\Omega^*\mathcal{L}, F, W)$ as well on the sub-complex $IC^*\mathcal{L}(\mathrm{Log}Z)$.

We introduce a relative version of the local purity theorem to reduce the proof to the non singular case.

The decomposition theorem is proved by induction on the dimension of $X$, first by a reduction to general hyperplane sections and then to isolated strata. The proof in chapter 10 is combined with the proof of the relative version of the local purity theorem.

Originally, the work on this section preceded the previous sections (see [EL 14] and [EY 14]).

### 9.1.0.1 Weight filtration on $\Omega^*\mathcal{L}$

Let $Y \subset X$ be a NCD and $(L, W^0, F)$ a graded admissible polarized variation of MHS on $X^* := X \setminus Y$. We define the weight $W$ on $\Omega^*\mathcal{L}$, extending $W^0$ over $X^* := X \setminus Y$, in terms of various local weight monodromy filtrations at points $x \in Y$. We prove the following theorem:

**Theorem 9.1.** Let $(L, W^0, F)$ be a graded polarized admissible VMHS on $X \setminus Y$. There exists a weight filtration $W$ on the logarithmic complex with coefficients in Deligne's extension $\mathcal{L}_X$ (§3.1) by complexes of analytic perverse sub-sheaves and a Hodge filtration $F$ by complexes of analytic sub-sheaves such that the bi-filtered complex

$$ \Omega^*\mathcal{L} := (\Omega_X^*(\mathrm{Log} Y) \otimes \mathcal{L}_X, W, F) \qquad (9.1) $$

induces a MHS on the cohomology groups $\mathbb{H}^i(X \setminus Y, L)$.

The construction of $W$ and the proof of the theorem will occupy the rest of the chapter.

### 9.1.0.2 Logarithmic complexes along a NCD

Various complexes with weight and Hodge filtrations related to a NCD sub-divisor $Z$ of $Y \subset X$ are defined by restriction to $Z$, with support in $Z$ or Verdier's duality.

Let $\mathcal{L} := L[\dim X]$ denote the associated perverse sheaf on $X \setminus Y$ with the conventional shifted degree. It is called a shifted VMHS.
Let $i_Z : Z \to X$, $j := (X \setminus Y) \to X$ and $j_Z := (X \setminus Z) \to X$. We construct the weight and Hodge filtrations on the sub-complex $IC^*\mathcal{L}(\mathrm{Log} Z) \subset \Omega^*\mathcal{L}$ isomorphic to $j_{Z^*}(j_{!*}^1\mathbf{L})|_{X\setminus Z}$, from which we deduce logarithmic complexes isomorphic to $j_{Z^*}(j_{!*}^p\mathbf{L})|_{X\setminus Z}$, $i_Z^*j_{Z^*}(j_{!*}^1\mathbf{L})|_{X\setminus Z}$, $i_Z^*j_{Z^*}(j_{!*}^p\mathbf{L})$ and $i_Z^*j_{Z^*}^p\mathbf{L}$ (see ch.3, §3.2.4). We refer to such complexes as bifiltered logarithmic complexes in the derived category of bifiltered sheaves of abelian groups.

### 9.1.0.3 Relative local purity

We modify the statement of the local purity theorem (ch. 1, Theorem 1.1), to explore its relation with the decomposition theorem.

Let $K \in D_c^b(V, \mathbb{Q})$ be a complex in the derived category of complexes with constructible cohomology on an analytic space $V$. The topological middle perversity truncations of $K$ define an increasing *perverse filtration* $p_\tau$ of $K$ on $V$ ([BBD 83] section 2 and proposition 2.1.17).

We deduce for each subset $S$ of $V$, an increasing filtration $p_\tau$ on the hypercohomology of the restriction of $K$ to $S$:

$$ p_{\tau_i} \mathbb{H}^k(S, K) := \mathrm{Im} \{ \mathbb{H}^k(S, p_{\tau_i} K) \to \mathbb{H}^k(S, K) \}. \quad (9.2) $$

Let $f : X \to V$ be a projective morphism and let $j_{!*}^1\mathbf{L}$ be the intermediate extension on $X$ of a (shifted) polarized variation of Hodge structures (VH) of weight $p_w$ on a smooth Zariski open subset of $X$.

Let $B_{X_v} := f^{-1}(B_v)$ be the inverse image of a small ball $B_v$ with center $v \in V$ and $X_v := f^{-1}(v) \subset B_{X_v}$. A *perverse filtration* $p_\tau$ on $\mathbb{H}^k(B_{X_v} \setminus X_v, j_{!*}^1\mathbf{L})$ is defined by the natural equality $\mathbb{H}^k(B_{X_v} \setminus X_v, j_{!*}^p\mathbf{L}) = \mathbb{H}^k(B_v \setminus v, f_{*j_{!*}^1}\mathbf{L})$:

$$ p_{\tau_i} \mathbb{H}^k(B_{X_v} \setminus X_v, j_{!*}^p \mathbf{L}) := p_{\tau_i} \mathbb{H}^k(B_v \setminus v, f_{*j_{!*}^p} \mathbf{L}) \quad (9.3) $$

$(f_*$ is the derived functor, in general we do not use henceforth the letter $R$ for a derived functor (see chapter 1, §2.4 and 2.4.0.2 for more conventions).

The hypercohomology $\mathbb{H}^k(B_{X_v} \setminus X_v, j_{!*}^p\mathbf{L})$ is independent of the choice of $B_v$ with small radius and carry a natural MHS such that the perverse filtration is a filtration by sub-MHS (ch.11.1 Proposition 10.2 and ch. 10 Proposition 10.1). We prove in chapter 10:

**Theorem 9.2 (Relative local purity).** Let $f : X \to V$ be a projective morphism of complex algebraic varieties, $v \in V$, $X_v := f^{-1}(v) \subset X$ and $B_{X_v} := f^{-1}(B_v)$ the inverse image of a small ball $B_v \subset V$.

Let $j_{!*}^p\mathbf{L}$ be a (shifted) polarized variation of HS on $X$ of weight $p_w$, the weight $w$ of the MHS on the graded-cohomology spaces with respect to the middle perversity filtration $p_\tau$ satisfy the inequalities: $w\mathrm{Gr}_i^{p_\tau}\mathbb{H}^r(B_{X_v} \setminus X_v, j_{!*}^p\mathbf{L})$:

$$ w \le p_w + r \text{ if } r - i \le -1; \quad w > p_w + r \text{ if } r - i \ge 0. \quad (9.4) $$
Remark 9.1. i) We say that $j_{1*}p^L$ is locally pure at $v$ relative to $f$.

ii) Let $K := f_*j_*p^L$. If the decomposition theorem apply to $K$ on $V \setminus v$, then $j_{1*}p^L$ is locally pure at $v$ relative to $f$ if and only if $p\mathcal{H}^i(K)$ is locally pure at $v$ for each $i$, since $Gr_i^{p\tau}\mathbb{H}^r(B_{X_v} \setminus X_v, j_{1*}p^L) \simeq \mathbb{H}^{r-i}(B_v \setminus v, p\mathcal{H}^i(K))$. In this case, we say that $K$ is locally pure at $v$.

iii) In the case of a finite morphism and a non singular variety $X$, $\dim X = \dim V = n$, $X_v$ is a finite number of points and $B_{X_v} \setminus X_v$ is a finite number of punctured discs of dimension $n$. Since $f_*j_*p^L$ is perverse, the $p\tau$-filtration is reduced to one term $p\mathcal{H}^0(f_*j_*p^L) = f_*j_*p^L$.

The statement is reduced to a condition on the weight $w$ of $\mathbb{H}^r(D^n \setminus 0, j_*p^L)$. In this special case, the theorem is proved in ([CaKSc 87], Corollary 1.13, [KaK 87], Theorem 4.0.1) since $\mathbb{H}^r(D^n \setminus 0, j_*p^L) = \mathbb{H}^r(D^n, j_{1*}p^L)$ for $r \le -1$.

### 9.1.0.4 The decomposition theorem

The combined new proof of the decomposition theorem (ch.1, Theorem 1.2) and the local purity (ch.1, Theorem 1.1), is by induction on the dimension of $X$ and reduction to isolated strata (ch.10 and ch.10.3), in terms of a Thom-Whitney stratification.

### 9.1.0.5 Local purity and Intersection isomorphisms

The relative local purity theorem with coefficients in a polarized variation of HS is related to the Intersection morphism in the case of a divisor on a variety $X$ contracting to $v \in V$ (ch. 10.3, §10.4).

In the case of constant coefficients, the intersection morphism is studied in ([CaMi 5] section 6.3) where a generalized Grauert contractibility criterion is proved. This statement is in fact equivalent to the purity theorem (see §10.4.1) and it is related to the proof of the decomposition theorem in [CaMi 5].

Since MHS on cohomology can be assumed in the constant coefficients case, the proofs in [CaMi 5] are different from the general case where the construction of mixed Hodge theory is simultaneous and included in the proof. The reader will find in [CaMi 5] numerous interesting examples.

### 9.1.0.6 Fibration by NCD over the strata

The proof of the decomposition theorem is always reduced to the case of a fiber by NCD above an isolated stratum. This reduction is local. By induction, this amounts to the case where the inverse of a stratum is locally a fibration by NCD.
We remark at the end of chapter 11 (§11.2) that it is possible to reduce all proofs to the case of a family of projective morphisms fibered by NCD above the strata of a global fibration of V.

### 9.1.0.7 Admissible perverse VMHS on $X \setminus Z$

We added in this chapter complements on admissible perverse VMHS on $X \setminus Z$ ([Sa 90] and §9.7) to answer a question by the referee. We discuss the subject with hints to the proof as it is a motivation to the introduction of the filtration $N * W$ and shows its relation to Verdier's classification.

## 9.2 The weight filtration W

Let $\mathbf{L}$ be a VHS. The relation between the weight filtration on $\Psi^*L$ (Proposition 5.1) and the weight filtration $W$ on $\Omega^*L$ is defined by (Equation 5.54):

$$ \mathrm{Gr}_l^W j_{Z*} j_Z^*(j_{!*} \mathbf{L}) = \mathrm{Gr}_{l-1}^W (\mathrm{Coker} N) \quad \text{if } l > {}^p w $$

where $N$ acts on the nearby cycle complex (Equation 9.43).

This chapter, is a definition, in the case of a VMHS $\mathbf{L}$, of the weight filtration $W$ on $\Omega^*L$ described by repeated star transformation $N * W$ of a filtration $W$ of $L$ defined by Kashiwara ([Ka 86], 3.4, §9.2.1).

The main results of [Ka 86] is a test of admissibility in codimension one and an important property of the relative monodromy useful to extend results from the pure case to the mixed case:

**Proposition 9.1 (Kashiwara ([Ka 86], Theorem 3.2.9)).** Let $L$ be an object of an abelian category with an increasing filtration $W$ and $N : (L, W) \to (L, W)$ a nilpotent endomorphism compatible with $W$. The relative monodromy filtration $M$ satisfy the following natural decomposition

$$ \mathrm{Gr}_l^M L \simeq \bigoplus_k \mathrm{Gr}_k^W \mathrm{Gr}_l^M L \qquad (9.5) $$

In particular, there exist natural sections $\mathrm{Gr}_r^M \mathrm{Gr}_k^W L \to \mathrm{Gr}_r^M L$.

We give at the end of the paragraph (9.7.1.3 below) informations on the proof and the consequences.

However, we find in [Ka 86] various local statements needed to define the weight filtration $W$ on $\Omega^*\mathcal{L}$ which was probably the projected result but do not figure there, neither the motivation of the definition $N * W$.

In this chapter the properties of weight filtration $W$ on $\Omega^*\mathcal{L}$ are discussed and the decomposition of $\mathrm{Gr}_*^W \Omega^*\mathcal{L}$ is proved, which leads to a new proof of the decomposition theorem (Theorem 10.1, see also Theorem 11.1).

We consider a graded polarized admissible variation of MHS on the underlying local system $\mathbf{L}$ on $X^* := X \setminus Y$, satisfying asymptotic properties at points of $Y$, summarized in the condition of admissibility (ch. 2, §2.3).
The Hodge filtration $F$ denotes a filtration $\mathcal{F}_X^*$ by sub-bundles of $\mathcal{L}_X^* := \mathbf{L} \otimes \mathcal{O}_{X^*}$. By definition, it extends into a filtration of Deligne's extension $\mathcal{L}_X$ by analytic sub-bundles $\mathcal{F}_X$.

The filtration by sub-local systems $W^0$ of $\mathbf{L}$ extends to a filtration by
perverse analytic sub-sheaves $W^0 \subset \mathcal{L}_X$ which restricts to a filtration by
sub-bundles on each stratum of $Y$.

By the admissibility of $\mathbf{L}$, the monodromy filtration relative to the exten-
sion $W^0$ along $Y$ exists. Both filtrations $W^0$ and $\mathcal{F}_X$ are combined to define
the global weight filtration $W$ on $\Omega^*\mathcal{L}$ (Definition 9.2), although the con-
struction below of $W$ is local. Such results are used in chapters 10 and 10.3
to prove the theorem 11.1.

Remark 9.2. The weight $W$ is a filtration by $\mathbb{Q}$-perverse sub-sheaves although the definition is by analytic sub-sheaves of $\Omega^i\mathcal{L} := \Omega_X^i(\mathrm{Log} Y) \otimes \mathcal{L}_X$ in each degree. The reader should be aware that the filtration $W$ on $\Omega^*\mathcal{L}$ is called weight since it induces the weight of the MHS on the hypercohomology.

In the case of intermediate extension the weight is trivial. However, we
will see that the local cohomology carry also weight filtrations as local orbits
or IMHS (Definition 2.14) and the local purity property is satisfied (it is the
original conjecture by Deligne to prove purity).

The weight of a nilpotent orbit of dimension $n$ is defined by $N_1 + \dots + N_n$,
otherwise we write $M(A, W)$ for the relative weight filtration defined by an
endomorphism $A$ compatible with $W$.

9.2.1 Local weight filtration on Ω*L

In view of the local description of perverse sheaves on a product of discs
([Ka 86] §2 and ch.3, §3.2.6) many constructions on logarithmic complexes,
such as the weight filtrations, will appear first on the complex of vector spaces
Ω*L (ch.3, §3.1.2) where L := L_X(x) at a point x ∈ Y, and will be carried
later on the logarithmic complex by the tilde embedding (Lemma 3.1).

The properties of IMHS (Definition 2.14) are used in the description of
the decomposition of $Gr_*^W \Omega^*L$.

9.2.1.1 The filtration N * W

Let $(L, W, N)$ denote an increasing filtration $W$ on a vector space $L$ with
a nilpotent endomorphism compatible with $W$ $(NW_i \subset W_i)$ such that the
relative monodromy filtration $M(N, W)$ exists.

A new filtration $N * W$ of $L$ is defined by Kashiwara ([Ka 86], 3.4):

$$
(N * W)_k := NW_{k+1} + M_k(N, W) \cap W_k = NW_{k+1} + M_k(N, W) \cap W_{k+1} \quad (9.6)
$$
where the last equality follows from ([Ka 86], Prop 3.4.1). The endomorphism $N : L \to L$ as well the identity $I : L \to L$ shift the degrees:

$$N : W_k \to (N * W)_{k-1}, \quad I : (N * W)_{k-1} \to W_k \qquad (9.7)$$

### 9.2.1.2 Properties of $N * W$

1) The couple $((L, W) \xrightarrow{N} (L, N * W))$ form a (graded) distinguished pair (proved in ([Ka 86], Lemma 3.4.2 and admitted here): we have the decomposition property of $N * W$

$$ \begin{align} & Gr_k^{N*W} L \simeq \mathrm{Im} (N : Gr_{k+1}^W L \to Gr_k^{N*W} L) \oplus \ker (I : Gr_k^{N*W} L \to Gr_{k+1}^W L) \nonumber \\ & \mathrm{Im} (N : Gr_{k+1}^W L \to Gr_k^{N*W} L) \xrightarrow{\sim} \mathrm{Im} (N : Gr_{k+1}^W L \to Gr_{k+1}^W L) \nonumber \\ & \ker (I : Gr_k^{N*W} L \to Gr_{k+1}^W L) \simeq \mathrm{Coker}(N : W_k Gr_{k+2}^M \to W_k Gr_k^M) \tag{9.8} \end{align} $$

where $M := M(N, W)$ (see also [Ka 86], Corollary 3.4.3)

2) We have the following properties of $M$ and the induced filtration $W$ on $Gr_k^{N*W} L$ by ([Ka 86], Lemma 3.4.2):

$$ \begin{align} M(N, W) &= M(N, N * W) \nonumber \\ W_{k+1} Gr_k^{N*W} L &= Gr_k^{N*W} L, \nonumber \\ Gr_{k+1}^W Gr_k^{N*W} L &\simeq \mathrm{Im}(N : Gr_{k+1}^W L \to Gr_{k+1}^W L), \tag{9.9} \\ W_k Gr_k^{N*W} L &= \ker (I : Gr_k^{N*W} L \to Gr_{k+1}^W L) \nonumber \\ \forall p \le k : Gr_p^W (Gr_k^{N*W} L) &\simeq \mathrm{Coker}(N : Gr_{k+2}^M Gr_p^W L \to Gr_k^M Gr_p^W L) \nonumber \end{align} $$

Since $N$ is strict with respect to $W$ (as an IMHS), we have ([Ka 86], Corollary 3.4.3):

$$ Gr_k^{N*W} L \simeq \mathrm{Im}(N : Gr_{k+1}^W L \to Gr_{k+1}^W L) \oplus \mathrm{Coker}(N : W_k Gr_{k+2}^M \to W_k Gr_k^M) \qquad (9.10) $$

**Lemma 9.1.** ([Ka 86], Proposition 5.3.1, Corollary 5.5.4).

Let $(L, W, F, N_1, \dots, N_n)$ be an IMHS, then $(L, N_1 * W, F, N_1, \dots, N_n)$ and $(L, M(N_1, W), F, N_i, i \neq 1)$ are IMHS.

*Hint to the proof.* By (§2.3.2, Remark 2.9), there exists on $(L, N_1, W)$ an adequate filtration $F$ such that $(L, N_1, W, F)$ is an IMHS, hence $N_1 * W$ is well defined.

Let $M := M(N_1, W)$. By the nilpotent orbit structure on $Gr_k^W L$, the HS defined $F(t) := \exp(i \sum_j t_j N_j)$ for $t_j \gg 0$ is polarized, hence $Gr_k^M Gr_k^W L$ is polarized and the polarization is independent of $t$. Then by (Equation 9.1), $Gr_l^M L$ is polarized. From which we deduce that $(L, M(N_1, W), F, N_i, i \neq 1)$ is a mixed nilpotent orbit. By (Equation 9.10) applied to $N_1$, Coker $N_1$ is a
nilpotent orbit, as well $N_1Gr_{k+1}^W L$ by the descent lemma (ch.5, lemma 5.1), hence the direct sum $Gr_k^{N_1*W} L$ is also a nilpotent orbit.

**Corollary 9.1.** *The decomposition of $Gr_k^{N_i*W} L$ (Equation 9.8) is in the abelian category of IMHS.*

The commutativity of the star transformation ([Ka 86], Proposition 5.5.5):

$$N_1 * N_2 * W = N_2 * N_1 * W \qquad (9.11)$$

is deduced by Kashiwara from the relation ([Ka 86], Theorem 5.5.1)

$$N_2 * M(N_1, W) = M(N_1, N_2 * W)$$

### 9.2.1.3 The filtration $W^J$ associated to an IMHS

Let $(L, W^0, F, N_i, i \in M)$ denote an IMHS and let $J \subset M$. By convention, we set for $J = \emptyset \in M$, $W^0 := W^0$ on $L$.

Since by induction, $(L, W^J, F, N_i, i \in M)$ is an IMHS and by the commutativity of the star transformation, we define by induction

$$W^J := N_{i_1} * (\dots (N_{i_j} * W) \dots) \text{ for } J = \{i_1, \dots, i_j\} \qquad (9.12)$$

$(W^J)$ is denoted by $\Psi_J W$ in [Ka 86], 5.8.2).

The increasing filtration $W^J$ on $L$ does not depend on the order of composition of the respective transformations $N_{i_k} *$ since by (Equation 9.11) for all $i_p, i_q \in J: N_{i_p} * (N_{i_q} * W) = N_{i_q} * (N_{i_p} * W)$. The relative weight filtration with respect to $W$ is denoted by

$$M(J, W) := M(N, W) \text{ for } N \in C(J) := \{\Sigma_{j \in J} t_j N_j, t_j > 0\}, \qquad (9.13)$$

Let $i \in J \subset M$. We have: $M(N_i, W^J \setminus i) = M(N_i, W^J)$ by ([Ka 86], Lemma 3.4.2) and in general, the following equalities:

$$\forall J, K \subset M : M(J \cup K, W) = M(J, M(K, W)) \qquad (9.14)$$

are proved in ([Ka 86], Proposition 5.2.5). The star transformation satisfy the relations:

$$\begin{align}
\forall J \subset K \subset M : M(K, W^J) &= M(K, W) \tag{9.15} \\
\forall J_1, J_2 \subset M : M(J_1, W^{J_2}) &= M(J_1, W)^{J_2} \nonumber
\end{align}$$

([Ka 86], Equations 5.8.6 and 5.8.5). In particular, let $N_J = \sum_{i \in J} N_i$

$$\forall i \in J, \quad M(N_J, W^J) = M(N_J, W) = M(N_i, M(N_J \setminus i, W)) \qquad (9.16)$$
### 9.2.1.4 The weight W on $\Omega^*L$

The filtrations $W^J$ fit together to define the weight filtration $W$ on $\Omega^*L$:

**Definition 9.1 (Weigt W on $\Omega^*L$).** i) Let $(L, W, F, N_i, i \in M)$ be an IMHS. On the de Rham complex $\Omega^*L$ (ch.3, Definition 3.3), the filtrations $W$ and $F$ are defined by the families $W_{k-|J|}^J L$ and $F^{k-|J|} L$ for $J \subset M$:

$$W_k \Omega^* L := s(W_{k-|J|}^J L, N_i, i \in M)_{J \subset M}, \quad F^p \Omega^* L := s(F^{k-|J|} L, N_i, i \in M)_{J \subset M}$$

ii) The weight and Hodge filtrations on $IC^*L$ (ch.3, Definition 3.4) are induced by the embedding $IC^*L \hookrightarrow \Omega^*L$

where $W^\emptyset = W^0$. For example, in dimension 2:

$$W_k \Omega^* L := (W_k^0 L \xrightarrow{N_1, N_2} W_{k-1}^1 L \oplus W_{k-1}^2 L \xrightarrow{N_1, -N_2} W_{k-2}^{1,2} L.$$

*Remark 9.3.* The induced morphism $N_i : W_k^J \to (N_i * W^J)_{k-1}$ drops the degree of the filtration $W^J$. For further use, it is important to add to the above data, the canonical inclusion $I : (N_i * W^J)_{k-1} \to W_k^J$.

Beware that the weight filtration $W^0$ on $L$ underlying the variation of MHS, extends as a constant filtration $W^0$ on $\Omega^*L$ but it is different from the weight $W$. Later we shall see that the local definition of $W$ on $\Omega^*L$ extends to a global weight filtration on the de Rham complex.

## 9.2.2 Decomposition of $Gr_W^* \Omega^*L$

The following proposition is the basic local result in the construction of the structure of MHC on the logarithmic de Rham complex

**Proposition 9.2 (local decomposition).** *The graded complex with respect to the filtration $W$ on $\Omega^*L$ (Definition 9.1) satisfy the decomposition property*

$$(\mathrm{Gr}_k^W \Omega^* L, F) \xrightarrow{\sim} \bigoplus_{K \subset M} (IC^* P_{k-|K|}^K L, F)[|-K|](-|K|) \quad (9.17)$$

into a direct sum of intersection complexes (ch.3, Definition 3.4) of nilpotent orbit $(P_{k-|K|}^K L, F, N_i, i \in M \setminus K)$ of weight $k - |K|$ for all $K \subset M$.

The component $(IC^*\mathrm{Gr}_k^{W^0} L, F)$ figures in the direct sum for $K = \emptyset$.

Remark that $P_{k-|K|}^K L, F[|-K|](-|K|)$ is of weight $k$. The decomposition follows from the combinatorial description ([Ka 86] §2 and ch.3, §3.2.6) of perverse sheaves satisfying the relation of distinguished pairs (Equation 9.8). Such decomposition is described by the following de Rham family attached to the IMHS $L$

$$DR(L) := \{L_J, W^J, F_J, I_J^K : L_J \to L_K, N_{J \setminus K} : L_K \to L_J\}_{K \subset J \subset M} \quad (9.18)$$
where: $L_J = L$, $F_J = F$ and $W^J$ is the filtration defined by (Equation 9.12).

The morphisms $I_J^K := \text{Id} : L \to L$, $N_{J \setminus K} := \prod_{i \in J \setminus K} N_i : L \to L$. satisfy compatibility relations ([Ka 86], (5.6.1) to (5.6.7)) and induce the graded morphisms:

$\text{Gr}_{N_{J \setminus K}} : \text{Gr}_a^{W^K} L_K \to \text{Gr}_{a-|J \setminus K|}^{W^J} L_J$, and $\text{Gr}_J^{W^K} : \text{Gr}_{a-|J \setminus K|}^{W^J} L_J \to \text{Gr}_a^{W^K} L_K$

For each $J \subset M$: $(L_J, W^J, F_J, N_J, j \in M)$ is an IMHS structure on $L$.

Hence we have a decomposition into a graded distinguished pair for each set of subsets $K \subset J \subset M$:

$$ \text{Gr}_{a-|J \setminus K|}^{W^J} L_J \simeq \text{Im} \text{Gr}_{N_{J \setminus K}} \oplus \text{Ker} \text{Gr}_J^{W^K}. \quad (9.19) $$

which follows from ([Ka 86], Corollary 5.5.4) and the decomposition in the case $|J \setminus K| = 1$ (Equation 9.8).

Hence, the distinguished pairs criteria apply to the graded de Rham family

$$ \text{Gr}_a^W \text{DR}(L) := \{\text{Gr}_{a-|J|}^W L_J, F_J, \text{Gr}_J^I K, \text{Gr}_{N_{J \setminus K}}\}_{K \subset J \subset M} \quad (9.20) $$

**Lemma 9.2 ([Ka 86], Lemma 5.6.2).** Let $\mathbf{L}$ and $\mathcal{L}_X$ be as in theorem 11.1. For each $J \subset M$, $x \in Y_J$, and $L := \mathcal{L}_X(x)$, we define:

$$ P_k^J L := \cap_{K \subset J, K \neq J} \text{Ker}(\text{Gr}_J^K : \text{Gr}_k^{W^J} L \to \text{Gr}_{k+|J \setminus K|}^{W^K} L) \subset \text{Gr}_k^{W^J} L \quad (9.21) $$

i) For $x \in Y_J^*$ and $N_J = \sum_{j \in J} N_j: P_k^J L$ is a polarized HS of weight $k$, direct summand of $\text{Gr}_k^{M(N_J, W^J)} \text{Gr}_k^{W^J} L$.

ii) For $x \in Y_M^*$ and $L := \mathcal{L}_X(x)$ (hence $x \in Y_J$ for all $J \subset M, J \neq M$):

$$ \text{Gr}_{k-|M|}^{W^M} L \xrightarrow{\sim} \bigoplus_{J \subset M} N_{M \setminus J} P_{k-|J|}^{J}(L) \quad (9.22) $$

where $P_k^0 L = \text{Gr}_k^{W^0} L$ and the direct sum contains the component $N_M \text{Gr}_k^{W^0} L$.

iii) There exist a filtered isomorphism

$$ (IC_k^* L, F) \simeq \bigoplus_{J \subset M} (IC^* P_{k-|J|}^J L, F)[-|J|] \quad (9.23) $$

*Proof.* i) For $i \in J$ and $K = J \setminus i$, the endomorphism $N_i : \text{Gr}_k^{W^J} L \to \text{Gr}_k^{W^J} L$ is equal to the composition morphism $\text{Gr}_k^{W^J} L \xrightarrow{I_{J,K}} \text{Gr}_{k+1}^{W^K} L \xrightarrow{N_i} \text{Gr}_k^{W^J} L$ (Equation 9.7), hence $N_i$ vanish on $P_k^J L \subset \text{Ker} I_{J,K}$ for all $i \in J$.

Since by the property of distinguished pair $P_k^J L$ is a direct summand of $\text{Gr}_k^{W^J} L$ ([Ka 86] Lemma 5.6.2 and Proposition 3.3.1), $\text{Gr}_0^{W(N_J)} P_k^J L$ is a direct summand of $\text{Gr}_k^{M(N_J, W^J)} \text{Gr}_k^{W^J} L = \text{Gr}_0^{W(N_J)} \text{Gr}_k^{W^J} L$.

Since $N_J$ vanish on $P_k^J L$, $P_k^J = \text{Gr}_0^{W(N_J)} P_k^J$ is a direct summand of $\text{Gr}_k^{M(N_J)} \text{Gr}_k^{W^J} L$ of weight $k$.
ii) The statement is proved by induction on length $|M| = n$ of $M$. For $n = 1$ the decomposition is reduced to the property of graded distinguished pair. Let $M' := \{1, \dots, n-1\}$ and $M = M' \cup \{n\}$.

Let $W^n = N_n * W$, we decompose the data $DR(L)$ into two data where $(W^n)^J = W^{J \cup n}$ for all $J \subset M'$:

$$DR(L, W, M') = (L_J, W^J, F_J, I_J^K : L_J \to L_K, N_{J \setminus K} : L_K \to L_J) \text{ and}$$

$$DR(L, W^n, M') = (L_J, W^{J \cup n}, F^{J \cup n}, I_{J \cup n}^K, N_{J \setminus K}) \text{ for } K \subset J \subset M'$$

By induction, the decomposition is valid for $M'$ on the below

$$ \begin{align} & \mathrm{Gr}_{k-|M'|}^{W^{M'}} L \xrightarrow{\sim} \bigoplus_{J \subset M'} N_{M' \setminus J} P_{k-|J|}^{J}(L, W) \nonumber \\ & I_{M}^{M'} \uparrow \downarrow N_{n} \qquad I_{J \cup n}^{J} \uparrow \downarrow N_{n} \tag{9.24} \\ & \mathrm{Gr}_{k-|M|}^{W^{M}} L \xrightarrow{\sim} \bigoplus_{J \subset M'} N_{M' \setminus J} P_{k-1-|J|}^{J}(L, W^{n}) \nonumber \end{align} $$

$$ \text{Moreover } \mathrm{Gr}_{k-|M|}^{W^{M}}(L, W) = \mathrm{Gr}_{k-1-|M'|}^{(W^n)^{M'}}(L, W^n). $$

The last column satisfy the property of distinguished pairs (Corollary 9.1) for each $J \subset M'$:

$$ \begin{align} & P_{k-1-|J|}^{J}(L, W^n) \simeq N_n P_{k-|J|}^{J}(L, W) \oplus \mathrm{Ker} \, I_{J \cup n}^{J} \tag{9.25} \\ & P_{k-|J \cup n|}^{J}(L, W) = \mathrm{Ker} \, (I_{J \cup n}^{J} : P_{k-1-|J|}^{J}(L, W^n) \to P_{k-|J|}^{J}(L, W)) \nonumber \end{align} $$

We deduce from the second line of the diagram the sum over all subsets of $M$, that is $J \subset M'$ and $J \cup n, n \notin J$:

$$ \begin{align} \mathrm{Gr}_{k-|M|}^{W^{M}} L &\simeq \bigoplus_{J \subset M'} (N_n N_{M' \setminus J} P_{k-|J|}^{J}(L, W) \oplus N_{M' \setminus J} P_{k-|J \cup n|}^{J \cup n}(L, W)) \nonumber \\ &= \bigoplus_{J \subset M} N_{M \setminus J} P_{k-|J|}^{J}(L, W) \tag{9.26} \end{align} $$

The proposition 9.2 follows from (Lemma 9.2 ii) and the definition of $IC^*L$ by identification of the terms of the complexes in the equation 9.17.

iii) The isomorphism is an interpretation of (Equation 9.22). Moreover it is compatible with $F$ since we have an isomorphism in the category of MHS

$$ \bigoplus_{J \subset M} (N_{M \setminus J} P_{k-|J|}^{J} L, W, F) \xrightarrow{\sim} (\mathrm{Gr}_{k-|M|}^{W^{M}} L, W, F) \tag{9.27} $$

The decomposition of the filtrations $W$ and $F$ follows (Equation 9.22) by (Remark 3.6vii).

Remark 9.4. i) The decomposition is related to the decomposition in (§5.1.2, Remark 5.9) of $\mathrm{Gr}_*^{W(N)} \Psi^*IL$.

ii) For each $i \in J$, let $N_{0,i} := \sum_{j \in J, j \neq i} N_j$, then: $M_k(N_0) = M_k(N_i, M(N_{0,i}))$ and Coker $(N_i : W_k \mathrm{Gr}_{k+2}^M L \to W_k \mathrm{Gr}_k^M L)$ in (Equation 9.8) may be written as $W_k^{J \setminus i} \mathrm{Prim}_k^{M(N_i, M(N_{0,i}))} \subset \mathrm{Gr}_k^{M(N_0)} L$, then: $P_k^J := \cap_{\mathrm{Gr}_k^{M(N_0)} L} W_k^{J \setminus i} \mathrm{Prim}_k^{M(N_i, M(N_{0,i}))} \subset \mathrm{Gr}_k^{M(N_0)} L$.
See also (chapter 11, Equation 11.22) for a different computation of $P_k^J$.

iii) The equation 9.25 leads to the inductive definition:

$$P_i^{J \cup n}(L, W) = \text{Ker}(I_{J \cup n}^J : P_i^J(L, W^n) \to P_{i+1}^J(L, W)).$$

### 9.2.3 Weight filtration on $IC^*L$

We denote by $W$ and $F$ the induced filtrations on $IC^*L \subset \Omega^*L$.

**Proposition 9.3.** Let $(L, W^0, F, N_i, i \in M)$ be an IMHS, then:

$$Gr_k^W (IC^*L, F) \xrightarrow{\sim} (IC^*(Gr_k^{W^0} L, F). \qquad (9.28)$$

*Proof.* We prove the isomorphism: $(Gr_{k-|J|}^{W^J} N_J L, F) \simeq (N_J Gr_k^{W^0} L, F)$ for all $J \subset M$. Since $(L, W^J, F, N_i, i \in M)$ is an IMHS, by induction on the length of $J$, it is enough to consider the case $J = \{j\}$ for $j \in M$.

By the remark (??), up to shift in indices, we may suppose the morphism: $N_j : (L, W, F, N_i, i \in M) \to (L, W^j, N_i, i \in M)$ a morphism of IMHS, then by (Proposition 2.7), the morphism $N_j$ is strict with respect to $W$ and $W^j$.

## 9.3 Global definition and properties of the weight $W$

Let $L$ be a *graded* - *polarized* admissible VMHS locally unipotent along $Y$.

The complex of sheaves $\Omega^*L$ is perverse with constructible cohomology sheaves with respect to the stratification defined by the NCD $Y$.

Recall that $Y := \cup_{i \in I} Y_i$ is the union of smooth irreducible components with index $I$, $Y_J := \cap_{i \in J} Y_i$ for $J \subset I$ and $Y_J^* := Y_J - \cup_{i \notin J} (Y_i \cap Y_J)$ ($Y_\emptyset^* := X^* = X \setminus Y$). We denote uniformly the embeddings of the subsets $Y_J^*$ in $X$ by $j : Y_J^* \to X$. On the big stratum $U := X \setminus Y$, the complex $\Omega^*L$ is a resolution of its cohomology $L$.

The weight filtration $W$ on $\Omega^*L$ by analytic sub-complexes of $\Omega^*L$, is defined locally in terms of the tilde embedding.

Let $M \subset I$, the filtration $W$ is determined in a neighborhood of $y$ in terms of the IMHS $(L = \mathcal{L}_X(y), W, F, N_i)$ at $y \in Y_M^*$, by a set of coordinates $y_i$ at $y$ including local equations $y_i$ for $i \in M$ of $Y$ at $y$ and the filtrations $W^jL$ of $L$ for $J \subset M$ (Equation 9.12), as follows:

**Definition 9.2 (Weight $W$ on $\Omega^*L$).** The term $W_r(\Omega_X^*(\text{Log } Y) \otimes \mathcal{L}_X)_y$ of the filtration at $y \in Y_M^*$, is the $\mathcal{O}_{X,y}$-sub-module generated by the germs of the sections $\tilde{v} \otimes \wedge_{j \in J} \frac{dy_j}{y_j} \wedge \Omega_{X,y}^* \in \Omega^*L$ where $\tilde{v} \in \mathcal{L}_{X,y}$ is the tilde embedding of $v \in W_{r-|J|}^J L$ for varying $J \subset M$ (ch.3, Lemma 3.1).

In particular, for $M = \emptyset$, at a point $y \in U$, $W_r$ is the $\Omega_{X,y}^*$-sub-module generated by the germs of the sections $v \in W_r \mathcal{L}_y$.
The definition of W is independent of the choice of coordinates on a neighborhood U(y), since if we change the coordinate of index i into $y'_i = fy_i$ where f is holomorphic and invertible at y, the difference $\frac{dy'_i}{y'_i} - \frac{dy_i}{y_i} = \frac{df}{f}$ is holomorphic at y. Let $\tilde{\alpha} \in W_{r-|J|}^{J} \mathcal{L}_{X,y}$, the difference of the sections $\tilde{\alpha} \otimes \wedge_{j \in J} \frac{dy'_j}{y'_j} - \tilde{\alpha} \otimes \wedge_{j \in J} \frac{dy_j}{y_j}$ is still a section of the $\Omega_{X,y}^*$-sub-module generated by the germs of the sections $W_{r-|J|}^{J} \mathcal{L}_{X,y} \otimes \wedge_{j \in (J-i)} \frac{dy_j}{y_j}$. We remark that the sections defined at y restrict to sections defined on $(U(y) \setminus Y_M^* \cap U(y))$.

### 9.3.0.1 Purity of the intersection complex $IC^*\mathcal{L}$

Recall the definition of $IC^*\mathcal{L} \subset \Omega^*\mathcal{L}$ (ch.3, Definition 3.6) and set:

$$ (IC^* \mathcal{P}, W, F) := (IC^* \mathcal{L}, W, F)[\dim X], \quad (\Omega^* \mathcal{P}, W, F) := (\Omega^* \mathcal{L}, W, F)[\dim X] $$
such that: $IC^* \mathcal{P} \simeq j_* \mathcal{P}$ with induced filtration $W$ and $\Omega^* \mathcal{P} \simeq j_* \mathcal{P}$.

In the case of a pure polarized VHS, we recall (Purity Theorem 3.1):

**Proposition 9.4.** Let $(\mathcal{P}, F)$ be a polarized VHS of weight $p$, then the sub-complex $(IC^* \mathcal{P}, F)$ of the logarithmic complex with induced filtration $F$ is a Hodge complex which defines a pure HS of weight $a+i$ on its hypercohomology $\mathbb{H}^*(X, IC^* \mathcal{P})$, equal to the Intersection cohomology $\mathrm{IH}^*(X, \mathcal{P})$.

### 9.3.0.2 Decomposition into intersections complexes of $P_k^J \mathcal{L}$

The local study (§9.2.2) ended with the local decomposition of the graded weight filtration into a direct sum of Intersection complexes (Equation 9.17). We develop now the corresponding global result.

By definition of the weight $W$ on $(\Omega^*\mathcal{L}, F)$, there exists at each $x \in Y$, an induced embedding of

$$ (\mathrm{Gr}_k^W IC^* L, F) \hookrightarrow (\mathrm{Gr}_k^W \Omega^* \mathcal{L}, F)_x, \quad L := \mathcal{L}(x). \quad (9.29) $$

**Definition 9.3.** $(IC_{Y_K}^* P_{k-|K|}^K \mathcal{L} \subset \mathrm{Gr}_k^W \Omega^* \mathcal{L}[|K|])$. There exists a unique complex $IC_{Y_K}^* P_{k-|K|}^K \mathcal{L}$, defined as a sub-complex of $\mathrm{Gr}_k^W \Omega^* \mathcal{L}[|K|]$ such that at $x \in Y_M^*$, $L := \mathcal{L}(x)$ and $\forall K \subset M$, the consecutive embeddings by (Equations 9.17 and 9.29):

$$ (IC^* P_{k-|K|}^K L, F)(-|K|)(-|K|) \subset (\mathrm{Gr}_k^W \Omega^* L, F) \subset (\mathrm{Gr}_k^W \Omega^* \mathcal{L}[|K|])_x $$

induce a quasi-isomorphism:

$$ IC^* P_{k-|K|}^K L) \xrightarrow{\sim} (IC_{Y_K}^* P_{k-|K|}^K \mathcal{L})_x \subset (\mathrm{Gr}_k^W \Omega^* \mathcal{L}[|K|])_x, \quad L := \mathcal{L}(x) \quad (9.30) $$

The definition and the corollary below follow from the local decomposition by (Equation 9.17).
**Corollary 9.2.** i) There exists a polarized VHS: $P_a^K \mathbf{L}$ of weight a on $Y_K^*$ for $K \subset M$, such that $(IC_{Y_K}^* P_a^K \mathcal{L}, F)$ is a resolution of the intermediate extension by $j_K: Y_K^* \to X$ of $P_a^K \mathbf{L}$:

$$(IC_{Y_K}^* P_a^K \mathcal{L}, F) \simeq (j_{K!*} P_a^K \mathbf{L}, F).$$

ii) For $x \in Y_K \setminus Y_K^*$ and $x \in Y_M^*$, $(P_k^K L, F, N_i, i \in M \setminus K)$ is a nilpotent orbit of weight k.

**Proposition 9.5.** i) The graded perverse sheaves for the weight filtration, satisfy the decomposition property into intermediate extensions for all k

$$(Gr_k^W \Omega^* \mathcal{L}, F) \xrightarrow{\sim} \oplus_{J \subset I} (IC_{Y_J}^* P_{k-|J|}^J \mathcal{L}[-|J|], F)(-|J|). \quad (9.31)$$

where $IC_{Y_J}^* P_{k-|J|}^J \mathcal{L} \simeq j_{J!*} P_{k-|J|}^J \mathbf{L}$.

The shifted intersection complex $j_{!*}Gr_k^{W^0} \mathbf{L}$ figures in the sum for $J = \emptyset$.

ii) The weight W is a filtration by sub-complexes of $j_* \mathbf{L}$ consisting of perverse sheaves defined over $\mathbb{Q}$.

i) The complex $IC_{Y_J}^* P_{k-|J|}^J(\mathcal{L})$ is a shifted intermediate extension on $Y_J$. We remark that $P_{k-|J|}^J(\mathbf{L})[-|J|]$ on $Y_J^*$ is a VHS of weight $k-2|J|$, its twist by $(-|J|)$ is a VHS of weight k and corresponds to $\wedge_{j \in J} \frac{dy_j}{y_j}$ in the definition of the weight on $\Omega^* \mathcal{L} := \Omega_X^*(\mathrm{Log} Y) \otimes \mathcal{L}_X$.

The decomposition in the category of derived filtered complexes (Equation 9.31) is based on the decomposition of filtered vector spaces (Equation 9.27).

ii) The various graded complexes $Gr_k^W(\Omega_X^*(\mathrm{Log} Y) \otimes \mathcal{L}_X)$ are direct sum of Intersections complexes defined over $\mathbb{Q}$. The filtration $W_k$ on the de Rham complex is defined over $\mathbb{Q}$ and perverse with respect to the stratification defined by $Y_J^*$, since the spaces $W_k \Omega^* L$ and $P_k^J(L)$ are defined over $\mathbb{Q}$. When $x$ vary on $Y_J^*$, $P_k^J L$ defines a VHS and $P_k^J \mathbf{L}$ is a nilpotent orbit at $x \in Y_J \setminus Y_J^*$.

### 9.3.0.3 Restriction to $IC^* \mathcal{P} \mathcal{L}$

The Intersection complex $(IC^* \mathcal{P} \mathcal{L}, W, F)$ of an admissible VMHS is a mixed Hodge complex, embedded as a sub-complex of $(\Omega^* \mathcal{P} \mathcal{L}, W, F)$ with induced filtrations.

**Proposition 9.6.** Let $\mathcal{L}, W, F$ be an admissible VMHS. The induced filtrations on the complex $(IC^* \mathcal{P} \mathcal{L}, W, F) \subset (\Omega^* \mathcal{P} \mathcal{L}, W, F)$ define a structure of a mixed Hodge complex such that for all k:

$$(IC^*(Gr_k^{W^0} \mathcal{P} \mathcal{L}), F) \simeq (Gr_k^W IC^* \mathcal{P} \mathcal{L}, F)$$

The proposition follows from Proposition 9.3 in the local case.
In general the Intersection complex of an extension of two local systems, is not the extension of the intersection complex of the local systems.

In the case of a pure VHS $p\mathcal{L}$, the weights of $\Omega^* p\mathcal{L}$ start with the subcomplex $\mathcal{IC}^* p\mathcal{L}$. The next result on weights satisfy a property similar to ([De 80], cor 3.3.5) in the case of varieties of positive characteristic.

**Corollary 9.3.** *The de Rham logarithmic mixed Hodge complex $(\Omega^* p\mathcal{L}, W, F)$ of a variation of HS of weight $\omega \ge p\omega$ induces on the hypercohomology $\mathrm{H}^i(X \setminus Y, p\mathcal{L})$ a MHS of weight $\omega \ge p\omega + i$.*

Indeed, $W_k = 0$ on the logarithmic complex for $k \le p\omega$.

## 9.4 The logarithmic complex $\mathcal{IC}^*\mathcal{L}(\mathrm{Log} Z)$

Let $Z = \cup_{i \in I_Z} I_{Y_i} \subset Y \subset X$ be a sub-divisor of the NCD $Y$ union of components of $Y$ with index in a subset $I_Z$ of $I$, $j_Z := (X \setminus Z) \to X$ and $\mathcal{L}$ a VMHS on $X \setminus Y$. The induced filtrations $W$ and $F$ define a structure of mixed Hodge complexe (MHC) on the logarithmic sub-complex $\mathcal{IC}^*\mathcal{L}(\mathrm{Log} Z) \subset \Omega^*\mathcal{L}$ isomorphic to $j_{Z*}j_{Z}^*j_{*\mathcal{L}} \in D_c^b(X, \mathbb{C})$ (ch.3, §3.2).

### 9.4.0.1 Local decomposition of $Gr_i^W(\mathcal{IC}^*\mathcal{L}(\mathrm{Log} Z))$

**Lemma 9.3.** Let $P_k^J(L)$ be defined by Equation 9.21 for all $J \subset M_Z := M \cap I_Z$, and $\mathcal{IC}^*(P_k^J L)$ the Intersection complex defined by the nilpotent orbit $(P_k^J(L), N_i, i \in M \setminus J)$, then:

$$ (\mathrm{Gr}_k^W \mathcal{IC}^* L(\mathrm{Log} Z), F) \xrightarrow{\sim} \bigoplus_{J \subset M_Z} \mathcal{IC}^* P_{k-|J|}^J L, F, i \in M) [-|J|](-|J|) \quad (9.32) $$

*Proof.* The statement is similar to Proposition 9.2 with $J \subset M_Z$ instead of $J \subset M$. The proof is similar to the decomposition case of $\Omega^*L$ in (§9.2.2).

In terms of (ch. 3, §3.2, Remark 3.4), we introduce for $J' \subset M'_Z := M \setminus M_Z$ the IMHS $(N_{J'}L, W^{J'}, F, N_i, i \in M_Z)$ and the following data $\mathrm{DR}(N_{J'}L)$:

$$ \{(N_{J'}L)_J, W^{J' \cup J}, F^{J' \cup J}, I_J^K : N_{J'}L \to N_{J'}L, N_{J' \setminus K} : N_{J'}L \to N_{J'}L\}_{K \subset J \subset M_Z} $$

with indices $J \subset M_Z$, constant term $(N_{J'}L)_J = N_{J'}L$, and induced filtrations $W^{J' \cup J}, F^{J' \cup J} = F$ on $N_{J'}L$.

For all $K \subset J \subset M_Z$, $N_{J \setminus K}$ is induced by $\prod_{i \in J \setminus K} N_i$ (resp. $I_J^K$ by $\mathrm{Id} : L \to L$). The property of distinguished pairs is satisfied, in particular for consecutive terms $N_i : N_K L \to N_J L$ for $J = K \cup i$. Set for each $J \subset M_Z$

$$ P_k^J(N_{J'}L, W^{J'}) := \cap_{K \subset J, K \neq J} \mathrm{Ker} (\mathrm{Gr}_{I_{J,K}} : \mathrm{Gr}_k^{W^{J' \cup J'}} N_{J'}L \to \mathrm{Gr}_{k+|J \setminus K|}^{W^{K \cup J'}} N_{J'}L). $$
As in (§9.2.2), we deduce the decomposition:

$$Gr_{k-|J'|}^{W} \Omega^{*}(N_{J'}L, N_{i})_{i \in M_{Z}} = \oplus_{J' \subset M_{Z}} IC^{*}(P_{k-|J \cup J'|}^{J}(N_{J'}L, W^{J'}), i \in M_{Z})[-|J|]$$

where $IC^*$ is limited to the variables $i \in M_Z$.

For each $J' \subset M'_Z$, we have a surjective induced morphism $N_{J'}$

$$P_{k-|J|}^{J}(L) \xrightarrow{N_{J'}} P_{k-|J \cup J'|}^{J}(N_{J'}L, W^{J'}) \simeq N_{J'} P_{k-|J|}^{J}(L)$$

and a surjective morphism $Gr_k^W \Omega^*(L, N_i)_{i \in M_Z} \xrightarrow{N_{J'}} Gr_{k-|J'|}^W \Omega^*(N_{J'}L, N_i)_{i \in M_Z}$ which splits into a direct sum of surjective morphisms

$$\oplus_{J \subset M_Z} (IC^*(P_{k-|J|}^J L, i \in M_Z) \xrightarrow{N_{J'}} \oplus_{J \subset M_Z} IC^*(N_{J'} P_{k-|J|}^J L, i \in M_Z))[-|J|]$$

By summing the double complex over $J \subset M_Z$ and $J' \subset M'_Z$, we find the statement of the lemma.

### 9.4.0.2 Structure of MHC on $IC^*\mathcal{L}(\mathrm{Log} Z) \subset \Omega^*\mathcal{L}$

We remark that (Equation 9.32) is similar to (Equation 9.23) except that the sum is over $J \subset M_Z$ instead of $J \subset M$, hence (Definition 9.3) and (Corollary 9.2) apply. We deduce from (Lemma 9.3):

**Theorem 9.3.** Let $Z := \cup_{i \in I_Z} Y_i$. With the notations of (Definition 9.3) and (Corollary 9.2), the bifiltered complex $(IC^*\mathcal{L}(\mathrm{Log} Z), W, F) \subset (\Omega^*\mathcal{L}, W, F)$, is a MHC on $X$.

The perverse weight graded sheaves satisfy the decomposition property into intermediate extensions for all $k$

$$\begin{gathered} (Gr_k^W IC^* \mathcal{L}(\mathrm{Log} Z), F) \xrightarrow{\sim} \oplus_{J \subset I_Z} (IC_{Z_J}^* P_{k-|J|}^J \mathcal{L}[-|J|], F)(-|J|) \tag{9.33} \\ j_{Z_*} P_{k-|J|}^J (\mathcal{L}) \simeq IC_{Z_J}^* P_{k-|J|}^J \mathcal{L} \end{gathered}$$

where $j_Z : Z_J^* \to Z_J$ is the inclusion for each $J \subset I_Z$. Moreover, $j_* \mathrm{Gr}_k^{W_0} \mathcal{L}$ for $Z = \emptyset$, otherwise the summands are supported by $Z_J \subset Z$ for $J \neq \emptyset$.

The theorem follows from (Lemma 9.3). We remark that $P_{k-|J|}^J (\mathcal{L})[-|J|]$ for $J \subset I_Z$ are perverse, as shifted local systems on $Z_J^*$ by $[\dim X - |J|] = \dim Z_J^*$.

**Definition 9.4.** The MHS on $\mathbb{H}^*(X \setminus Z, j_* \mathcal{L})$ is defined by the bifiltered complex $(IC^* \mathcal{L}(\mathrm{Log} Z), W, F)$.

**Remark 9.5.** i) For any NCD $Z$ such that $Z \cup Y$ is still a NCD, we may always suppose that $\mathcal{L}$ is a variation of MHS on $X - (Y \cup Z)$ (by enlarging $Y$), then $Z$ is a union of components of $Y \cup Z$.

ii) For any subset $Z \subset X$, the MHS on $\mathbb{H}^*(X \setminus Z, j_* \mathcal{L})$ is defined by blowing up $Z$ in a non singular $X$ into a NCD without modifications above $X \setminus Z$.
#### 9.4.0.3 Transversality

We denote by $Z' := \cup_{i \notin I_Z} I_i Y_i \subset Y$ the sub-NCD of $Y$ complement to $Z$, such that $Y = Z \cup Z'$. We say that $Z'$ is transversal to $Z$. The equation 9.33 is interpreted as follows:

**Lemma 9.4.** Let $X \setminus Y \xrightarrow{j''} X \setminus Z' \xrightarrow{j_{Z'}} X$ and $X \setminus Y \xrightarrow{j'} X \setminus Z \xrightarrow{j_Z} X$ such that $j_Z \circ j' = j = j_{Z'} \circ j''$. We have the commutation isomorphism:

$$j_{Z*}j'_{!*}{}^p\mathbf{L} \simeq j_{Z'!*}j''_{*}{}^p\mathbf{L}.$$

The proof is local at each point of the intersection $Z \cap Z'$, by induction on the strata of the stratification defined by $Z$ and it is reduced to the case of one point in the intersection.

In the case of the remark (ch. 3, Remark 3.4) where $|M| = n$ and a point $0 \in D^n \subset \mathbb{C}^n$, we check that $(j_{Z'!*}j''_{*}{}^p\mathbf{L})_0$ is obtained by truncation in degree $\le -1$ of $(j_{Z*}j'_{!*}{}^p\mathbf{L})_0$. However, by the same remark the complex $(j_{Z*}j'_{!*}{}^p\mathbf{L})_0$ has no cohomology in degrees $\ge 0$ (after the shift in degrees on ${}^p\mathbf{L}$).

## 9.5 Hodge theory on perverse cohomology along a divisor

For basic results on perverse heaves along a divisor we refer to (ch. 3, §3.2.4), and to (ch. 2, §2.4) for induced filtrations on perverse hypercohomology. In the case of a NCD $Z \subset Y \subset X$, let $i: Z \to X$. We deduce from the bifiltered complex $(IC^*\mathcal{L}(\mathrm{Log} Z), W, F)$ various complexes (called logarithmic) with weight and Hodge filtrations, realizing the following classes of complexes in the derived category:

$$i^!j_{!*}{}^p\mathbf{L}, i^*j_{!*}{}^p\mathbf{L}, j_{Z'}j_{Z}^*j_{!*}{}^p\mathbf{L} \text{ and } i^*j_{Z*}j_{Z}^*j_{!*}{}^p\mathbf{L}.$$

In particular we construct a MHS on the hypercohomology of the boundary of a tubular neighborhood of $Z$ with its central fiber $Z$ deleted (§3.2.5).

Thus, we recover most of the results of (Proposition 5.7) without introducing the monodromy action.

**Theorem 9.4 (W and F).** Let $Z \subset Y$ be a locally principal divisor NCD in $X$, and ${}^p\mathbf{L}$ a PVHS of weight $p_w$ on $X \setminus Y$ and let $D$ denote Verdier dual

1. There exist unique global filtrations $W$ and $F$ on $j_{Z*}j_{Z}^*j_{!*}{}^p\mathbf{L}$ such that

$$ \begin{aligned} W_{p_w}j_{Z*}j_{Z}^*j_{!*}{}^p\mathbf{L} &= j_{!*}{}^p\mathbf{L}, \\ W_lj_{Z*}j_{Z}^*(j_{!*}{}^p\mathbf{L}) &= W_lIC^*{}^p\mathcal{L}(\mathrm{Log} Z) \text{ if } l > p_w \text{ and } 0 \text{ if } l < p_w \end{aligned} \quad (9.34) $$
2. There exist unique global filtrations W and F on $i^*j_! p^L[-1]$ (resp. $i'_j p^L[1]$) such that

$$ 
\begin{align} 
i_* Gr_l^W (i'_j p^L[1]) &\simeq Gr_l^W j_Z j_Z^*(j_! p^L) \quad \text{if } l > p_w \text{ and } 0 \text{ if } l \le p_w, \notag \\ 
i^* j_! p^L[-1](p_w) &\simeq D(i'_j p^L[1]) : Gr_l^W (i^* j_! p^L[-1]) = 0 \quad \text{if } l \ge p_w, \notag \\ 
Gr_l^W (i^* j_! p^L[-1]) &= DGr_{2^{p_w-l}} (i'_j p^L[1]) \quad \text{if } l < p_w, \notag \\ 
W_l(i^* j_! p^L) &:= (W_{l-1}(i^* j_! p^L[-1]))[1] \tag{9.35}
\end{align} 
$$

3. Dually, $i^*j_! p^L[-1]$ embeds into $j_Z j_Z^* j_! p^L$ and the filtration W on $j_Z j_Z^* j_! p^L$ satisfy $Gr_{p_w}^W j_Z j_Z^* j_! p^L = j_! p^L$ and:

$$ Gr_l^W j_Z j_Z^* j_! p^L \simeq i_* Gr_l^W (i^* j_! p^L[-1]) \quad \text{if } l < p_w. $$

4. If X is projective, the filtrations W and F on

$$ i^* j_! p^L[-1], \quad i'_j p^L[1], \quad j_Z j_Z^* j_! p^L, \quad j_Z j_Z^* j_! p^L $$

define respectively a structure of mixed Hodge complex.

Example 9.1. i) For a pure $p^L$ of weight $p_w$, $i_* i'_j p^L$ is supported by Z, has weights $w \ge p_w$, and $i_* i^* j_! p^L$ has weights $w \le p_w$ such that: $\forall \ell \ge p_w$:

$$ D(Gr_l^W i'_j p^L) \simeq Gr_{2^{p_w-\ell}}^W i^* j_! p^L = Gr_{2^{p_w-\ell-1}}^W i^* j_! p^L[-1]. $$

The Intersection morphism I induces morphisms:

$$ Gr_i^W i_* i'_j p^L \to Gr_i^W j_! p^L \to Gr_i^W i_* i^* j_! p^L \text{ for } i = p_w \text{ and } 0 \text{ for } i \ne p_w. $$

ii) Let $p^L$ be a polarized VHS on $\mathbb{C}^*$ with a nilpotent endomorphism $N$ on $L$. The duality at 0 between $Gr_r^W i^*(\mathcal{L}(\mathrm{Log} Z)/\mathcal{L})$ and $Gr_{-r+1}^W i^*(j_! p^L)[-1]$ corresponds to the duality between $Gr_{r+1}^W \mathrm{Ker} N$ and $Gr_{-r-1}^W L/NL$.

For $p^L$ on $(\mathbb{C}^*)^n$, this duality may be interpreted as a duality of perverse sheaves for the action of $N$ on the functor of nearby cycles or in terms of a strict simplicial coverings of a NCD.

### 9.5.0.1 Weights of the MHS

As a MHC, the weights of $\mathcal{L}(\mathrm{Log} Z)$ are $\ge p_w$ by construction. The weights of $i^*j_! p^L$ are $\ge p_w$ since the weights of $\mathcal{L}(\mathrm{Log} Z)/\mathcal{L}$ are $\ge p_w + 1$.

**Corollary 9.4.** The weights $w$ satisfy the following inequalities:

$$ 
\begin{align*}
w \mathbb{H}^i(X \setminus Z, j_! p^L) &\ge p_w + i, w \mathbb{H}_c^i(X \setminus Z, j_! p^L) \le p_w + i \\
w \mathbb{H}_Z^i(X, j_! p^L) &\ge p_w + i, w \mathbb{H}^i(Z, j_! p^L) \le p_w + i \\
\mathbb{H}^i(Z, Gr_j^W i^* j_! p^L) &\simeq \mathbb{H}^{-i}(Z, (Gr_{-j}^W i'_j p^L(p_w))(-j))^*
\end{align*} 
$$
We have the following duality isomorphisms for the weights of $\mathbb{H}^i(Z, j_{1*}^p \mathbf{L})$:

$$ \mathbb{H}^i(Z, \mathrm{Gr}_j^W i^* j_{1*}^p \mathbf{L}) \simeq \mathbb{H}^i(Z, \mathrm{Gr}_j^W D_i^! j_{1*}^p \mathbf{L}(p_w)) \\ \simeq \mathbb{H}^i(Z, D(\mathrm{Gr}_{-j}^W i^! j_{1*}^p \mathbf{L}(p_w))(-j)) \simeq \mathbb{H}^{-i}(Z, (\mathrm{Gr}_{-j}^W i^! j_{1*}^p \mathbf{L}(p_w))(-j))^* $$

where $w(\mathrm{Gr}_{-j}^W i^! j_{1*}^p \mathbf{L}(p_w))(-j) = -p_w - j + 2j$, hence $w\mathbb{H}^{-i}(Z, (\mathrm{Gr}_{-j}^W i^! j_{1*}^p \mathbf{L}(p_w))(-j))^* = p_w + i - j \le p_w + i$ as it vanishes if $j < 0$.

**Lemma 9.5.** Let $\mathbf{p}\mathbf{L}$ be a shifted polarized VHS of weight $p_w$ on a Zariski open subset $X \setminus Y$ and $Z$ a closed subvariety of a projective variety $X$.

i) There exists a long exact sequence of MHS

$$ \rightarrow \mathbb{H}^i(X, j_{1*}^p \mathbf{L}) \rightarrow \mathbb{H}^i(X, j_{1*}^p \mathbf{L}) \rightarrow \mathbb{H}^i(X \setminus Z, j_{1*}^p \mathbf{L}) \rightarrow \mathbb{H}_Z^{i+1}(X, j_{1*}^p \mathbf{L}) \rightarrow \dots \tag{9.36} $$

with weights: $w(\mathbb{H}_Z^i(X, j_{1*}^p \mathbf{L})) \ge p_w + i$, $w(\mathbb{H}^i(X, j_{1*}^p \mathbf{L})) = p_w + i$, and $w(\mathbb{H}^i(X \setminus Z, j_{1*}^p \mathbf{L})) \ge p_w + i$.

ii) We have a dual exact sequence of MHS

$$ \rightarrow \mathbb{H}_c^i(X \setminus Z, j_{1*}^p \mathbf{L}) \rightarrow \mathbb{H}^i(X, j_{1*}^p \mathbf{L}) \rightarrow \mathbb{H}^i(Z, j_{1*}^p \mathbf{L}) \rightarrow \mathbb{H}_c^{i+1}(X \setminus Z, j_{1*}^p \mathbf{L}) \rightarrow \dots \tag{9.37} $$

with weights: $w(\mathbb{H}_c^i(X \setminus Z, j_{1*}^p \mathbf{L})) \le p_w + i$, $w(\mathbb{H}^i(X, j_{1*}^p \mathbf{L})) = p_w + i$, and $w(\mathbb{H}^i(Z, j_{1*}^p \mathbf{L})) \le p_w + i$.

The lemma is proved first for $X$ smooth projective, $Y$ a NCD and $Z \subset Y$ a sub-NCD. The exact sequence is deduced from the distinguished triangle: $i_*^i j_*^p \mathbf{L} \rightarrow j_*^p \mathbf{L} \rightarrow j_{Z_*}^p(\mathbf{L}|_{X \setminus Z})$ and the inequalities on the weights follow from the corollary above. In the general case, the lemma will be deduced from the case of NCD by the decomposition theorem.

### 9.5.0.2 Compatibility of Thom isomorphism with MHS

Thom isomorphism (ch. 3, §3.2.1) is compatible with weight and Hodge filtrations up to a twist.

**Lemma 9.6.** Let $Z \subset X$ such that $Y \cup Z$ is a NCD and let a smooth hypersurface $H$ intersects transversally $Y \cup Z$ such that $H \cup Y \cup Z$ is a NCD. There is an isomorphism compatible with the weight and Hodge filtrations up to a twist

$$ IC^*(i_H^*\mathcal{L})(\mathrm{Log} Z \cap H) \xrightarrow{\sim} i_H^*IC^*(\mathcal{L})(\mathrm{Log} Z \cap H) \xrightarrow{\sim} i_H^1IC^*\mathcal{L}(\mathrm{Log} Z)[2](1) \tag{9.38} $$

By transversality, we have an isomorphism of the restriction $i_H^*IC^*\mathcal{L}(\mathrm{Log} Z)$ with the logarithmic complex $IC^*(i_H^*\mathcal{L})(\mathrm{Log} Z \cap H)$ constructed directly on $H$. The connecting isomorphism in the distinguished triangle

$$ (i_H)_* i_H^1 IC^* \mathcal{L}(\mathrm{Log} Z) \rightarrow IC^* \mathcal{L}(\mathrm{Log} Z) \rightarrow IC^* \mathcal{L}(\mathrm{Log} Z \cup H) \xrightarrow{[1]} $$
defines an isomorphism of the quotient complex with the cohomology with support

$$(IC^*\mathcal{L}(\mathrm{Log} Z \cup H)/IC^*\mathcal{L}(\mathrm{Log} Z)) \xrightarrow{\sim} Ri_H^!IC^*\mathcal{L}(\mathrm{Log} Z)[1]$$

The isomorphism $i_H^!IC^*\mathcal{L}(\mathrm{Log} Z)[2](1) \xrightarrow{\sim} IC^*i_H^*\mathcal{L}(\mathrm{Log} Z \cap H)$ is induced by the residue $\mathrm{Res}_H : IC^*\mathcal{L}(\mathrm{Log} Z \cup H)[1](1) \to i_H^*IC^*(i_H^*\mathcal{L})(\mathrm{Log} Z \cap H)$ compatible with the filtrations up to a shift in degrees since it vanishes on $IC^*\mathcal{L}(\mathrm{Log} Z)$.

**Corollary 9.5.** Under the hypothesis of the lemma, let $i : H \to X$ be a general embedding of complex varieties of codimension $d$, transversal to the strata of a stratification $S$ of $X$ including $Z$ and $Y$.

Thom isomorphism $\mathbb{H}^i(H \setminus Z \cap H, j_! p_L) \simeq \mathbb{H}_{H \setminus Z \cap H}^{i+2d}(X \setminus Z, j_! p_L)(d)$, and

Gysin morphism $\mathbb{H}^i(H \setminus Z \cap H, j_! p_L) \simeq \mathbb{H}^{i+2d}(X \setminus Z, j_! p_L)(d)$

are compatible with MHS.

In general, let $i_W : W \to X$ be a closed embedding of a non singular variety of codimension $d$ transverse to $Z$, $Y$ and to the strata of an adequate stratification $S$ of $X$, then: $i_W^!j_! p_L[2d] \simeq i_W^!j_! p_L$.

The following isomorphism is compatible with MHS:

$$\mathbb{H}^i(W \setminus (Z \cap W), j_! p_L) \simeq \mathbb{H}_{W \setminus (Z \cap W)}^{i+2d}(X \setminus Z, j_! p_L).$$

### 9.5.1 Hodge theory on $i^*j_Z^*j_Z^*j_! p_L$

We have a distinguished triangle

$$j_Z^*j_Z^*j_! p_L \xrightarrow{\mathrm{can}} j_Z^*j_Z^*j_! p_L \to i_*i^*j_Z^*j_Z^*j_! p_L \xrightarrow{[1]} \quad (9.39)$$

**Definition 9.5.** The structure of MHC on $K' := i^*j_Z^*j_Z^*p_L$ is defined by the mixed cone $C_M(\mathrm{can})$ with the following weight filtration:

$$W_rC_M(\mathrm{can}) = W_{r-1}j_Z^*j_Z^*j_! p_L[1] \oplus W_rj_Z^*j_Z^*j_! p_L$$

We recover the results of (Lemma 5.3). The terms of the weight spectral sequence are $E_1^{p,q} = \mathbb{H}^{p+q}(X, \mathrm{Gr}_{-p}^W C_M(\mathrm{can}))$ where

$$\mathrm{Gr}_r^W C_M(\mathrm{can}) = \mathrm{Gr}_{r-1}^W j_Z^* j_Z^* j_! p_L[1] \oplus \mathrm{Gr}_r^W j_Z^* j_Z^* j_! p_L$$

$$E_1^{p,q} = \mathbb{H}^{p+q}(X, \mathrm{Gr}_{-p}^W i_! j_! p_L[1]) \quad \text{if } -p > p_w + 1$$

$$E_1^{p,q} = \mathbb{H}^{p+q}(X, j_! p_L[1] \oplus \mathrm{Gr}_{p_w+1}^W i_! j_! p_L[1]) \quad \text{if } -p = p_w + 1$$

$$E_1^{p,q} = \mathbb{H}^{p+q}(X, \mathrm{Gr}_{p_w-1}^W i_! j_! p_L[-1])[1] \oplus j_! p_L) \quad \text{if } -p = p_w$$

$$E_1^{p,q} = \mathbb{H}^{p+q}(X, \mathrm{Gr}_{-p-1}^W i_! j_! p_L[-1])[1]) \quad \text{if } -p < p_w.$$

The terms with value in $j_! p_L[1]$ for $-p = p_w + 1$ and $j_! p_L$ for $-p = p_w$ vanish at rank 2. In particular, for $-p = p_w + 1$ and $q = p_w, p+q = -1$ the terms:
1) $E_1^{-p_w-1, p_w} = j_{1*}^p \mathbf{L}[1] \oplus \mathrm{Gr}_{p_w+1}^W i^1 j_{1*}^p \mathbf{L}[1]$ where

$\mathrm{Gr}_{p_w+1}^W i^1 j_{1*}^p \mathbf{L}[1] = \mathrm{Gr}_{p_w-1}^W \mathrm{Coker} N | \psi_f j_{1*}^p \mathbf{L}$ gives the term:

$$E_2^{-p_w-1, p_w} = \mathbb{H}^{-1}(X, \mathrm{Gr}_{p_w-1}^W \mathrm{Coker} N | \psi_f j_{1*}^p \mathbf{L}) \implies \mathrm{Gr}_{p_w}^W \mathbb{H}^{-1}(X, i^* j_{Z*} j_Z^* j_{1*}^p \mathbf{L})$$

2) $E_1^{-p_w, p_w} = \mathrm{Gr}_{p_w}^W j_{Z*}^j j_Z^* j_{1*}^p \mathbf{L}[1] \oplus j_{1*}^p \mathbf{L}$ where

$\mathrm{Gr}_{p_w}^W j_{Z*}^j j_Z^* j_{1*}^p \mathbf{L}[1] = (\mathrm{Gr}_{p_w-1}^W i^* j_{1*}^p \mathbf{L}[-1])[1] \oplus j_{1*}^p \mathbf{L}$ gives the term:

$$E_2^{-p_w, p_w} = \mathbb{H}^1(X, \mathrm{Gr}_{p_w-1}^W \mathrm{Ker} N | \psi_f j_{1*}^p \mathbf{L}) \implies \mathrm{Gr}_{p_w}^W \mathbb{H}^0(X, i^* j_{Z*} j_Z^* j_{1*}^p \mathbf{L})$$

The duality between the terms $E_2^{p,q}$ above induces the duality at the limit. The limits vanish according to the local purity theorem (crucial case).

The terms $E_2^{p,q}$ coincide with the terms of the weight spectral sequence of $C_M(I)$ (§5.2.2) since $j_{1*}^p \mathbf{L}$ in the terms $E_1^{p,q}$ disappear at rank 2.

**Remark 9.6.** For any deformation retract neighborhood $B_Z$ of $Z$, let $B_Z^* := B_Z \setminus Z$ with $Z$ deleted in $B_Z$, the MHS on the hypercohomology of $B_Z^*$ is defined by

$$\mathbb{H}^i(B_Z^*, j_{1*}^p \mathbf{L}) \simeq \mathbb{H}^i(Z, i^* j_{Z*} ((j_{1*}^p \mathbf{L})_{|X\setminus Z}))$$

We have an exact sequence of MHS

$$\dots \to \mathbb{H}^i(X \setminus Z, j_{1*}^p \mathbf{L}) \to \mathbb{H}^i(B_Z^*, j_{1*}^p \mathbf{L}) \to \mathbb{H}^{i+1}(X, j_{1}(j_{1*}^p \mathbf{L}_{|X\setminus Z})) \to \dots$$

and a duality isomorphism

$$i^* j_{Z*} (j_{1*}^p \mathbf{L}_{|X\setminus Z}) [-1] \xrightarrow{\sim} D(i^* j_{Z*} (j_{1*}^p \mathbf{L}_{|X\setminus Z})).$$

## 9.6 The bi-filtered relative complexes $\mathcal{IC}_f^*\mathcal{L} \subset \Omega_f^*\mathcal{L}$

Let $f: X \to V$ be a smooth proper morphism of smooth complex algebraic varieties, and let $Y$ be a NCD in $X$. The relative logarithmic complex $\Omega_f^*\mathcal{L} := \Omega_{X/V}^*(\mathrm{Log}Y) \otimes \mathcal{L}_X$ is defined in ([De 70], §2.22).

**Definition 9.6.** i) A normal crossing divisor $Y$ in a non singular algebraic $X$ is said “relative horizontal” if for each point $v$ in a big stratum of $V$, the fiber $Y_v \subset X_v$ is a NCD with smooth components in the smooth fiber $X_v$ and the restriction of $f$ to $Y$ is a topological fiber bundle over the big strata.

ii) A NCD inverse image of a strict subset $W \subset V$ is called vertical NCD.

iii) The fibration is adapted to a local system $\mathcal{P}\mathbf{L}$ defined on a Zariski open algebraic subset if the restriction of the cohomology groups of the intermediate extension $j_{1*}^p \mathbf{L}$, to the various strata of $X$, are locally constant.

The fibration is adapted to a NCD $Y$ in $X$, if $Y$ is a union of strata of the underlying whitney stratification $S$ of $X$.

Let $Y$ be relative horizontal over $V$. For each point $v \in V$, the fiber $Y_v$ is a NCD in $X_v$ and the various intersections $Y_{i_1, \dots, i_p}$ of $p$- components are smooth over $V$ and $Y \to V$ is a topological fiber bundle.
Let $U := X \setminus Y$ and $j: U \to X$. The sheaf of modules $i_{X_v}^*\mathcal{L}_X$ induced by the canonical extension $\mathcal{L}_X$ on each fiber $X_v$, is isomorphic to the canonical extension $(i_{U_v}^*\mathcal{L})_{X_v}$ of the induced local system $i_{U_v}^*\mathcal{L}$.

The cohomology spaces $H^i(U_v, \mathcal{L})$ (resp. $\mathbb{H}^i(X_v, j_*\mathcal{L})$) for $v \in V$, form a variation of MHS. The logarithmic complex $\Omega_{X/V}^*(\mathrm{Log}Y) \otimes \mathcal{L}_X$ satisfy, in the case of horizontal NCD: $i_{X_v}^*\Omega_{X/V}^*(\mathrm{Log}Y) \otimes \mathcal{L}_X \simeq \Omega_{X_v}^*(\mathrm{Log}Y_v) \otimes (i_{U_v}^*\mathcal{L})_{X_v}$.

When $\mathbf{L}$ underlies an admissible graded polarized VMHS: $(\mathbf{L}, W, F)$, its restriction to the open subset $U_v$ in $X_v$ is also admissible. The image of the filtrations $W$ and $F$, by the map $\Omega^*\mathcal{L} \to \Omega_f^*\mathcal{L}$

$$F := \mathrm{Im}(R^i f_* F_X \to R^i f_*(\Omega_{X/V}^*(\mathrm{Log}Y) \otimes \mathcal{L}_X)),$$

$$W := \mathrm{Im}(R^i f_* W_X \to R^i f_*(\Omega_{X/V}^*(\mathrm{Log}Y) \otimes \mathcal{L}_X))$$

define a variation of MHS on $R^i(f \circ j)_*\mathbf{L}$ inducing at each point $v \in V$ the corresponding weight $W$ and Hodge $F$ filtrations of the MHS on $(\mathbb{H}^i(U_v, \mathbf{L}), W, F)$.

**Proposition 9.7.** Let $f: X \to V$ be a proper smooth morphism of non singular complex varieties.

i) The direct image $R^i(f \circ j)_*\mathbf{L}$ is a local system on $V$ and

$$R^i(f \circ j)_* \mathbf{L} \otimes \mathcal{O}_V \simeq R^i f_*(\Omega_{X/V}^*(\mathrm{Log}Y) \otimes \mathcal{L}_X)$$

ii) The connecting morphism in Katz-Oda's construction [KaOd 68] coincides with the connection on $V$ defined by the local system $R^i(f \circ j)_*\mathbf{L}$

$$R^i f_*(\Omega_{X/V}^*(\mathrm{Log}Y) \otimes \mathcal{L}_X) \xrightarrow{\nabla_V} \Omega_V^1 \otimes R^i f_*(\Omega_{X/V}^*(\mathrm{Log}Y) \otimes \mathcal{L}_X)$$

iii) The filtration $F$ satisfy Griffith's transversality with respect to $\nabla_V$, $W$ is locally constant and $(R^i(f \circ j)_*\mathbf{L}, W, F)$ is a graded polarized variation of MHS on $V$.

Deligne's proof of ([De 70], proposition 2.28) extends in (i), as well the connecting morphism [KaOd 68] in (ii). Since $Y$ is horizontal $R^i(f \circ j)_*\mathbf{L}$ is a local system and on each fiber $W$ and $F$ induce a MHS, hence $W$ is locally constant and $F$ is a filtration by analytic sub-bundles.

### 9.6.1 The relative Intersection complex $\mathrm{IC}_f^*\mathcal{L}$

Let $\mathrm{IC}_f^*\mathcal{L} := \mathrm{Im}(\mathrm{IC}^*\mathcal{L} \to \Omega_f^*\mathcal{L})$ with image filtrations $W$ and $F$:

$$W := \mathrm{Im}(f_*W \to f_*(\mathrm{IC}_f^*\mathcal{L})), \quad F := \mathrm{Im}(f_*F \to f_*(\mathrm{IC}_f^*\mathcal{L})).$$

We deduce the filtrations on the $i$-th direct image $(R^i f_* j_* \mathbf{L}, W, F)$:
$$F^p R^i f_* j_1_* \mathbf{L} := (\mathrm{Im}(R^i f_* F^p \mathcal{I}C^* \mathcal{L}) \to R^i f_* \mathcal{I}C^* \mathcal{L}),$$
$$W_q R^i f_* j_1_* \mathbf{L} := \mathrm{Im}(R^i f_* W_q \to R^i f_* \mathcal{I}C^* \mathcal{L}))$$

where $W_q$ is locally constant on $V$, and $F^p$ is an analytic sub-bundle on $V$.

**Proposition 9.8.** i) The direct image $R^i f_* j_1_* \mathbf{L}$ of the intersection complex is a local system on $V$ and

$$R^i f_* j_1_* \mathbf{L} \otimes \mathcal{O}_V \simeq R^i f_* (\mathcal{I}C_f^* \mathcal{L})$$

ii) The filtrations $W$ and $F$ define a structure of variation of MHS on $R^i f_* j_1_* \mathbf{L}$.

We remark that $R^i f_* \mathcal{I}C_f^* \mathcal{L}$ is locally constant since $f$ is a topological fibration and induces a topological fibration on $X \setminus Y$. Moreover, at each $v \in V$, the Hodge decomposition $Gr_q^W \mathbb{H}^i(X_v, j_1_* \mathbf{L}) = \sum_{p+q=i} \mathbb{H}^{p,q}(X_v, Gr_q^W j_1_* \mathbf{L})$ shows that the Hodge terms $R^{p,q} f_* \mathcal{I}C_f^* \mathcal{L}$ are locally free $\mathcal{O}_V$-modules ([De 68], Théorème 5.5).

**Lemma 9.7.** Let $Y$ be a relative NCD over the maximal strata of $V$ and $S$ a stratum of $Y$ such that $X_S$ is a relative NCD over $S$:

i) $R^i f_* i_{X_S}^! j_1_* \mathbf{L}$ (resp. $R^i f_* i_{X_S}^* j_1_* \mathbf{L}$) are polarized VMHS on $S$ of weight $w \ge p w + i$ (resp. $w \le p w + i$).

ii) The image of the intersection morphism $\mathbf{pL}_S^i := \mathrm{Im} I$ is a polarized VHS of weight $p w + i$.

*Proof.* Set $Z := X_S$ and $\mathcal{I}C_f^* \mathcal{P}\mathcal{L}(\log Z) := \mathrm{Im}(\mathcal{I}C_f^* \mathcal{P}\mathcal{L}(\log Z) \to \Omega_f^* \mathcal{P}\mathcal{L})$. Let $v \in S$ and $N_v$ a normally embedded section to $S$ in $V$ at $v$. Then $Z_v := X_v$ is a NCD in $X_{N_v}$ and for each $v \in S$:

$$(f_* \mathcal{I}C_f^* \mathcal{P}\mathcal{L}(\log Z))_v \simeq \mathcal{I}C_{f|_{X_{N_v}}}^* \mathcal{P}\mathcal{L}|_{X_{N_v}} (\log X_v) \subset \Omega_{f|_{X_{N_v}}}^* \mathcal{P}\mathcal{L}|_{X_{N_v}}$$
$$(f_* i_{X_S}^! \mathbf{pL})_v \simeq i_{X_v}^! (\mathbf{pL}|_{X_{N_v}}), (f_* i_{X_S}^* \mathbf{pL})_v \simeq i_{X_v}^* (\mathbf{pL}|_{X_{N_v}})$$

By Thom-Whitney stratifications properties, $R^i f_* i_{X_S}^! j_1_* \mathbf{pL}$ and $R^i f_* i_{X_S}^* j_1_* \mathbf{pL}$ are local systems. The dimensions $\dim F^p \cap W_q$ of the filtrations defined by the MHS on $\mathbb{H}^i(i_{X_v}^! (\mathbf{pL}|_{X_{N_v}}), \mathbb{Q})$ are constant when $v \in S$ vary since no jump in the dimension of the fiber of the coherent analytic sheaf : $(W_q \cap F^p) R^i f_* i_{X_S}^* j_1_* \mathbf{pL}$ occurs.

**Remark 9.7.** The above results in the relative version are applied generically over the strata (ch. 11, §11.2) since refinements of Thom-whitney stratifications may be adapted.
## 9.7 Admissible perverse VMHS on $X \setminus Z$

It has been suggested by the referee to give an exposition of the theory of perverse PVMHS (Chapter 7, Definition 7.2) in terms of Grothendieck's six functors (see M. Saito's theory of mixed Hodge modules [Sa 90] §2 and [Shi 93]).

Such treatment is not in the spirit of this text.

Instead, we add this paragraph to give an informal study of the definition of the direct image of an admissible PVMHS, from which we deduce various cases (Remark 9.10).

In the same time, this is a motivation of the definition of $N*W$ (§9.2.1.1). Below a VMHS is always defined on a local system, a PVMHS is defined on a perverse sheaf, while in general we extend the notation of mixed Hodge complex (MHC) to a complex of sheaves (K, W, F) with the property that its global hypercohomology is a MHC, that is the hypercohomology on a complete complex algebraic is a MHS with the induced filtrations in each degree.

### 9.7.1 PVMHS on $j_*p\mathcal{L}$

*Hypothesis.* We resume the notations of chapter 7. Let $Y \subset X$ be a NCD, $Z \subset Y$ be a sub-NCD in $Y$ and $x \in Z_M^* \subset Y$, $L := \mathcal{L}_X(x)$ (Definition 3.1). An admissible VMHS $(\mathcal{L}, W^0, F)$ (Definition 2.16) on $U := X \setminus Y$ defines at $x$ an IMHS (Remark 2.10) where the filtration $W^0$ on the limit $L$ (preserved by $N_i$) is the natural extension of the filtration $W^0$ of $\mathcal{L}$. The relative monodromy filtration $M(\sum_{i \in M} N_i, W^0)$ exists by definition of admissibility and defines a limit MHS on $L$:

($\mathcal{L}, W^0, \mathcal{F}$) defines the IMHS: $(L, W^0, F, P, N_i, i \in M)$ where $L := \mathcal{L}_X(x)$.

Let $Z$ be defined by an algebraic map $f: X \to \mathbb{C}$ on $X$, then $W^0p\mathcal{L}$ defines a filtration: $W^0 = W^0(p\psi_f j_* p\mathcal{L}) := p\psi_f j_* W^0 p\mathcal{L}$. By admissibility, the relative filtration $M(N, W^0)$ exists on $p\psi_f j_* p\mathcal{L}$ (resp. $p\varphi_f j_* p\mathcal{L}$).

Let $W := M(N, W^0)[-1]$ on $p\psi_f j_* p\mathcal{L}$ such that $W_r := M(N, W^0)_{r+1}$.

**Proposition 9.9 (Perverse VMHS on $p\psi_f j_* p\mathcal{L}$ and $p\varphi_f j_* p\mathcal{L}$).** With the notations of (Equation 5.36), let $p\mathcal{L}$ be a shifted admissible VMHS quasi-unipotent along $Y$.

i) There exists a natural structure of perverse VMHS on $p\psi_f j_* p\mathcal{L}$ (resp. $p\varphi_f j_* p\mathcal{L}$) supported by the NCD $Z$ such that the canonical morphism (denoted can) is a filtered morphism while the morphism: $N$ and the variation (denoted var) are of type $(-1, -1)$.

ii) Let $W^0 = W^0(p\psi_f j_* p\mathcal{L}) := p\psi_f j_* W^0 p\mathcal{L}$ and $W := M(N, W^0)[-1]$, then $\text{Gr}_i^W p\psi_f j_* p\mathcal{L}$ (resp. $\text{Gr}_i^{M(N, W^0)} p\varphi_f j_* p\mathcal{L}$) decomposes into a direct sum of intermediate extension of polarized VHS of weight $i$ on the various intersection of components of $Z$.
**Proof.** The proof is local. We suppose also **L** locally unipotent along Y. We consider the de Rham family attached to L := L(x) and defined by the IMHS Ψ<sub>J</sub>L for various J ⊂ M<sub>Z</sub> as in (Equation 5.18).

The relative monodromy filtration $M(N, W^0)$ is well defined on the de Rham complex $\Psi^*IL$ and satisfy the decomposition of the relative monodromy (Equation 9.5)

$$ \mathrm{Gr}_i^{M(N, W^0)} \Psi^*IL \simeq \oplus_k \mathrm{Gr}_i^{M(N, W^0)} \mathrm{Gr}_k^{W^0} \Psi^*IL \quad (9.40) $$

As in (Equation 5.5), we define L[N] with the filtrations F and the relative filtration M(∑<sub>i∈M<sup>J</sup></sub> N<sub>i</sub>, W<sub>0</sub>). Then, for each J ⊂ M<sub>Z</sub>, we define similarly a MHS on

$$ (\Psi_J L = (L[N]/\mathrm{Im}A_J, W_J^0, M^J, F_J), W_J^0 = W_J^0(\Psi_J L) := \Psi_J W^0 L \quad (9.41) $$

Let B<sup>J</sup> := N + ∑<sub>i∈M<sup>J</sup></sub> N<sub>i</sub>. Then, the relative monodromy filtration M<sup>J</sup> := M(B<sup>J</sup>, W<sup>0</sup>) exists on Ψ<sub>J</sub>L.

Let B<sup>J</sup><sub>k</sub> denote the nilpotent morphism induced on Gr<sup>W<sub>J</sub><sup>0</sup></sup><sub>k</sub>Ψ<sub>J</sub>L. By definition M<sup>J</sup><sub>j+k</sub> induces the monodromy filtration W<sub>j</sub>(B<sup>J</sup><sub>k</sub>) on Gr<sup>W<sub>J</sub><sup>0</sup></sup><sub>k</sub>Ψ<sub>J</sub>L. Hence Gr<sup>M<sup>J</sup></sup><sub>j+k</sub>Gr<sup>W<sub>J</sub><sup>0</sup></sup><sub>k</sub>Ψ<sub>J</sub>L is of weight j + k − 1. Let W<sup>J</sup> := M<sup>J</sup>[-1] such that W<sup>J</sup><sub>r</sub> := M<sup>J</sup><sub>r+1</sub>, then Gr<sup>W<sub>J</sub><sup>0</sup></sup><sub>j+k</sub>Gr<sup>W<sub>J</sub><sup>0</sup></sup><sub>k</sub>Ψ<sub>J</sub>L is of weight j + k on Gr<sup>W<sub>J</sub><sup>0</sup></sup><sub>k</sub>Ψ<sub>J</sub>L.

Thus, (W<sup>J</sup>, M<sup>J</sup>, F<sub>J</sub>) induces a MHS on each Gr<sup>W<sub>J</sub><sup>0</sup></sup><sub>i</sub>Ψ<sub>J</sub>L and defines a MHS on Ψ<sub>J</sub>L since Gr<sup>W<sub>J</sub><sup>0</sup></sup><sub>r</sub>Ψ<sub>J</sub>L ≃ ⊕<sub>i</sub>Gr<sup>W<sub>J</sub><sup>0</sup></sup><sub>r</sub>Gr<sup>W<sub>J</sub><sup>0</sup></sup><sub>i</sub>Ψ<sub>J</sub>L.

The case of ϕ<sup>u</sup><sub>f</sub>j<sub>!&Sigma;</sub><sup>L</sup> is deduced by the vanishing cycle lemma (5.1).

**Lemma 9.8** ($j_{Z*}j_{Z*}^*j_{!*}^*P^L$ and $\varphi_f^u(j_{Z*}j_{Z*}^*j_{!*}^*P^L)$). Let $i: Z \to X$ and

$$ i^*j_{Z*}j_{Z*}^*j_{!*}^*P^L \simeq \mathrm{Cone} \left( p_f \psi_f^u(j_{!*}^*P^L) \xrightarrow{N} p_f \psi_f^u(j_{!*}^*P^L) \right) (-1) $$

be the mixed cone by (Equation 2.16, [Br 82], Proposition 2.3.6) and let $\varphi_f j_{!*}^*P^L$ be defined as the mixed cone over the specialization sp: $\varphi_f^u(j_{Z*}j_{Z*}^*j_{!*}^*P^L) := C_M(sp: i^*j_{Z*}j_{Z*}^*j_{!*}^*P^L \to \psi_f^u(j_{Z*}j_{Z*}^*j_{!*}^*P^L))$, then:

$$ \mathrm{var}: \varphi_f^u(j_{Z*}j_{Z*}^*j_{!*}^*P^L) \to \psi_f^u(j_{Z*}j_{Z*}^*j_{!*}^*P^L) (-1) \quad (9.42) $$

is an isomorphism.

We have the triangles where $\psi_f^u(j_{!*}^*P^L) = \psi_f^u(j_{Z*}j_{Z*}^*j_{!*}^*P^L)$

$$ \begin{tikzcd} i^*j_{Z*}j_{Z*}^*j_{!*}^*P^L \arrow[r, "sp"] & \psi_f^u(j_{Z*}j_{Z*}^*j_{!*}^*P^L) \arrow[r, "can"] & \varphi_f^u(j_{Z*}j_{Z*}^*j_{!*}^*P^L) \arrow[r, "[1]"'] & (9.43) \\ \big\downarrow_{\mathrm{Id}} & & \big\downarrow_{\mathrm{Id}} & \\ i^*j_{Z*}j_{Z*}^*j_{!*}^*P^L \arrow[r, "sp"] & \psi_f^u(j_{Z*}j_{Z*}^*j_{!*}^*P^L) \arrow[r, "N"] & \psi_f^u(j_{Z*}j_{Z*}^*j_{!*}^*P^L) \arrow[r, "[1]"'] & (-1) \arrow[downarrow] \end{tikzcd} $$
where $N$ induces the vertical isomorphism $\text{var}$. Hence The morphism $\text{var}$ is an isomorphism and this is a characterization of the extension $j_{Z*}j_Z^*j_{!*}^p\mathbf{L}$ in Verdier's classification (§2.2.3.1 4)).

To define the limit Hodge filtrations on $j_{!*}\mathbf{L}$ as well on $\Psi^*\mathcal{I}\mathcal{L} \simeq \psi_f^u j_{!*}\mathbf{L}$ (ch.4, 4.2) we used the complexes $\mathcal{I}\mathcal{C}^*\mathcal{L}, F) \subset (\mathcal{I}\mathcal{C}^*\mathcal{L}(\log Z), F) \subset (\mathcal{O}^*\mathcal{L}, F)$ (ch. 3, §3.8) isomorphic respectively to $j_{!*}\mathbf{L} \subset j_{Z*}j_Z^*j_{!*}^p\mathbf{L} \subset j_{!*}j_{!*}^p\mathbf{L}$.

**Example 9.2.** i) Let $\mathbf{L}$ be on $\mathbb{C}^*$, $i: 0 \to \mathbb{C}, j: \mathbb{C}^* \to \mathbb{C}$, then:

1) $\psi_z^u \mathbf{L} = L$ at $0$, $i^*j_*L \simeq (L \xrightarrow{N} L(-1))$. The action of $N$ on $i^*j_*L$ induces $0$ on cohomology. Then $\varphi_z^u j_* \mathbf{L} := \text{cone} (i^*j_* \mathbf{L} \xrightarrow{sp} \psi_z^u \mathbf{L})$ is isomorphic to

$$ \varphi_z^u j_* \mathbf{L} := (L \xrightarrow{d_{-1}:=(-N,id)} L(-1) \oplus L) \text{ in degrees } -1 \text{ and } 0 $$

with unique cohomology in degree 0

$$ H^0 := (L(-1) \oplus L)/\operatorname{Im} d_1 \xrightarrow{\text{var}\simeq} \psi_z^u \mathbf{L}(-1) = L(-1) : (b, c) \mapsto b + Nc. $$

2) The canonical map $\text{can}: \psi_z^u \mathbf{L} \to \varphi_z^u j_* \mathbf{L}$ is defined in degree 0 by

$$ \text{Id} : L \xrightarrow{(0,id)} L(-1) \oplus L : a \mapsto (0, a). $$

3) The variation map $\text{var}_\psi = \text{var} \circ \text{can} : L \to L(-1) : a \to Na$.

ii) In the case of $j_{!*}\mathbf{L}$: $\varphi_z^u j_{!*} \mathbf{L} \simeq (L \xrightarrow{d_{-1}:=(-N,id)} \operatorname{Im} N \subset L(-1) \oplus L)[1]$, $\text{var}: \varphi_z^u j_{!*} \mathbf{L} \simeq \operatorname{Im} N \subset L(-1) : (b, c) \mapsto b + Nc \in \operatorname{Im} N \subset L(-1)$.

4) This example apply to perverse sheaves, as we can see for example on the associated de Rham family.

### 9.7.1.1 ($j_{Z*}j_Z^*j_{!*}^p\mathbf{L}, W, F$) and $N * W$

**Proposition 9.10.** Let $\mathbf{p}\mathbf{L}$ be a polarized VMHS on the complement of a NCD $Y$ of an algebraic variety $X$, ilmp its intermediate extension, $Z \subset X$ a NCD and $(j_Z^*j_{!*}^p\mathbf{L}, W^0, F)$ on $X$.

i) There exists a weight filtration $W$ on the complex

($j_{Z*}j_Z^*j_{!*}^p\mathbf{L}, F) \simeq (\mathcal{I}\mathcal{C}^*\mathcal{L}(\log Z), F)$ by perverse sub-sheaves on $X$ such that ($j_{Z*}j_Z^*j_{!*}^p\mathbf{L}, W, F$) is a PVMHS and $W_k$ is a perverse extension of $j_Z^*W_k^0j_{!*}^p\mathbf{L}$.

ii) There exists a short exact sequence of perverse sheaves

$$ 0 \to (j_{!*}^p\mathbf{L}, W, F) \to (j_{Z*}j_Z^*j_{!*}^p\mathbf{L}, W, F) \to i_{Z*}i_Z^1j_{!*}^p\mathbf{L}[1] \to 0 $$

$$ \text{iii) Moreover: } (\mathrm{Gr}_k^W j_{!*}^p \mathbf{L}, F) = (j_{!*} \mathrm{Gr}_k^{W^0} \mathbf{L}, F) $$

The proof of the proposition follows from (Corollary 9.9) below, as we need to describe the weight filtration $W$ first.

Let $Z$ be a locally principal NCD, the proof is reduced to the case where $Z$ is principal defined by an equation $f$, in which case the weight and Hodge filtrations are defined on $\psi_f^u(j_{!*}^p\mathbf{L})$ and $\varphi_f^u(j_{!*}^p\mathbf{L})$. Such filtrations defined locally, will induce a unique filtration on Coker $N$, hence on $j_{Z*}j_Z^*j_{!*}^p\mathbf{L}/j_{Z*}j_Z^*j_{!*}^p\mathbf{L}$ which is a characterization of $W$.
### Interpretation of $N * W^0$ in terms of Verdier's classification (§2.2.3).
Let $Z$ be defined by an equation $f$. We deduce from the filtered perverse sheaf $(\mathcal{P}\mathbf{L}, W^0)$, a perverse filtration $W^0j_Z^*j_{j_*}^*\mathcal{P}\mathbf{L} := j_Z^*j_{j_*}W_0$. Let

$$W^0 := p_{\psi_f^u}(W^0 j_Z^* j_{j_*}^* \mathcal{P}\mathbf{L}) \text{ on } p_{\psi_f^u}(j_Z^* j_{j_*}^* \mathcal{P}\mathbf{L}).$$

To apply Verdier's classification, we need to define a sequence

$$W_k^0 p_{\psi_f^u}(j_Z^* j_{j_*}^* \mathcal{P}\mathbf{L}) \xrightarrow{a} W_k \xrightarrow{b} W_k^0 p_{\psi_f^u}(j_Z^* j_{j_*}^* \mathcal{P}\mathbf{L}) \quad (9.44)$$

with the correct weight and Hodge filtrations on $W_k$. Below, there will be weight filtrations on various spaces all denoted by $W$.

It is convenient to define the filtration $N * W^0$ by perverse sub-sheaves of $p_{\psi_f^u}(j_Z^* j_{j_*}^* \mathcal{P}\mathbf{L})$ with a change of indices. Set:

$$*W_k^0 := N W_k^0 + M_{k-1}(N, W^0) \cap W_k^0 \subset p_{\psi_f^u}(W^0)$$

where $*W^0 = N * W^0[1]$ such that $*W_{k+1}^0 := (N * W^0)_k$.

### Remark 9.8. $*W^0$ is defined here in the abelian category of perverse sheaves.
Equivalently it is defined on the de Rham family of vector spaces associated to $p_{\psi_f^u}(W^0)$. It is related to the weight on $j_{Z*}j_{Zj_*}^*\mathcal{P}\mathbf{L}$ by Verdier's correspondence and the isomorphism $j_{Z*}j_{Zj_*}^*\mathcal{P}\mathbf{L}/j_{j_*}^*\mathcal{P}\mathbf{L} \simeq \text{Coker } N$.

We have by (Equations 9.8 and 9.9):

$$M(N, *W^0) = M(N, N * W^0)[1] = M(N, W^0)[1]$$

since $M(N, N * W^0) = M(N, W^0)$. We have now: $W_k^0 \xrightarrow{N} *W_k^0 \xrightarrow{I} W_k^0$.

From now on, we write $\varphi_f^u$ for $p_{\varphi_f^u}(j_{Z*}j_{Zj_*}^*\mathcal{P}\mathbf{L})$ and $p_{\psi_f^u}$ for $p_{\psi_f^u}(j_{Zj_*}^*\mathcal{P}\mathbf{L})$. We have $\text{Ker}(\varphi_f \xrightarrow{\text{can}} \psi_f) = \text{Ker}(N : \psi_f \to \psi_f)$. Set

$$W_k^0(\varphi_f^u) := \text{var}^{-1}(W_k^0 \psi_f^u) \text{ and } NW_k^0(\varphi_f^u) := \text{var}^{-1}(NW_k^0 \psi_f^u), \text{ then:}$$

$$NW_k^0(\varphi_f^u) = \text{Im}(C : W_k^0 \to \varphi_f^u) \xrightarrow{\text{var}\simeq} NW_k^0(\psi_f).$$

### The weights.
As perverse sheaves, $\varphi_f^u$ and $\psi_f^u$ are isomorphic, but with different weights.

The sub-quotient sheaf $(\text{Gr}_{k-1}^{M(N, W^0)}, F)$ of $p_{\psi_f^u}$ is pure Hodge of weight $k-2$.

Set $(W_{k-2} p_{\psi_f^u}(j_Z^* j_{j_*}^* \mathcal{P}\mathbf{L}), F) := (M_{k-1}(N, W^0), F)$, then $(W, F)$ is a PVMHS of $p_{\psi_f^u}(j_Z^* j_{j_*}^* \mathcal{P}\mathbf{L})$, while $(W', F) := (W, F)(-1)$ is a PVMHS of $p_{\varphi_f^u}(j_{Z*}j_{Zj_*}^*\mathcal{P}\mathbf{L})$ such that by (Equation 9.42):

$$(p_{\varphi_f^u}, W', F) \xrightarrow{\text{var}\simeq} (p_{\psi_f^u}, W, F)(-1) \quad (9.45)$$

### Remark 9.9.
In order to apply the results of [Ka 86] and the classification of Verdier in their terminology, we keep the convention $W_k^0(\psi_f^u) := \psi_f^u(W_k^0)$ instead of $W_k^0(\psi_f^u) := \psi_f^u(W_{k+1}^0)$ in ([Shi 93] §3.1).

We introduce the subspace
$$ W_k p^{\varphi_f^u} := NW_k^0 + W'_k \cap W_k^0 : W_k p^{\varphi_f^u} \xrightarrow{\text{var}\simeq} (*W_k^0) p^{\psi_f^u} \qquad (9.46) \\ (p^{\psi_f^u}, W^0) \xrightarrow{C} (p^{\varphi_f^u}, W) \xrightarrow{V} (p^{\psi_f^u}, W^0) $$

satisfying $V \circ C = N$ and $C \circ V = N$. Set $W_k := W_k p^{\varphi_f^u}(j_{Z^*}j_{Z^*j_*}^*p^L)$ in (Equations 9.44 and 9.46). The extension $W_k p^{\varphi_f^u}(j_{Z^*}j_{Z^*j_*}^*p^L)$ above, corresponds to a term $W_k$ of the weight filtration on $(j_{Z^*}j_{Z^*j_*}^*p^L, F)$. The filtrations $W'$ and $F$ are essential in the definition of $W_k$ and in the proof below.

**Lemma 9.9.** *The sequence*

$$ \psi_f^u(W_k^0 j_Z^* j_{j_*}^* p^L) \xrightarrow{C_k} \phi_k := W_k p^{\varphi_f^u} j_{Z^*} j_{j_*}^* p^L \xrightarrow{V_k} \psi_f^u(W_k^0 j_Z^* j_{j_*}^* p^L) $$

defines by Verdier's classification, the perverse extension $W_k j_{Z^*}j_{Z^*j_*}^*p^L$ of $W_k^0j_{Z^*j_*}^*p^L$ such that $(W, F)$ is a PVMHS.

*Proof.* We write $W_k p^{\varphi_f^u}$ for $W_k p^{\varphi_f^u}(j_{Z^*}j_{Z^*j_*}^*p^L)$. In terms of the sequence in (Equation 9.31) and with the definition of $W'$ (Equation 9.45) and by (Equations 9.8 and 9.9):

$$ \mathrm{Gr}_k^W p^{\varphi_f^u} \simeq \mathrm{Im} \left( C_k : \mathrm{Gr}_k^{W^0} p^{\psi_f^u} \to \mathrm{Gr}_k^W p^{\varphi_f^u} \right) \oplus \ker \left( V_k : \mathrm{Gr}_k^W p^{\varphi_f^u} \to \mathrm{Gr}_k^{W^0} p^{\psi_f^u} \right) \qquad (9.47) $$

$$ \forall l < k, \mathrm{Gr}_k^W \mathrm{Gr}_k^{W^0} p^{\varphi_f^u} = 0 $$

$$ \mathrm{Gr}_k^W \mathrm{Gr}_k^{W^0} p^{\varphi_f^u} \simeq \mathrm{Im} \left( N : \mathrm{Gr}_k^{W^0} p^{\psi_f^u} \to \mathrm{Gr}_k^{W^0} p^{\psi_f^u} \right) $$

$$ \forall l \ge k+1, \mathrm{Gr}_k^W \mathrm{Gr}_k^{W^0} \simeq \mathrm{Coker} \left( N : \mathrm{Gr}_{l+2}^{W'} \mathrm{Gr}_k^{W^0} p^{\varphi_f^u} \to \mathrm{Gr}_l^{W'} \mathrm{Gr}_k^{W^0} p^{\varphi_f^u} \right). \qquad (9.48) $$

Hence the corresponding perverse filtration $W_k$ on $j_{Z^*}j_{Z^*j_*}^*p^L$ satisfies the equation: $(\mathrm{Gr}_k^W j_{Z^*}j_{Z^*j_*}^*p^L, F) = (j_{Z^*j_*}j_Z^*\mathrm{Gr}_k^{W^0}j_{j_*}^*p^L, F) \oplus_{p \le k} \mathrm{Coker} (N : \mathrm{Gr}_{k+2}^{W'}\mathrm{Gr}_p^{W^0}p^{\varphi_f^u} \to \mathrm{Gr}_k^{W'}\mathrm{Gr}_p^{W^0}p^{\varphi_f^u})$

Moreover, the filtration $W$ induces on $(\mathrm{Gr}_k^{W^0}j_{Z^*}j_{Z^*j_*}^*p^L, F)$ the weight filtration on $(j_{Z^*}\mathrm{Gr}_k^{W^0}j_{Z^*j_*}^*p^L, F)$

Finally, by the splitting of $(\mathrm{Gr}_k^{W^0} p^{\varphi_f^u}, F) \simeq \oplus_p \mathrm{Gr}_k^{W'} \mathrm{Gr}_p^{W^0} p^{\varphi_f^u}, F)$ (Equation 9.5), we deduce a perverse HS of weight $k$ on

$$ (\mathrm{Gr}_k^W j_{Z^*}j_{Z^*j_*}^*p^L, F) \simeq (j_{Z^*j_*}j_Z^*\mathrm{Gr}_k^{W^0}j_{j_*}^*p^L, F) \oplus_{p<k} \mathrm{Gr}_k^W j_{Z^*}j_{Z^*j_*}^*p^L, F). $$

The above results extend to the singular case as follows:

**Corollary 9.6.** Let $(p^L, W^0, F)$ be a graded polarized admissible VMHS on the non singular set $V^*$ of an algebraic variety $V$, $j_{j_*}^*p^L$ its intermediate extension and $g: V \to \mathbb{C}$ an algebraic map with central fiber $Z := g^{-1}(0)$:

i) the nearby (resp. vanishing) cycles complex $p^{\psi_gj_{j_*}^*p^L}$ (resp. $p^{\varphi_gj_{j_*}^*p^L}$) is a PVMHS on $Z$ is a PVMHS on $Z$.

ii) let $U := V \setminus Z, j_Z : U \to V$, then $p^{\varphi_gj_{Z^*j_*}^*p^L}$ is a PVMHS on $Z$.

iii) the direct image $j_{Z^*}j_{Z^*j_*}^*p^L$ is a PVMHS on $X$.
**Proof.** We consider a desingularization $\pi : V' \to V$ such that $Z' := \pi^{-1}(Z)$ is a NCD. Let $f := g \circ \pi, u : V^* \to V'$ and $u_Z : V' \setminus Z' \to V'$. We suppose $Y' := \pi^{-1}(V \setminus V^*)$ a NCD.

i) The PVMHS: $p\psi_f u_Z^* u_{1*}^p \mathbf{L}$ and $p\varphi_f u_{Z*} u_Z^* u_{1*}^p \mathbf{L}$ are well defined by assumption of admissibility. We apply the weight spectral sequence to the derived direct image $\pi_*(\mathcal{O}_X)$ (Equation 7.4, Proposition 7.2):

$$ pE_1^{p,q} := p\mathcal{H}^{p+q}(\pi_*Gr_{-p}^W p\psi_f u_{1*}^p \mathbf{L}) \implies Gr_q^W p\psi_g p\mathcal{H}^{p+q}(\pi_*u_{1*}^p \mathbf{L}) $$

which degenerate at rank 2, to deduce the PVMHS. In the case $p+q=0$, the limit is the weight filtration on $p\psi_g p\mathcal{H}^0(\pi_*u_{1*}^p \mathbf{L}) \supset p\psi_g j_{1*}^p \mathbf{L}$.

The case of $p\varphi_f u_{Z*} u_Z^* u_{1*}^p \mathbf{L}$ is similar.

ii) We have: $\text{var} : p\varphi_g j_Z^* j_{1*}^p \mathbf{L} \simeq p\psi_g j_{1*}^p \mathbf{L}(-1)$.

iii) In the case $j_{Z*} j_Z^* j_{1*}^p \mathbf{L}$, we apply the weight spectral sequence to the derived direct image $\pi_*(\mathcal{O}_X)$ (Equation 7.4):

$$ pE_1^{p,q} := p\mathcal{H}^{p+q}(\pi_*Gr_{-p}^W u_{Z*} u_Z^* u_{1*}^p \mathbf{L}) \implies Gr_q^W p\mathcal{H}^{p+q}(\pi_*u_{Z*} u_Z^* u_{1*}^p \mathbf{L}) $$

which degenerates at rank 2.

Since $u_Z$ and $j_Z$ are affine and $\pi$ projective, we have:

$$ \pi_* u_{Z*} u_Z^* u_{1*}^p \mathbf{L} = j_{Z*} j_Z^* \pi_* u_{1*}^p \mathbf{L}. \text{ We consider } \pi_* W \text{ induced on} $$

$$ j_{Z*} j_Z^* \pi_* u_{1*}^p \mathbf{L} \supset p_{\tau \le 0} j_{Z*} j_Z^* \pi_* u_{1*}^p \mathbf{L} \text{ and then projected on:} $$

$$ p\mathcal{H}^0(j_{Z*} j_Z^* \pi_* u_{1*}^p \mathbf{L}) = j_{Z*} j_Z^* p\mathcal{H}^0(\pi_* u_{1*}^p \mathbf{L}) \supset j_{Z*} j_Z^* j_{1*}^p \mathbf{L}. $$

**Corollary 9.7.** The direct definition of $W^0$ on $V$ corresponds by Verdier's classification (Equation 9.46) to the structure of PVMHS on $j_{Z*} j_Z^* j_{1*}^p \mathbf{L}$.

**Proof.** (Equation 9.46) is defined directly on $V$ by (Corollary 9.6 i) and ii)). In both constructions, the induced structure on $j_{Z*}(Gr_*^{W^0} j_Z^* j_{1*}^p \mathbf{L}, W)$ coincide.

### 9.7.1.2 ($j_{Z*}j_Z^*V, W, F$) for $V$ a PVMHS and $N \ast W$

**Hypothesis.** We assume now that $(V, W, F)$ is a PVMHS on a complex algebraic variety $V$, $Z \subset V$ a locally principal divisor, $U := V \setminus Z$, $j_Z : U \to V$ and $i_Z : Z \to V$.

To construct the PVMHS on $(j_{Z*}j_Z^*V, W, F)$, we suppose $Z$ principal defined by a morphism $f$ defined on the algebraic complex variety $V$, $Z := f^{-1}(0)$. Indeed, the various constructions for different local equations $f$ will glue together. The functor $\psi_f$ (resp. $\phi_f$) is well defined on $(V, W)$.

Let $W_i$ denote the filtration on $\psi_f V : W_i \psi_f V := \psi_f W_i V$ and the filtration $F$ is well defined on $\psi_f \text{Gr}_i^W (V, F)$ since $\text{Gr}_i^W (V, F)$ is perverse pure Hodge.

**Definition 9.7.** A PVMHS $(V, W, F)$ on $V \setminus Z$ is admissible along $Z$ if:

1) There exists a filtration $F$ on $\psi_f(V, W)$ inducing, for each index $i$, the filtration $F$ on $\psi_f \text{Gr}_i^W (V, F)$

2) Let $W$ on $\psi_f(V)$ be defined by $W_i \psi_f(V) := \psi_f W_i V$. The relative monodromy filtration $M := M(N, W)$ exists on $\psi_f(V)$.
**Corollary 9.8.** Let $\tilde{W} := M[-1]$ be the filtration defined by $\tilde{W}_{k-1} = M_k$ and $j_Z : U \to V$, then $(\psi_f j_Z^* \mathbf{V}, \tilde{W}, W, F)$ is a PVMHS on $Z$. Moreover: $(\varphi_f j_{Z*} j_Z^* \mathbf{V}, \tilde{W}, W, F) \xrightarrow{\text{var}\approx} (\psi_f j_{Z*} j_Z^* \mathbf{V}, \tilde{W}, W, F)(-1)$ defines a structure of PVMHS on $\varphi_f$.

*Hint to the proof.* By (Equation 9.5): $\mathrm{Gr}_r^M(\psi_f \mathbf{V}, F) \simeq \oplus_k(\mathrm{Gr}_r^M \mathrm{Gr}_k^W \psi_f K, F)$. Hence the proof is reduced to the case of a direct sum of intermediate extensions of perverse VHS (Definition of $\mathcal{I}\mathcal{H} 7.1$) then (Corollary 9.6) apply.

**Corollary 9.9.** Let $Z \subset V$ be a locally principal divisor and $j$ the embedding of $U := V \setminus Z$ in $V$. The derived direct image of an admissible PVMHS $(\mathbf{V}, W^0, F)$ on $U$, is a PVMHS $(j_* \mathbf{V}, W, W^0, F)$.

*Hint to the proof.* We suppose $Z$ locally defined by an equation $f$ (as local constructions on a local covering of $Z$ glue together). The relative monodromy filtration $M_k := M_k(N, \psi_f W^0)$ on $\psi_f j_{Z*} \mathbf{V}$ defines the weight filtration $W$ such that $(W_{k-1}, F) := (M_k, F)$ from which we deduce by the isomorphism $\mathrm{var}: \varphi_f \to \psi_f(-1)$ the definition of $(\varphi_f j_{Z*} j_Z^* \mathbf{V}, W', F)$

In this case (Equation 9.46) is well defined (see also Corollary 9.7). Verdier's extension (§2.2.3, equation 2.23 the case 5) applies and defines the weight filtration $W$ on $(j_{Z*} j_Z^* \mathbf{V}, W, F)$ as extension of $W^0$.

Moreover, the filtration induced by $W$ on $(j_{Z*} Gr_*^{W^0} j_Z^* \mathbf{V}, W, F)$ coincide with the filtration constructed by desingularization in the case of $(j_{Z*} Gr_*^{W^0} j_Z^* \mathbf{V}, W, F)$. By (Equations 9.47 and 9.48), the filtrations $W$ is unique.

**Corollary 9.10.** The induced filtration $W$ on $(j_{Z*} j_Z^* \mathbf{V}, F)$ satisfy:

$$(\mathrm{Gr}_i^W j_{i*} \mathbf{V}, F) \simeq (j_{i*} \mathrm{Gr}_i^{W^0} \mathbf{V}, F)$$

*Remark 9.10.* As in (§5.2.1.4, Corollary 9.4 and Theorem 9.4), we deduce from the open embedding case the filtration $W$ for various functors:

1) The filtrations $(W, F)$ on $j_1 \mathbf{V}$ may be defined by duality with $j_*(\mathbf{V}, W^0, F)$. A direct proof is based on the filtration $(N_! W)_k := W_{k-1} + M_k(N, W) \cap N^{-1} W_{k-1}$ dual to $N_* W$ ([Ka 86] Equation 3.4.2).

2) The filtrations $(W, F)$ on $(i_Z^! \mathbf{V}, W, F)$[1] may be defined as a mixed cone of the morphism $\rho$ in the natural sequence $i_Z^! \mathbf{V} \to \mathbf{V} \xrightarrow{\rho} j_{Z*} j_Z^* \mathbf{V}$.

3) The filtrations $(W, F)$ on $i_Z^* \mathbf{V}$ may be defined by duality.

### 9.7.1.3 Proof of (Equation 9.5)

The result of Kashiwara involves sections $\mathrm{Gr}_r^M \mathrm{Gr}_k^W L \to \mathrm{Gr}_r^M L$ (Equation 9.5). The following description in ([Sa 90]) of the sections is clear.

Since the filtration induced by $M$ on $\mathrm{Gr}_k^W$ coincides the shifted monodromy filtration $W(N)[k]$, we can define the following primitive object $P\mathrm{Gr}_{i+k}^M \mathrm{Gr}_k^W L$, denoted by $P_{i+k,k}$
$$ P_{i+k,k} := \mathrm{Ker}(\mathrm{Gr}_{i+k}^M \mathrm{Gr}_k^W N^{i+1} : \mathrm{Gr}_{i+k}^M \mathrm{Gr}_k^W L \to \mathrm{Gr}_{-i-2+k}^M \mathrm{Gr}_k^W L) $$

The definition of the section is by induction and compatible with the action of $N$, such that the definition is reduced to the case of the primitive subspace $s_{i+k,k} : P_{i+k,k} \to \mathrm{Gr}_{i+k}^M W_k L$ as follows. If we suppose $s_{j+l,l}$ defined for $l \le k-2$ for all $j$, we define the subspace $V_{i+k,k}$ by the sum:

$$ V_{i+k,k} = \sum_{j,m,l \in S_k} \mathrm{Im} \, N^m s_{j+l,l} \subset \mathrm{Gr}_{-i-2+k}^M W_k L \text{ where} $$

$$ S_k := \{j, m \ge 0, l \le k-2 \text{ such that } j \ge m, j+l < i+k, j+l-2m = k-i-2\} $$

$$ \text{and } P_{j+l,l} \xrightarrow{s_{j+l,l}} \mathrm{Gr}_{j+l}^M W_k L \xrightarrow{N^m} \mathrm{Gr}_{j+l-2m}^M W_k L. \text{ Then,} $$

$$ s_{i+k,k} : P_{i+k,k} \xrightarrow{\sim} \mathrm{Ker}(N^{i+1} : \mathrm{Gr}_{i+k}^M W_k L \to \mathrm{Gr}_{-i-2+k}^M W_k L / V_{i+k,k}). $$

The section $s_{i+k,k}$ extends (by the primitive decomposition) to define the section on $\mathrm{Gr}_{i+k}^M \mathrm{Gr}_k^W L$ ([Sa 90] Proposition 1.5).
<figure><img src="image_14.png" /></figure>
Chapter 10
Relative local purity and
Decomposition

We prove the decomposition theorem (ch. 1, Theorem 1.2) and the local purity
(ch. 4, Theorem 4.1) in terms of a Thom-Whitney stratification, by induction
on $n := \dim X$ the dimension of $X$ and reduction to isolated strata.

Let $\pi : \tilde{X} \to X$ be a desingularization of $X$. The decomposition theorem for $f : X \to V$ follows directly from the case of $\pi$ and $f \circ \pi$ ([De 68] Proposition 2.16), hence to start the proof of Theorem 1.2, we suppose $X$ non singular and $f$ surjective such that $\dim V \le \dim X$.

The proof of the decomposition (§10.3, Theorem 10.1), is reduced, by in-
duction (§10.0.0.1), to the case of isolated strata of $v \in V$, and it is related
to the properties of the cohomology of the NCD above $v$ stated in terms of
the relative local purity (§10.6).

10.0.0.1 Induction statement $\mathcal{D}(n)$

$\mathcal{D}(n)$: Theorem 9.2 on relative local purity and the decomposition theorem (ch.1, Theorem 1.2) apply to projective morphisms $f : X \to V$ on projective varieties $X$ of dimension $\le n$.

The assertion $\mathcal{D}(0)$ is obvious. To prove $\mathcal{D}(n-1) \Rightarrow \mathcal{D}(n)$, we assume first that $X$ is nonsingular. Recall that $\mathcal{L}$ is a PVHS of weight $w'$ on a Zariski-open subset $U \subset X$ such that $Y = X \setminus U$ is a NCD, $^p\mathcal{L} := \mathcal{L}[\dim X]$ is of weight $^pw := w' + \dim X$ and $j_{1*}^p\mathcal{L} \simeq IC^*{}^p\mathcal{L}$ (§3.1.4) is endowed with a Hodge filtration $F$.

10.0.1 The case dim $V = 0$ and dim $X = n$

We distinguish the case where $\dim V = 0$ which is admitted since the de-
composition is reduced to the purity of the HS on $\mathbb{H}^*(X, j_*\mathcal{L})$ proved in
[KaK 87] and [CaKSc 87] by comparison with $L^2$-cohomology (see Theorem 3.1). A proof by reduction to the case where $X$ is over a curve is in [Sa 88].

Hard Lefschetz theorem is proved by the classical reduction to an hyperplane section ([KaK 87]. In particular, $\mathbb{H}^i(X, j_*^p \mathbf{L})$ is a polarized HS of weight $^p w + i$ as a direct sum of primitive polarized HS. By the property of auto-duality [KaK 86], all HS coincide with the HS defined in this article by the subcomplex $(IC^* \mathcal{P}\mathcal{L}, F) \subset (\Omega^* \mathcal{P}\mathcal{L}, F)$.

The statement of relative local purity is trivial in this case since $B_{X_v}^* = \emptyset$.

### 10.0.2 Intermediate statement $D(n, V^*)$ in the case $0 < \dim V \le n$

*Hypothesis.* Let $f: X \to V$ be defined on a non singular complex variety $X$ of dimension $n$, $K := f_*j_*^p \mathbf{L}$, $S$ a Thom-Whitney stratification of $f$ adapted to the intermediate extension of a shifted polarized VHS $j_*^p \mathbf{L}$ on $X$.

We denote by $V_0 \subset V$ the union of zero-dimensional strata of $S$, $i_{V_0}: V_0 \to V$, $V^* := V \setminus V_0$ and $k: V^* \to V$.

Assuming $D(n-1)$ and $\dim V > 1$, the first step is to prove the decomposition theorem over $V^*$. The assumption $D(n)$ over $V^*$, denoted by $D(n, V^*)$, is proved by reduction to hyperplane sections of $V$ transversal to the strata of $V$.

**Lemma 10.1** ($D(n, V^*)$). Under the Hypothesis above, assuming $D(n-1)$, and $\dim V > 1$:

i) The restriction of the perverse cohomology to $V \setminus V_0$ decomposes in each degree $i$ into a direct sum of intermediate extensions of shifted polarized VHS $^p \mathbf{L}_S^i$ (ch. 1, Equation 1.7) of weight $^p w + i$ on the strata $S$ of dimension $0 < \dim S \le n$

$$ {}^p \mathcal{H}^i(k^*K) \xrightarrow{\sim} \oplus_{S \in \mathcal{S}, S \subset V \setminus V_0} k^* i_{S!_*} {}^p \mathbf{L}_S^i \quad (10.1) $$

ii) The iterated cup-product $\eta$ with the class of an hyperplane section of $X$, induces an isomorphism on the restrictions to $V \setminus V_0$

$$ \eta^i : {}^p \mathcal{H}^{-i} (f_* j_*^p \mathbf{L})_{|V \setminus V_0} \xrightarrow{\sim} {}^p \mathcal{H}^i (f_* j_*^p \mathbf{L})_{|V \setminus V_0} \quad (10.2) $$

*Proof.* Let $f: X \to V$ be a surjective morphism, $K := f_*j_*^p \mathbf{L}$ on $V$ and let $N$ denote a normally embedded subvariety of $V$ of dimension $d < \dim V$ intersecting a stratum $S$ of dimension $s > 0$ at a point $u$, such that $X_N$ intersect transversally the strata of $X$. If $X$ is non singular, then $X_N$ is also non singular for a general $N$. The induction apply to $f \circ i_{X_N}$ since we have good reductions:

$$ i_N^* K \simeq (f \circ i_{X_N})_* j_*^p \mathbf{L}, \quad i_N^* {}^p \mathcal{H}^i(K) \simeq {}^p \mathcal{H}^i((f \circ i_{X_N})_* j_*^p \mathbf{L}). \quad (10.3) $$
Remark that $i_S^t \mathbf{L}_S^i$ is uniquely defined by the decomposition as $i_S^t \mathbf{L}_S^i = i_S^* H^{-s}(\mathcal{H}^i(K))$.

Since $R^{i-s} f_{S*}(i_{X_S}^1 j_*^t \mathbf{L})$ and $R^{i-s} f_{S*}(i_{X_S}^* j_*^t \mathbf{L})$) are variation of MHS of weights $w \ge p w + i - s$ (resp. $w \le p w + i - s$), the image $i_S^t \mathbf{L}_S^i$ of $I_S$ is a PVHS of weight $w = p w + i - s$ (ch. 9, Lemma 9.7) including a reduction to the case where $X_S$ is a fibration by NCD).

The isomorphisms in (10.2) follow by induction since $\dim X_N < \dim X$.

**Example 10.1.** By definition of the big stratum $U$ in $V$, the restriction $f|: X_U \to U$ of the morphism $f$ to $X_U := f^{-1}(U)$ is smooth and the restrictions of the cohomology sheaves $\mathcal{H}^i(K)$ are locally constant on $U$. Hence: $i_S^t \mathcal{H}^i(K)|_U = \mathcal{H}^{i-\dim V}(K)|_U[\dim V]$ is a shifted polarized VHS on $U$. At a point $u \in U$: $(R^i f_* j_*^t \mathbf{L})_u \simeq \mathbb{H}^i(X_u, j_*^t \mathbf{L})$ is isomorphic to the intersection cohomology of the restriction of $\mathbf{L}$ to the fiber $X_u$, hence Hard Lefschetz apply.

# 10.1 Proof of the relative local purity

In the next step, we deduce the relative local purity at points $v \in V_0$ from the decomposition over $V \setminus V_0$ (Lemma $D(n, V^*)$ 10.1). As the statement is local at points in $V_0$, we can suppose $V_0$ reduced to one point $v$.

## 10.1.0.1 Perverse filtration by sub-MHS on the Link

Since we assume $X$ non singular, we may also assume that $i_S^t \mathbf{L}$ is a VHS on the complement of a NCD $Y \subset X$ so to apply the results of chapters 3 and 9. Let $f: X \to V$, $v \in V$ a closed point. We also assume that $X_v := f^{-1}(v)$ and $X_v \cup Y$ are NCD in $X$.

Let $i: X_v \to X$, the complex $(K', W, F) := (i^*j_{X_v^*}(j_*^t \mathbf{L})|_{(X-X_v)}, W, F)$ (ch. 9, §9.5.1, Definition 9.35) defines a structure of MHC on the space of global sections $R\Gamma(X_v, K', W, F)$.

The topological interpretation of this intrinsic cohomology space in terms of balls with center $v$ is more suggestive. A small ball $B_v \subset V$ with center $v$ is defined as the trace of a ball $B \in \mathbb{C}^N$ with center 0. The link at a point $v \in V$ is a topological invariant represented by the boundary of a ball $B_v$ of $V$. As $B_v^*$ retracts on the boundary, we use the hypercohomology of $B_v^*$.

The inductive limit of the hypercohomology when the radius of $B_v$ is small is an intrinsic group representing the hypercohomology of the link.

Let $X_v := f^{-1}(v)$, $X_{B_v} := f^{-1}(B_v)$ its tubular neighborhood and $X_{B_v^*} = X_{B_v} \setminus X_v$. There exists an isomorphism

$$ \mathbb{H}^r(X_{B_v^*}, j_*^t \mathbf{L}) \xrightarrow{\sim} \mathbb{H}^r((X_v, i^*(j_{X_v})_* j_{X_v}^* j_*^t \mathbf{L})) \quad (10.4) $$
Set $k: V \setminus \{v\} \to V, j_{X_v}: X \setminus Z \to X$ and $i_v: v \to V$. We denote again by $W$ and $F$ the filtrations $(f_{|X_v})_*W, (f_{|X_v})_*F$ on $(f_{|X_v})_*K'$. The isomorphism $(f_{|X_v})_*K' \simeq i_{v*}k_*((f_*j_{!*}^p\mathbf{L})_{|(V-v)})$, defines by transport of structure, a filtration $^{p\tau*}$ on $(f_{|X_v})_*K'$ deduced from the perverse filtration $^{p\tau}$ on $(f_*j_{!*}^p\mathbf{L})_{|(V\setminus v)}$

$$^{p\tau^*}(f_{|X_v})_*K' \simeq {}^{p\tau^*}(i_{v*}k_*(f_*j_{!*}^p\mathbf{L})_{|(V\setminus v)}) := i_{v*}k_*({}^{p\tau}(f_*j_{!*}^p\mathbf{L})_{|(V\setminus v)}) \quad (10.5)$$

Thus we have three filtrations $W, F, {}^{p\tau*}$ on the complex $(f_{|X_v})_*K'$ inducing filtrations $W, F$ and $^{p\tau*}$ on the cohomology space $\mathbb{H}^j(X_v, i^*j_{X_v*}(j_{!*}^p\mathbf{L})_{|(X\setminus X_v)}$.

**Lemma 10.2 (Compatibility with MHS).** i) With the notations of (Definition 9.35), let $K' := i^*j_{X_v*}j_{X_v*}^*p\mathbf{L}$. Under the decomposition over $V^*$ ($D(n, V^*)$) the filtration $^{p\tau*}$, induced by the filtration $^{p\tau}$ on $f_*j_{!*}^p\mathbf{L}$, is compatible with the MHS on $\mathbb{H}^j(X_v, j_{!*}^p\mathbf{L})$:

The induced filtrations $W$ and $F$ on $Gr_i^{p\tau^*}(f_{|X_v})_*K'$ define a MHS and we have an isomorphism of MHS:

$$H^j(v, Gr_i^{p\tau^*}(f_{|X_v})_*K', W, F) \simeq Gr_i^{p\tau^*} \mathbb{H}^{i+j}(X_v, K', W, F) \quad (10.6)$$

ii) Equivalently, we have an isomorphism of graded polarized MHS:

$$\mathbb{H}^j(B_v^*, {}^p\mathcal{H}^i(K), W, F) \simeq Gr_i^{p\tau} \mathbb{H}^{i+j}(B_{X_v}^*, j_{!*}^p\mathbf{L}, W, F).$$

*Proof.* The filtration $^{p\tau*}$ in the lemma corresponds to the perverse filtration $^{p\tau}$ on the hypercohomology of $B_{X_v}^*$ (Equation 11.5).

Let $B^j := \mathbb{H}^j(B_{X_v}^*, j_{!*}^p\mathbf{L}, W, F)$ with the filtration $^{p\tau}$, $B := \oplus_j B^j$ and $K := f_*j_{!*}^p\mathbf{L}$. The morphism $\eta: K \to K[2]$, defined by cup product with the Chern class $c_1$ of an ample line bundle, induces a morphism on the perverse cohomology sheaves ([BBD 83], Théorème 5.4.10).

We deduce a morphism of MHS $\eta: B \to B(1)$. Let $A$ be the following increasing filtration on $B$:

$$\begin{align}
A_i &:= \oplus_{j<-i} {}^{p\tau}_{j-1} B^j \oplus {}^{p\tau}_{-i} B^{-i} \oplus {}^{p\tau}_{j>i} B^j \nonumber \\
A_{i-1} &:= \oplus_{j<-i} {}^{p\tau}_{j-1} B^j \oplus {}^{p\tau}_{-i-1} B^{-i} \oplus {}^{p\tau}_{j>i} B^j \nonumber \\
A_{i-2} &:= \oplus_{j+2<-i+2} {}^{p\tau}_{j+1} B^{j+2} \oplus {}^{p\tau}_{-i+2} B^{-i+2} \oplus {}^{p\tau}_{j+2>i-2} B^{j+2} \tag{10.7}
\end{align}$$

$$\eta: A_i \to A_{i-2}, \eta: \mathrm{Gr}_i^A = \mathrm{Gr}_{-i}^{p\tau} B^{-i} \to \mathrm{Gr}_{i-2}^A = \mathrm{Gr}_{-i+2}^{p\tau} B^{-i+2}$$

We deduce: $\eta^i: \mathrm{Gr}_i^A = \mathrm{Gr}_{-i}^{p\tau} B^{-i} \simeq \mathrm{Gr}_i^{p\tau} B^i(i) = \mathrm{Gr}_{-i}^A$. Hence $A_i$ is the unique increasing filtration defined by the nilpotent endomorphism $\eta$ on $B$ in the category of MHS. The lemma follows by the fact that $A$ is necessarily a filtration by MHS.

Once the above MHS is defined, we can express the conditions of local purity and give a meaning to the purity theorem in [DeG 81].

We prove now (Theorem 9.2) in the case $\dim X = n$:
**Proposition 10.1 (Local purity).** The weight $w$ satisfy the inequalities :

$$
\begin{align}
w > {}^{p}w + i + j \quad \text{on} \quad H^j(B_v \setminus v, {}^p\mathcal{H}^i(f_*j_*!^p\mathbf{L})) & \text{ if } j \ge 0, \nonumber \\
w \le {}^{p}w + i + j \quad \text{on} \quad H^j(B_v \setminus v, {}^p\mathcal{H}^i(f_*j_*!^p\mathbf{L})) & \text{ if } j \le -1 \tag{10.8}
\end{align}
$$

Remark 10.1. Equivalently, the inequalities may be written as:

$$
\begin{align}
w > {}^{p}w + r \quad \text{on} \quad {}^{p}\tau_{\le r} \mathbb{H}^{r}(B_{X_v} \setminus X_v, j_{1*}^{p}\mathbf{L}), \notag \\
w \le {}^{p}w + r \quad \text{on} \quad \mathbb{H}^{r}(B_{X_v} \setminus X_v, j_{1*}^{p}\mathbf{L}) / {}^{p}\tau_{\le r} \mathbb{H}^{r}(B_{X_v} \setminus X_v, j_{1*}^{p}\mathbf{L}). \tag{10.9}
\end{align}
$$

The proof in [DeG 81] is based on the local invariant cycle theorem. Our
proof is based on duality and the polarization of the Intersection cohomology
$\mathbb{H}^*(X, j_{1*}^p \mathbf{L})$.

In the case of a variety $V$ with an isolated singularity and $X$ is its desingu-
larization, the polarization occurs also in the proof in the case of constant
coefficients by ([Na 85], Prop. 5.1).

As in [DeG 81], the proof of the inequalities below is by induction on an
hyperplane section of $X$, except the case: $Gr_{pw}^W Gr_0^{p\tau} \mathbb{H}^{-1}(B_{X_v} \setminus X_v, j_{1*}^p \mathbf{L}) = 0$
$(i=0$ and $j=-1$, equivalently by duality $j=0$ and $i=0$). The reader may
skip the induction and go directly to the crucial case (§10.2).

10.1.0.2 Duality

**Lemma 10.3.** The dual of $Gr_{p w+l}^W \mathbb{H}^{-1}(B_v \setminus v, {}^p\mathcal{H}^i(f_*j_*!^p\mathbf{L}))$ for all $i$ and $l \le i-1$ is $Gr_{p w-l}^W \mathbb{H}^0(B_v \setminus v, {}^p\mathcal{H}^{-i}(f_*j_*!^p\mathbf{L}))$ for all $i$ and $-l \ge -i+1$.

Proof. Let $k: V \setminus \{v\} \to V, K := f_*j_*!^p\mathbf{L}$ and $K(v) := i_v^*k_*k^*K$.
We have quasi-isomorphisms of derived complexes:

$$
K(v) \xrightarrow{\sim} \Gamma(B_v \setminus v, K) \xrightarrow{\sim} \Gamma(B_{X_v} \setminus X_v, j_*!^p \mathbf{L}).
$$

The duality isomorphism $D(K(v))[1] \xrightarrow{\sim} K(v)$ where $D$ stands for Verdier dual, is deduced from the auto-duality of of $j_*!^p \mathbf{L}$, Verdier's duality formula for the direct image by a proper morphism $f$ and the duality between $k_*$ and $k_!$ (resp. $i_*^!$ and $i_!^!$).

The dual of the distinguished triangle $k_!k_*^*K \to k_*k_*^*K \to i_{v_*}i_v^!k_*k_*^*K$ is
the distinguished triangle: $i_{v_*}i_v^!k_*k_*^*K \to k_!k_*^*K \to k_*k_*^*K$.

By identifying the terms of the triangles, we deduce the isomorphisms

$$
D(K(v))[1] \xrightarrow{\sim} i_{v_*} i_v^! k_! k_*^* K[1] \xrightarrow{\sim} K(v).
$$

The duality isomorphism is compatible with the weight filtration:

$$
D(Gr_{p w-q}^{W} H^{-j}(K(v))) \xrightarrow{\sim} Gr_{p w+q}^{W} H^{j}(DK(v)) \xrightarrow{\sim} Gr_{p w+q}^{W} H^{j-1}(K(v)).
$$

and corresponds to the duality of the Intesection cohomology of the boundary:
$\partial B_{X_v} = f^{-1}(\partial B_v)$ with coefficients in the restriction of $j_*!^p \mathbf{L}$.
Since the perverse filtration is compatible with duality and weight, we deduce an isomorphism: $D(p\mathcal{H}^i(K)) \xrightarrow{\sim} p\mathcal{H}^i(K)$ and

$$D(Gr_{pW-q}^W Gr_{-i}^{p\tau} H^{-j}(K(\ast v))) \xrightarrow{\sim} Gr_{pW+q}^W Gr_i^{p\tau} H^{j-1}(K(\ast v)).$$

Hence, the dual of $Gr_{pW+l}^W \mathbb{H}^j(B_v \setminus v, p\mathcal{H}^i(f_*j_*!p\mathbf{L}))$ for all $i, l \le i-1$ and $j \le -1$ is $Gr_{pW-l}^W \mathbb{H}^{j-1}(B_v \setminus v, p\mathcal{H}^{-i}(f_*j_*!p\mathbf{L}))$ for all $i, -l \ge -i+1$ and $-j-1 \ge 0$.

The proof of (Proposition 10.1) is reduced by the above duality, to one of the two cases in degree $j \ge 0$ or $j \le -1$.

### 10.1.1 Preliminaries on hyperplane sections of X

Let $H$ be a general hyperplane section of $X$ transversal to all strata of a stratification $S$ of $X$. We prove here by induction the inequalities of the equation 10.8, except the case $i=0, j=0, w=pw$ (see §10.2).

Recall that $!p\mathbf{L}$ is defined on the complement $X \setminus Y$ of a NCD $Y$.

We suppose $H \cap Y$ a NCD. Let $j: (X \setminus Y) \to X, j': H \setminus (H \cap Y) \to H$, and $i_H: H \to X$. By transversality, the restriction $i_H^*j_*!p\mathbf{L}[-1] \simeq j'_*!p\mathbf{L}|_H[-1])$ is perverse. Let $K_H = f_*i_H^*i_H^*j_*!p\mathbf{L}[-1]$.

By induction, the local purity theorem apply to the perverse cohomology sheaves of $K_H$ since $\dim H < \dim X$. We use Artin-Lefschetz vanishing theorem to deduce the local purity for the perverse cohomology sheaves of $K$ in degrees $i \ne 0$.

**Lemma 10.4.** Let $j_H: (X \setminus H) \to X$ and $K_c = f_*(j_H)!j_H^*j_*!p\mathbf{L}$.

i) The complex $K_c \in pD_V^{\ge 0}$, equivalently: $p\mathcal{H}^i(K_c) = 0$ for $i < 0$.

ii) Dually $K(\ast) := f_*(j_H)!j_H^*j_*!p\mathbf{L} \in pD_V^{\le 0}$ ($p\mathcal{H}^i(K(\ast)) = 0$ for $i > 0$).

*Proof.* As $f \circ j_H: (X \setminus H) \to V$ is affine, by a version of Artin-Lefschetz vanishing theorem, the functor $f_* \circ j_H!$ transforms $pD_{X-H}^{\ge 0}$ into $pD_V^{\ge 0}$ (the functor is left t-exact [BBD 83], corollary 4.1.2), hence $K_c \in pD_V^{\ge 0}$. The statement $p\mathcal{H}^i(K(\ast)) = 0$ for $i > 0$ follows by duality.

Let $K = f_*j_*!p\mathbf{L}$. The restriction morphism $\rho: K \to K_H[1]$, and dually the Gysin morphism $G: K_H[-1] \to K$, induce morphisms

$$\rho_i : p\mathcal{H}^i(K) \to p\mathcal{H}^{i+1}(K_H), \quad G_i : p\mathcal{H}^{i-1}(K_H) \to p\mathcal{H}^i(K), \qquad (10.10)$$

The cup product with the Chern class $c_1$ of an ample line bundle (or equivalently relative ample) defines a morphism $\eta: K \to K[2]$ inducing morphisms of perverse sheaves ([BBD 83], Théorème 5.4.10)

$$\eta_i = G_{i+2} \circ \rho_i : p\mathcal{H}^i(K) \to p\mathcal{H}^{i+2}(K)$$
**Corollary 10.1.** i) The restriction $\rho_i : {}^p\mathcal{H}^i(K) \to {}^p\mathcal{H}^{i+1}(K_H)$ is an isomorphism for each integer $i < -1$ and a monomorphism for $i = -1$.

Dually, the Gysin morphism $G_i : {}^p\mathcal{H}^{i-1}(K_H) \to {}^p\mathcal{H}^i(K)$ is an isomorphism for $i > 1$, and an epimorphism for $i = 1$.

ii) The filtrations $W$ and $F$ define a MHS for $i < 1$ on:

$$Gr_i^{p\tau} \mathbb{H}^{i+j}(B_{X_v}, j_*!\mathbf{P}\mathbf{L}, W, F) \simeq Gr_i^{p\tau} \mathbb{H}^{i+j}(B_{H_v} \setminus H_v, i_H^*j_*!\mathbf{P}\mathbf{L}[-1]).$$

We have a dual statement for $i > 1$.

**Proof.** Here monomorphism (resp. epimorphism) stands for the perverse kernel is zero (resp. the perverse cockernel is zero). We deduce from the distinguished triangle $(j_H)_!j_H^*j_*!\mathbf{P}\mathbf{L} \to j_*!\mathbf{L} \to i_{H_*}i_H^*j_*!\mathbf{P}\mathbf{L} \xrightarrow{(1)}$, and its derived direct image by $f_*$, an exact sequence of perverse cohomology ([BBD 83], Théorème 1.3.6)

$$\dots \to {}^p\mathcal{H}^i(K_c) \xrightarrow{\text{can}_i} {}^p\mathcal{H}^i(K) \xrightarrow{\rho_i} {}^p\mathcal{H}^{i+1}(K_H) \xrightarrow{\partial_i} {}^p\mathcal{H}^{i+1}(K_c) \to \dots \quad (10.11)$$

The corollary follows from Lemma 10.4 as $ {}^p\mathcal{H}^i(K_c) = 0 $ for $i < 0$.

We consider the diagrams:

$$\begin{array}{ccc}
{}^p\mathcal{H}^{-1}(K_H)(-1) & \xrightarrow{G_0} & {}^p\mathcal{H}^0(K) \xrightarrow{\rho_0} {}^p\mathcal{H}^1(K_H) \\
{}^p\mathcal{H}^{-1}(K)|_{V\setminus v} & \xrightarrow{\rho_{-1}} & {}^p\mathcal{H}^0(K_H)|_{V\setminus v} \xrightarrow{G_1} {}^p\mathcal{H}^1(K)|_{V\setminus v}(1)
\end{array}
\quad (10.12)$$

**Lemma 10.5 (Induction).** The morphism $G_0$ is injective, $\rho_0$ is an epimorphism and by induction $D(n-1)$ (§10.0.0.1)

$${}^p\mathcal{H}^0(K) \xrightarrow{\sim} \text{Im } G_0 \oplus \text{Ker } \rho_0. \quad (10.13)$$

Respectively, $\rho_{-1}$ is a monomorphism, $G_1$ is an epimorphism and by induction (Lemma D(n, V*) 10.1)

$${}^p\mathcal{H}^0(K_H)|_{V\setminus v} \xrightarrow{\sim} \text{Im } \rho_{-1} \oplus \text{Ker } G_1 \quad (10.14)$$

**Proof.** Since $\rho_0 \circ G_0 = \eta_H$ in the equation 10.12 is an isomorphism on $H$ by induction $D(n-1)$, we deduce the decomposition in Equation 10.13.

Since $G_1 \circ \rho_{-1} = \eta$ in the equation 10.12 is an isomorphism on $V \setminus v$ by induction (Lemma D(n, V*) 10.1), we deduce the decomposition in the equation 10.14.

Still we have no information on $\text{Ker } \rho_0$, for example if $f(H)$ is closed of dimension strictly less than $\text{dim } V$. This will be the crucial case in the proof.

### 10.1.1.1 Reduction to an hyperplane section

Let $H_v := X_v \cap H$, $B_v^* := B_v \setminus v$, $B_{X_v}^* := B_{X_v} \setminus X_v$ and $B_{H_v}^* := B_{X_v}^* \cap H$.
$$
\begin{equation} \tag{10.15}
\begin{split}
\mathbb{H}^j(B_v^*, {}^p\mathcal{H}^i(K), W, F) &\xrightarrow{\rho_i} \mathbb{H}^j(B_v^*, {}^p\mathcal{H}^i(K_H[1]), W, F) \\
&\xleftarrow[D_X \simeq]{\downarrow} \mathbb{H}^j(B_{X_v}^*, j_*{}^p\mathbf{L}, W, F) \\
&\xrightarrow{\rho_i} \mathbb{H}^j(B_{H_v}^*, i_*{}^p\mathbf{L}[1], W, F)
\end{split}
\end{equation}
$$

where the isomorphisms (by the decomposition) $D_X$ and $D_H$ transport the filtration $W$ and $F$ by induction over $V^*$ and $\rho_i$ respects the filtrations $W$ and $F$. This diagram will be used to check that $W$ and $F$ define MHS satisfying local purity by (including a new proof of the compatibility of $p_\tau$ with MHS) by induction on an hyperplane section $H$. We resume the notations of the proof of (Corollary 10.1).

Let $K := f_*j_*{}^p\mathbf{L}$, $K_H = f_*i_{H_*}i^*_{H_*}j_*{}^p\mathbf{L}[-1]$, $\rho: K \to K_H[1]$, and dually the Gysin morphism $G: K_H[-1] \to K$.

**Corollary 10.2.** The local purity in proposition 10.1 applies to ${}^p\mathcal{H}^i(K)$ with the MHS defined by $W$ and $F$, except eventually for the component $\text{Ker } \rho_0 \subset {}^p\mathcal{H}^0(K)$ in Equation 10.13.

*Proof.* The conditions on the weights $wH^j(B_v \setminus v, {}^p\mathcal{H}^{i+1}(K_H))$: $w > {}^pw - 1 + i + 1 + j$ if $j \ge 0$ are assumed by induction, since $K_H$ is of weight $pw - 1$.

Since for $i < -1$ the restriction morphisms $\rho_i$ are isomorphisms, the filtrations $W$ and $F$ on $H^j(B_v \setminus v, {}^p\mathcal{H}^i(K))$ define a MHS satisfying the conditions on the weight by transport of structure.

In the case $i = -1$, $\rho_{-1}$ induces an isomorphism onto a direct summand $\text{Im } \rho_{-1}$ of ${}^p\mathcal{H}^{-1}(f_*(i_H)_*i^*_{H_*}j_*{}^p\mathbf{L})|_{V-v}$ (Equation 10.14), hence the MHS is carried to $\mathbb{H}^j(B_v^*, {}^p\mathcal{H}^{-1}(K), W, F)$ and satisfy local purity. The case $i \ge 1$ follows by duality.

In the case $i = 0$, local purity holds for $\text{Im } G_0$ since $G_0$ is an isomorphism by (Lemma 10.5).

Only the case of $\text{Ker } \rho_0 \subset {}^p\mathcal{H}^0(K)$ is not deduced by induction from $K_H$.

**Lemma 10.6 (The case $i = 0, j \ne 0, -1$).** Let $j_*{}^p\mathbf{L}$ be of weight $a$. The MHS on

$$
Gr_0^{p\tau} H^j(B_{X_v}^*, j_*{}^p\mathbf{L}) \simeq H^j(B_v^*, {}^p\mathcal{H}^0(K))
$$

is of weight $\omega > {}^pw + j$ for $j > 0$ and dually of weight $\omega \le {}^pw + j$ for $j < -1$.

*Proof.* Let $H$ be a general hyperplane section of $V$ containing $v$, $H_v = B_v \cap H$ and $H_v^* = B_v^* \cap H$. Since $H$ is general, it is normally embedded outside $v$ so that the perverse cohomology commute with the restriction to $H \setminus v$.

We consider the exact sequence

$$
\mathbb{H}_{H_v^*}^j(B_v^*, {}^p\mathcal{H}^0(K)) \xrightarrow{\alpha_j} \mathbb{H}^j(B_v^*, {}^p\mathcal{H}^0(K)) \to \mathbb{H}^j(B_v \setminus H_v, {}^p\mathcal{H}^0(K))
$$
Since $\mathbb{H}_{H_v^*}^j(B_v^*, {}^p\mathcal{H}^0(K)) \simeq \mathbb{H}^{j-2}(H_v^*, {}^p\mathcal{H}^0(K))(-1)$ by Thom isomorphism (see Corollary 9.5), we deduce $w > {}^p w + j$ on $\mathbb{H}_{H_v^*}^j(B_v^*, {}^p\mathcal{H}^0(K))$ as $w > {}^p w + j - 2$ on $\mathbb{H}^{j-2}(H_v^*, {}^p\mathcal{H}^0(K))$ by induction.

By Artin-Lefshetz hyperplane section theorem, extended and applied to the Stein open subset $B_v \setminus H_v$ with coefficients in ${}^p\mathcal{H}^0(K) \in {}^p D_c^{\le 0}(V, \mathbb{Q})$: $\mathbb{H}^j(B_v \setminus H_v, {}^p\mathcal{H}^0(K)) \simeq 0$ for $j > 0$.

Then, $\alpha_j$ is an isomorphism for $j > 1$ and surjective for $j = 1$, and $w > {}^p w + j$ on $\mathbb{H}^j(B_v^*, {}^p\mathcal{H}^0(K))$ by the surjectivity of $\alpha_j$ for $j > 0$.

### 10.1.2 The large inequality: $w \ge {}^p w$ for $j = 0$

We prove by induction:

**Lemma 10.7.** i) $\mathbb{H}^0(B_v^*, {}^p\mathcal{H}^0(K))$ is of weight $w \ge {}^p w$.

ii) Dually: $\mathbb{H}^{-1}(B_v^*, {}^p\mathcal{H}^0(K))$ is of weight $w \le {}^p w$.

*Proof.* Let $H_1 \not\supseteq v$ be a general hyperplane section not containing $v$, $k : (V \setminus v) \to V$. We deduce from the distinguished triangle:

$$k_!k^* {}^p\mathcal{H}^0(K) \to k_*k^* {}^p\mathcal{H}^0(K) \to i_{v_*}i_v^*k_*k^* {}^p\mathcal{H}^0(K),$$

the exact sequence of hypercohomology on $U_1 := V \setminus H_1$

$$\mathbb{H}^0(U_1, k_*k^* {}^p\mathcal{H}^0(K)) \xrightarrow{\gamma_0} \mathbb{H}^0(i_v^*k_*k^* {}^p\mathcal{H}^0(K)) \to \mathbb{H}^1(U_1, k_!k^* {}^p\mathcal{H}^0(K)) = 0$$

where $\mathbb{H}^1(U_1, k_!k^* {}^p\mathcal{H}^0(K)) = 0$ since $U_1$ is affine ([BBD 83] Théorème 4.1.1) as $k_!k^* {}^p\mathcal{H}^0(K) \in {}^p D_c^{\le 0}$.

We deduce the lemma since, the weight $w$ of $\mathbb{H}^j(U_1, k_*k^* {}^p\mathcal{H}^0(K))$ satisfy $w \ge {}^p w$ on the open set $U_1 \setminus v$, and $\gamma_0$ is surjective.

## 10.2 The crucial case

It remains to prove:

$$Gr_{p_w}^W Gr_0^{p\tau} \mathbb{H}^{-1}(B_{X_v}^*, j_{!*} \mathbf{L}) \xrightarrow{\sim} Gr_{p_w}^W \mathbb{H}^{-1}(B_v^*, {}^p\mathcal{H}^0(K)) \xrightarrow{\sim} 0. \quad (10.16)$$

and by duality $Gr_{p_w}^W Gr_0^{p\tau} \mathbb{H}^0(B_{X_v}^*, j_{!*} \mathbf{L}) \xrightarrow{\sim} 0$.

Let $k : (V \setminus V_0) \to V$, $i_{V_0} : V_0 \to V$ and $K := Rf_{*}j_{!*} \mathbf{L}$. As the proof below is local, it applies to all isolated strata $v \in V_0$ of dimension 0 at the same time. We can suppose $V_0$ reduced to a unique stratum $i_v : v \to V$.

The proof is based on the idea that the cohomology of $B_v^*$ fits in two exact sequences associated to distinguished two triangles involving $i_v^*k_*k^*K$

$$k_!k^*K \to k_*k^*K \to i_{v_*}i_v^*k_*k^*K \text{ and } i_v^!K \to i_v^*k_*k^*K \quad (10.17)$$
assumption on the complement of $v$ or on $H$, and $l_X$ is a morphism by definition of $p_{\tau \le 0}$.

All the morphisms in the diagram are commutative as the columns represent $\rho \circ \gamma$ respectively on $X$, on $V$ and $\gamma$ on $H$.
We are interested by the level $Gr_{p_w}^W$ of all the spaces in the diagram.

Since $u \in Gr_{p_w}^W \mathbb{H}^{-1}(B_v^*)(\text{Ker } \rho_0)$ (Equation 10.20) is killed by $\rho$:
$Gr_{p_w}^W(\rho_v^* \circ r_1)(u) = 0$ on the first line. By the commutativity of the diagram:
$Gr_a^W(\rho \circ \alpha_X \circ \partial_X)(u) = Gr_{p_w}^W(\alpha_H \circ \partial_H \circ \rho_v^* \circ r_1)(u) = 0$
where the isomorphism on the last line gives the equality.

iii) To prove $\gamma(u) = 0$, we just apply Poincaré duality to the diagram

<figure><img src="image_15.png" /></figure>

where $A^*$ is the dual of $A$, $I := A^* \circ A$.

Poincaré duality is expressed by a scalar product $P$ on $\mathbb{H}^0(X, j_{1*} p_L)$ and a non-degenerate pairing $P_v$

$$P_v : Gr_{p_w}^W \mathbb{H}_{X_v}^0 (X, j_{1*} p_L) \otimes Gr_{p_w}^W \mathbb{H}^0 (X_v, j_{1*} p_L) \to \mathbb{C}.$$

while the duality between $A$ and $A^*$ is defined by the relation

$$P(Ab, c) = P_v(b, A^*c).$$

for all $b \in Gr_{p_w}^W \mathbb{H}_{X_v}^0(X, j_{1*} p_L)$ and $c \in Gr_{p_w}^W \mathbb{H}^0(X, j_{1*} p_L)$.

Let $C$ be the Weil operator defined by the $HS$ on $Gr_a^W \mathbb{H}_{X_v}^0(X, j_{1*} p_L)$, respectively on $Gr_{p_w}^W \mathbb{H}^0(X, j_{1*} p_L)$ and $u \in Gr_{p_w}^W \mathbb{H}^{-1}(B_{X_v}^*, j_{1*} p_L)$, then

$$P(C.A(\partial u), A(\partial \bar{u})) = P_v(C \partial u, A^* \circ A(\partial \bar{u})) = P_v(C.\partial(u), I(\partial \bar{u})) = 0 \quad (10.22)$$

since $I \circ \partial = 0$ by the long exact sequence defined by the second triangle in the equation 10.17. Since the polarization $Q$ on the primitive part of $\mathbb{H}^0(X, j_{1*} p_L)$ is defined by the relation $Q(a, b) := P(Ca, \bar{b})$, we deduce: $\gamma(u) = A(\partial u) = 0$ by polarization as $\gamma(u)$ is already primitive.

iv) The end of the proof of the crucial case $i = 0, j = 0, w = p_w$ is based on two lemmas.

**Lemma 10.9.** The morphism $Gr_{p_w}^W \mathbb{H}_c^0(X \setminus X_v, j_{1*} p_L) \xrightarrow{\alpha_X} \mathbb{H}^0(X, j_{1*} p_L)$ is injective.

In the long exact sequence $H^{-1}(X_v, j_{1*} p_L) \to \mathbb{H}_c^0(X \setminus X_v, j_{1*} p_L) \xrightarrow{\alpha_X} \mathbb{H}^0(X, j_{1*} p_L)$, the weight $w$ of $H^{-1}(X_v, j_{1*} p_L)$ satisfy $w < p_w$ since $X_v$ is closed, hence the morphism $\alpha_X$ in the lemma is injective since $Gr_{p_w}^W \mathbb{H}^0(X, j_{1*} p_L) = \mathbb{H}^0(X, j_{1*} p_L)$.
**Lemma 10.10.** i) The connecting morphism $\partial_X$ in the diagram 10.18 induces an injective morphism:

$$Gr_{p_w}^W Gr_0^{p_\tau} \mathbb{H}^{-1}(B_{X_v}^*, j_{1*}^p \mathbf{L}) \xrightarrow{\partial_X} Gr_{p_w}^W Gr_0^{p_\tau} \mathbb{H}_c^0(X \setminus X_v, j_{1*}^p \mathbf{L}).$$

ii) Moreover $Gr_{p_w}^W p_{\tau_0} \mathbb{H}^{-1}(B_{X_v}^*, j_{1*}^p \mathbf{L}) \xrightarrow{\partial_X} Gr_{p_w}^W Gr_0^{p_\tau} \mathbb{H}_c^0(X \setminus X_v, j_{1*}^p \mathbf{L})$ is injective.

Let $V^* := V \setminus v$. We prove first:

Sub-lemma: 1) $\mathbb{H}_c^1(V^*, p\mathcal{H}^0(K))$ is pure of weight $p_w + 1$.

2) By duality, $\mathbb{H}^{-1}(V^*, p\mathcal{H}^0(K))$ is pure of weight $p_w - 1$.

*Proof of the sub-lemma.* 1) Let $H_1$ be a general hyperplane section of $V$ not containing $v$ intersecting the strata transversally with non singular inverse image $X_1 \subset X$.

Let $k: V \setminus H_1 \to V$. The long exact sequence of hypercohomology with coefficients in $k_1k^*p\mathcal{H}^0(K)$:

$$\mathbb{H}_{H_1}^1(V, k_1k^*p\mathcal{H}^0(K)) \xrightarrow{\varphi} \mathbb{H}^1(V, k_1k^*p\mathcal{H}^0(K)) \to \mathbb{H}^1(V \setminus H_1, k_1k^*p\mathcal{H}^0(K))$$

where $\mathbb{H}^1(V \setminus H_1, k_1k^*p\mathcal{H}^0(K)) = 0$, since $V \setminus H_1$ is affine and $k_1k^*p\mathcal{H}^0(K) \in pD_c^0$ ([BBD 83] Théorème 4.1.1); hence $\varphi$ is surjective.

Since $v \notin V^*$, the decomposition apply to $X_1 \to H_1$ and to $(X \setminus X_v) \to V^*$. Then, Thom isomorphism on $X \setminus X_v$ corresponds to a Thom isomorphism

$$\mathbb{H}^{-1}(H_1, p\mathcal{H}^0(K))(-1) \xrightarrow{\sim} \mathbb{H}_{H_1}^1(V, p\mathcal{H}^0(K)) = \mathbb{H}_{H_1}^1(V, k_1k^*p\mathcal{H}^0(K))$$

of pure HS of weight $p_w + 1$. We deduce that the middle term:

$$\mathbb{H}^1(V, k_1k^*p\mathcal{H}^0(K)) = \mathbb{H}_c^1(V^*, p\mathcal{H}^0(K)) \text{ is pure of weight } p_w + 1.$$

2) By duality, $\mathbb{H}^{-1}(V^*, p\mathcal{H}^0(K))$ is pure of weight $p_w - 1$.

*Proof.* i) We deduce from the sub-lemma and the long the long exact sequence:

$$\mathbb{H}^{-1}(V^*, p\mathcal{H}^0(K)) \to \mathbb{H}^{-1}(B_v^*, p\mathcal{H}^0(K)) \xrightarrow{\partial_V} \mathbb{H}_c^0(V^*, p\mathcal{H}^0(K))$$

that the kernel of $\partial_V$ has weight $< p_w$, hence:

$$Gr_{p_w}^W \mathbb{H}^{-1}(B_v^*, p\mathcal{H}^0(K)) \xrightarrow{\partial_V} Gr_{p_w}^W \mathbb{H}_c^0(V^*, p\mathcal{H}^0(K))$$

which translates by the decomposition over $V^* := V \setminus v$ into the statement i) of the lemma.

ii) follows by the isomorphism (Equation 10.20):

$$Gr_{p_w}^W p_{\tau_0} \mathbb{H}^{-1}(B_{X_v}^*, j_{1*}^p \mathbf{L}) \simeq Gr_{p_w}^W Gr_0^{p_\tau} \mathbb{H}^{-1}(B_{X_v}^*, j_{1*}^p \mathbf{L}).$$

**Corollary 10.3.** $Gr_{p_w}^W Gr_0^{p_\tau} \mathbb{H}^{-1}(B_{X_v}^*, j_{1*}^p \mathbf{L}) = 0$.

*Proof.* Let $u \in Gr_{p_w}^W p_{\tau_0} \mathbb{H}^{-1}(B_{X_v}^*, j_{1*}^p \mathbf{L}) = Gr_{p_w}^W Gr_0^{p_\tau} \mathbb{H}^{-1}(B_{X_v}^*, j_{1*}^p \mathbf{L})$ by (Equation 10.20). We consider the sequence

$$Gr_{p_w}^W p_{\tau_0} \mathbb{H}^{-1}(B_{X_v}^*, j_{1*}^p \mathbf{L}) \xrightarrow{\partial_X} Gr_{p_w}^W p_{\tau_0} \mathbb{H}_c^0(X - X_v, j_{1*}^p \mathbf{L}) \xrightarrow{\alpha_X} \mathbb{H}^0(X, j_{1*}^p \mathbf{L})$$

Since $\gamma(u) = 0$ (Lemma 10.8) and $\alpha_X$ is injective on $Gr_{p_w}^W$ by Lemma (10.9), we deduce that

$$\partial_X u = 0 \in Gr_{p_w}^W p_{\tau_0} \mathbb{H}_c^0(B_{X_v}^*, j_{1*}^p \mathbf{L}).$$
Since the class $\partial_X u = 0 \in Gr_0^{p_\tau} Gr_{p_w}^W \mathbb{H}_c^0(B_{X_v}^*, j_{1*} p_L)$, we deduce by Lemma (10.10) $u = 0 \in Gr_{p_w}^W Gr_{p_u}^{p_\tau} \mathbb{H}_c^{-1}(B_{X_v}^*, j_{1*} p_L) \simeq Gr_{p_w}^W Gr_0^{p_\tau} \mathbb{H}_c^{-1}(B_{X_v}^*, j_{1*} p_L)$ which ends the proof of the corollary and the the crucial case.

**Example 10.2.** ([Na 85] Proposition 5.1) In the case of isolated singularity $x \in X$, let $\tilde{X}$ denote a desingularization of $X$ of dimension $n$ such that the inverse image $Z$ of $x$ is a NCD. Local purity at $x$ has interesting cohomological consequences on $Z$:

If $k < n$, the MHS on $H_x^k(X, \mathbb{Q})$ is of weight $< k$ and the morphism $H_Z^k(\tilde{X}, \mathbb{Q}) \to H^k(Z, \mathbb{Q})$ is injective.

The MHS on $H_x^n(X, \mathbb{Q})$ is of weight $< n$ and the morphism $H_Z^k(\tilde{X}, \mathbb{Q}) \to H^k(Z, \mathbb{Q})$ is an isomorphism.

If $k > n$, the MHS on $H_x^k(X, \mathbb{Q})$ is of weight $\ge k$ and the morphism $H_Z^k(\tilde{X}, \mathbb{Q}) \to H^k(Z, \mathbb{Q})$ is surjective.

## 10.3 The decomposition theorem at v

We complete the proof of the decomposition in the case of a non singular complex variety $X$, of dimension $n := \dim X$, $1 \le \dim V \le n$.

We suppose $D(n, V^*)$ satisfied on $V - V_0$ (Lemma 10.1) and the relative local purity at isolated strata in $V_0$ (Proposition 10.1). Since the proof is local, we suppose $V_0$ reduced to an isolated stratum $v \in V$.

In this section, we extend the decomposition to $v$. The case of a singular variety $X$ follows by desingularization (see §10.3.3).

### 10.3.1 Extension of the perverse cohomology across v

The description below of the extensions of perverse cohomology sheaves across $v$ is based on properties of the intermediate extension across $v$ and in general along a stratification ([BBD 83], 2.1.11 and ch.2, §2.2).

**Lemma 10.11.** Let $S$ be a fixed stratum of $V$ of dimension $l$, $L'$ a local system on $S$, $p_L' := L'[l]$, $i_v : v \to V$ a closed point in the closure of $S$, $k : (V \setminus v) \to V$ and $j : S \to V$. $i$)

1. We have a short exact sequence of perverse sheaves

$$0 \to j_{1*} p_L' \to p\mathcal{H}^0(k_* k^* j_{1*} p_L') \xrightarrow{h} R^0 k_* k^* j_{1*} p_L' \to 0$$

2. $p\mathcal{H}^i(k_* k^* j_{1*} p_L') = 0$ if $i < 0$, $p\mathcal{H}^i(k_* k^* j_{1*} p_L') = R^i k_* k^* j_{1*} p_L'$ if $i > 0$.

In particular: $H^0(p\mathcal{H}^0(k_* k^* j_{1*} p_L')) \xrightarrow{\sim} R^0 k_*(k^* j_{1*} p_L')$.
ii) Let $j_! \mathbf{p}\mathbf{L}^j, j \in \mathbb{Z}$, be a family of intermediate extensions of local systems on $S$ and let $K' := \oplus_j j_! \mathbf{p}\mathbf{L}^j [-j]$ denote their direct sum.
We have short exact sequences of perverse sheaves for $i \in \mathbb{Z}$

$$0 \to j_! \mathbf{p}\mathbf{L}^i \to {}^p\mathcal{H}^i(k_* k^* K') \xrightarrow{h} {}^p\mathcal{H}^i(k_* k^* {}^p\mathbf{L}^j) \to 0 \quad (10.23)$$

where the last term $R^i k_* k^* {}^p\mathbf{L}^j$ is a sum of vector spaces supported on $v$ in degree zero.

iii) There exist isomorphisms of the cohomology in degree 0

$$H^0({}^p\mathcal{H}^i(k_* k^* K')) \simeq R^i k_* k^* {}^p\mathbf{L}^j \xrightarrow{\sim} \oplus_{j \le i} R^i k_* (k^* j_! \mathbf{p}\mathbf{L}^j [-j]). \quad (10.24)$$

iv) A morphism of perverse sheaves $\varphi: \mathcal{P} \to {}^p\mathcal{H}^i(k_* k^* K')$ factors through $j_! \mathbf{p}\mathbf{L}^i$ if and only if $h \circ \varphi = 0$.

**Proof.** We recall that $S$ is a non singular locally closed subvariety but $V$ may be singular along $S$.

i) The perverse long exact sequence defined by the distinguished triangle:

$$\tau_{\le -1} k_* k^* j_! \mathbf{p}\mathbf{L}' \to k_* k^* j_! \mathbf{p}\mathbf{L}' \to \tau_{\ge 0} k_* k^* j_! \mathbf{p}\mathbf{L}' \quad (10.25)$$

is reduced to the statement in i) since: $j_! \mathbf{p}\mathbf{L}' := \tau_{\le -1} k_* k^* j_! \mathbf{p}\mathbf{L}'$ is perverse, ${}^p\mathcal{H}^i(j_! \mathbf{p}\mathbf{L}') = 0$ for $i \ne 0$ and $\tau_{\ge 0} k_* k^* j_! \mathbf{p}\mathbf{L}'$ is supported by $v$.

ii) The statement follows from the case of a unique local system in i) since

$${}^p\mathcal{H}^i(k_* k^* K') = {}^p\mathcal{H}^i(k_* k^* {}^p\mathbf{L}^j) \simeq \oplus_{j \le i} {}^p\mathcal{H}^{i-j}(k_* k^* j_! \mathbf{p}\mathbf{L}^j)$$

as for $j > i$: ${}^p\mathcal{H}^i(k_* k^* j_! \mathbf{p}\mathbf{L}^j [-j]) = {}^p\mathcal{H}^{i-j}(k_* k^* j_! \mathbf{p}\mathbf{L}^j) = R^{i-j} k_* k^* j_! \mathbf{p}\mathbf{L}^j$.

The statements iii) and iv) are clear.

Instead of the triangle in equation 10.25, we could refer to the distinguished triangle

$$i_{v_*} i_v^! j_! \mathbf{p}\mathbf{L}' \to j_! \mathbf{p}\mathbf{L}' \to R k_* k^* j_! \mathbf{p}\mathbf{L}' \xrightarrow{[1]} \quad (10.26)$$

since in the statement i): $R^0 k_* k^* j_! \mathbf{p}\mathbf{L}' \simeq H^1(i_{v_*} i_v^! j_! \mathbf{p}\mathbf{L}')$.

**Remark 10.2.** [Dual statements for $k_* k^*$] We deduce from the long exact sequence of perverse cohomology defined by the triangle dual to (10.26):

$$k_* k^* j_! \mathbf{p}\mathbf{L}' \to j_! \mathbf{p}\mathbf{L}' \to i_{v_*} i_v^* j_! \mathbf{p}\mathbf{L}' \xrightarrow{[1]} \quad (10.27)$$

i) A short exact sequence:

$$0 \to i_{v_*} H^{-1}(i_v^* j_! \mathbf{p}\mathbf{L}') \to {}^p\mathcal{H}^0(k_* k^* j_! \mathbf{p}\mathbf{L}') \to j_! \mathbf{p}\mathbf{L}' \to 0$$

${}^p\mathcal{H}^i(k_* k^* j_! \mathbf{p}\mathbf{L}') = i_{v_*} H^{i-1}(i_v^* j_! \mathbf{p}\mathbf{L}')$ for $i < 0$, ${}^p\mathcal{H}^i(k_* k^* j_! \mathbf{p}\mathbf{L}') = 0$ for $i > 0$.
Moreover: $i_{v_*} H^{-1}(i_v^* j_! \mathbf{p}\mathbf{L}') \xrightarrow{\sim} H^0(i_v^* {}^p\mathcal{H}^0(k_* k^* j_! \mathbf{p}\mathbf{L}'))$.

ii) ${}^p\mathcal{H}^i(k_* k^* K') = \oplus_{j \ge i} {}^p\mathcal{H}^{i-j}(k_* k^* j_! \mathbf{p}\mathbf{L}^j)$ and we have short exact sequences
$$0 \to \bigoplus_{j \ge i} i_{v_*} H^{i-j-1} (i_v^* j_{j_*} {}^p \mathbf{L}^j) \xrightarrow{h'} {}^p \mathcal{H}^i (k_* k^* K') \to j_{j_*} {}^p \mathbf{L}^i \to 0$$

where the first term is supported on v in degree zero.

iii) A morphism of perverse sheaves $\varphi$ defined on ${}^p\mathcal{H}^i(k_*k^*K')$ factors through $j_{j_*} {}^p\mathbf{L}^i$ if and only if $\varphi \circ h' = 0$.

### 10.3.2 Proof of the decomposition at v

The proof of the decomposition at $v$ in the case $\dim X = n$ and $X_v$ is a NCD follows from $D(n, V^*)$ (Lemma 10.1). The result is a consequence of the properties of the cohomology of the NCD above $v$ following the relative local purity (Proposition 10.1).

**Theorem 10.1.** With the Hypothesis of §10.0.2, let:

$$K := f_* j_{j_*} {}^p \mathbf{L}, k : V^* \to V, i_v : \{v\} \to V.$$

i) The decomposition of ${}^p\mathcal{H}^i(k^*K)$ (Equations 10.1 and 10.2) extends into a decomposition over $V$, including the term ${}^p\mathbf{L}_v^i$ (ch. 1, Equation 1.7)

$${}^p\mathcal{H}^i(K) \simeq {}^p\mathbf{L}_v^i \oplus k_{j_*} k^* {}^p\mathcal{H}^i(K) \quad (10.28)$$

ii) Hard Lefschetz: Let $\eta : {}^p\mathcal{H}^i(K) \to {}^p\mathcal{H}^{i+2}(K)$ denote the cup-product with the class of a relative hyperplane section of $X$. The Lefschetz isomorphisms $k^*\eta^i$ on the open subset $V \setminus v$, extend to Lefschetz isomorphisms $\eta^i$ on $V$:

$$k^* \eta^i : {}^p\mathcal{H}^{-i}(K)|_{V \setminus v} \xrightarrow{\sim} {}^p\mathcal{H}^i(K)|_{V \setminus v} \implies \eta^i : {}^p\mathcal{H}^{-i}(K) \xrightarrow{\sim} {}^p\mathcal{H}^i(K) \quad (10.29)$$

*Proof.* Given the distinguished triangle $i_{v_*} i_v^! K \xrightarrow{\alpha} K \xrightarrow{\rho} k_* k^* K \xrightarrow{[1]}$, let

$${}^p\mathcal{H}^{i-1}(k_*k^*K) \xrightarrow{p\delta^{i-1}} {}^p\mathcal{H}^i(i_{v_*}i_v^!K) \xrightarrow{p\alpha^i} {}^p\mathcal{H}^i(K) \xrightarrow{p\rho^i} {}^p\mathcal{H}^i(k_*k^*K) \xrightarrow{p\delta^i} \quad (10.30)$$

be the associated long exact sequence of perverse cohomology on $V$

**Lemma 10.12.** Under the decomposition $k^* {}^p\mathcal{H}^i(K)[-i] \simeq \bigoplus_{S \subset V \setminus v} i_{S_{j_*}} {}^p\mathbf{L}_S^i$ on $V \setminus v$. The perverse image $\text{Im} {}^p\rho^i$ (Equation 10.30) is isomorphic to:

$$\text{Im} {}^p\rho^i \xrightarrow{\sim} k_{j_*} k^* {}^p\mathcal{H}^i(K) \xrightarrow{\sim} \bigoplus_S i_{S_{j_*}} {}^p\mathbf{L}_S^i.$$

*Proof of the lemma.* We use the description of the intermediate extension of a perverse sheaf $P$ on $V \setminus v$ in terms of the functor $k_!$ (extension by 0) and the derived direct image $k_* : k_{j_*} P = \text{Im} ({}^p\mathcal{H}^0(k_! P) \to {}^p\mathcal{H}^0(k_* P))$ is the image in the abelian category of perverse sheaves ([BBD 83], section 2.1.7).

We fix a Whitney stratification $S$ of $V$, morphisms $\sigma_j : k^* {}^p\mathcal{H}^j(K) \to k^* K[j]$ and a decomposition $k^* K \simeq \bigoplus_j k^* {}^p\mathcal{H}^j(K)[-j]$ on $V \setminus v$.

We apply (Lemma 10.11) in the case of a direct sum of local systems over all strata $S \neq v$ of to prove
1) There exists a monomorphism: $k_{1*}k^*p\mathcal{H}^i(K) \subset p\mathcal{H}^i(k_*k^*K)$:

The morphism of functors $\phi: k_1 \to k_*$ induces the following morphisms

$$k_{1*} k^* p\mathcal{H}^i(K) = \mathrm{Im} \left( p\mathcal{H}^0(k_1 k^* p\mathcal{H}^i(K)) \xrightarrow{p\phi_i^i} p\mathcal{H}^0(k_* k^* p\mathcal{H}^i(K)) \right) = \\ \oplus_j \mathrm{Im} \left( p\mathcal{H}^i(k_1 p\mathcal{H}^j(k^*K)[-j]) \xrightarrow{p\phi_i^j} p\mathcal{H}^i(k_* p\mathcal{H}^j(k^*K)[-j]) \right) = \\ \mathrm{Im} \left( p\mathcal{H}^i(k_1 k^* K) \xrightarrow{p\phi_i} p\mathcal{H}^i(k_* k^* K) \right) \subset p\mathcal{H}^i(k_* k^* K)$$

The first equality follows by definition of $k_{1*}$. The second equality follows as in the direct sum over $j$, only the term for $j=i$ is significant since:

$p\mathcal{H}^i(k_1 k^* i_{S_{1*}} p\mathcal{L}_S^j[-j]) = 0$ for $j < i$, and $p\mathcal{H}^i(k_* k^* i_{S_{1*}} p\mathcal{L}_S^j[-j]) = 0$ for $j > i$. The last equality follows from the decomposition of $k^*K$ (Equations 10.1 and 10.2).

2) $k_{1*}k^*p\mathcal{H}^i(K) \subset \mathrm{Im} p\rho^i$:

We deduce from the (non canonical) morphism $\sigma_i: k^*p\mathcal{H}^i(K) \to k^*K[i]$ on $V \setminus v$ and from the composition of morphisms $k_1 k^* p\mathcal{H}^i(K) \to p\mathcal{H}^i(K) \to k_* k^* p\mathcal{H}^i(K)$:

$$p\mathcal{H}^0(k_1 k^* p\mathcal{H}^i(K)) \xrightarrow{a} p\mathcal{H}^i(K) \xrightarrow{p\mathcal{H}^0(\rho^i)} p\mathcal{H}^0(k_* k^* p\mathcal{H}^i(K)) \xrightarrow{\sigma_i^0} p\mathcal{H}^0(k_* k^* K[i])$$

where $p\phi_i^i = p\mathcal{H}^0(\rho^i) \circ a, p\rho^i = \sigma_i^0 \circ p\mathcal{H}^0(\rho^i)$ since $p\mathcal{H}^0(k_* k^* K[i]) = p\mathcal{H}^i(k_* k^* K)$ and $\sigma_i^0$ is an isomorphism by the decomposition hypothesis on $V \setminus v$. Hence:

$$k_{1*} k^* p\mathcal{H}^i(K) = \sigma_i^0 (\mathrm{Im} p\phi_i^i) \subset \mathrm{Im}(p\rho^i) \subset p\mathcal{H}^0(k_* k^* K[i])$$

3) $\mathrm{Im} p\rho^i = k_{1*} k^* p\mathcal{H}^i(K) = \oplus_{S_i \subset V \setminus v} i_{S_{1*}} p\mathcal{L}_S^i$:

By (Lemma 10.11 iv), it is enough to prove that the induced morphism below $\rho_0^i := H^0(i_v^* p\rho^i)$ vanish in degree 0:

**Lemma 10.13.** The morphism $\rho_0^i$ induced by $i_v^* p\rho^i$ on the cohomology in degree zero

$$H^0(i_v^* p\mathcal{H}^i(K)) \xrightarrow{\rho_0^i=0} H^0(i_v^* p\mathcal{H}^i(k_* k^* K)) = R^i k_* (k^* p_{\tau \le i} K)$$

vanish.

*Proof of the lemma.* This result involves the intricate relation between decomposition and local purity. The equality with the last term on the right follows by (Lemma 10.11 iii) and (Equation 10.24) applied to various strata.

i) Let $B_v \subset V$ be a small ball with center $v$. We prove:

$$H^0(i_v^* p\mathcal{H}^i(K)) = \mathbb{H}^0(B_v, p\mathcal{H}^i(K)) \simeq \mathbb{H}^0(B_v, p_{\tau \le 0}(K[i])) = \mathbb{H}^i(B_v, p_{\tau \le i} K) \quad (10.31)$$

Indeed, we deduce from the distinguished triangle:

$$p_{\tau < 0}(K[i]) \to p_{\tau \le 0}(K[i]) \to p\mathcal{H}^i(K) \xrightarrow{[1]}$$

an exact sequence:
$\mathbb{H}^0(B_v, \mathcal{P}_{\tau<0}(K[i])) \to \mathbb{H}^0(B_v, \mathcal{P}_{\tau\le0}K[i]) \to \mathbb{H}^0(B_v, \mathcal{P}^i(K)) \to \mathbb{H}^1(B_v, \mathcal{P}_{\tau<0}K[i])$
where the two terms $\mathbb{H}^r(B_v, \mathcal{P}_{\tau<0}(K[i])) = 0$ for $r = 0, 1$ by definition of $\mathcal{P}_{\tau<0}$.

ii) Since $(R^i k_*(k^* \mathcal{P}_{\tau \le i} K))_v = \mathbb{H}^i(B_v \setminus v, \mathcal{P}_{\tau \le i} K) = \mathcal{P}_{\tau \le i} \mathbb{H}^i(B_{X_v} \setminus X_v, j_* \mathbf{P}_L)$ by the decomposition of the restriction of $K := f_* j_* \mathbf{P}_L$ to $V \setminus v$, the proof of the lemma is reduced to the following interpretation in terms of $B_v \subset V$ and $B_{X_v} := f^{-1}(B_v)$:

$$ \mathbb{H}^i(B_v, \mathcal{P}_{\tau \le i} K) \xrightarrow{\rho_0^i=0} \mathcal{P}_{\tau \le i} \mathbb{H}^i(B_{X_v} \setminus X_v, j_* \mathbf{P}_L) \quad (10.32) $$

We lift the problem to $B_{X_v}$ and deduce a factorization of the morphism $\rho_0^i$ through the morphism $\rho'$ below:

$$ \mathbb{H}^i(B_v, \mathcal{P}_{\tau \le i} K) \to \mathcal{P}_{\tau \le i} \mathbb{H}^i(B_{X_v}, j_* \mathbf{P}_L) \xrightarrow{\rho'} \mathcal{P}_{\tau \le i} \mathbb{H}^i(B_{X_v} \setminus X_v, j_* \mathbf{P}_L) $$

where $\mathcal{P}_{\tau \le i} \mathbb{H}^i(B_{X_v}, j_* \mathbf{P}_L) := \mathrm{Im}(\mathbb{H}^i(B_v, \mathcal{P}_{\tau \le i} K) \to \mathbb{H}^i(B_{X_v}, j_* \mathbf{P}_L))$.

Any element $u$ of $\mathbb{H}^i(B_{X_v}, j_* \mathbf{P}_L) \simeq \mathbb{H}^i(X_v, j_* \mathbf{P}_L)$ is of weight $w \le p_w + i$ since $X_v$ is closed (§9.5.0.1, Corollary 9.4), while by the relative local purity (§9.1.0.3, Theorem 9.2) and (Proposition 10.1):

The weight $w$ of $\mathcal{P}_{\tau \le i} \mathbb{H}^i(B_{X_v} \setminus X_v, j_* \mathbf{P}_L)$ satisfy: $w > p_w + i$

hence $\rho'(u) = 0$ and $\rho_0^i = 0$, which ends the proof of the lemmas 10.13 and 10.12.

**Remark 10.3.** The space $\mathcal{P}_{\tau \le i} \mathbb{H}^i(B_{X_v}, j_* \mathbf{P}_L)$ is a sub-MHS. In the above proof this result is not needed yet and will be proved later (Lemma 10.17). However, there exists an independent proof (Proposition 11.3).

**Lemma 10.14.** Let $\mathbf{P}_L^i := \mathrm{Im}(\mathbb{H}_{X_v}^i(X, j_* \mathbf{P}_L) \to \mathbb{H}^i(X_v, j_* \mathbf{P}_L))$, then: $\mathrm{Im} \mathcal{P}^i \simeq i_* \mathbf{P}_L^i$.

*Proof.* By (Equation 10.30): $\mathrm{Ker} \mathcal{P}^{\delta^{i-1}} = \mathrm{Im} \mathcal{P}^{\rho^{i-1}} = \oplus_{S \subset V \setminus v} i_{S_*} \mathbf{P}_S^{i-1}$ and by (Equation 10.23), we extract from the exact sequence (Equation 10.30) the sub-exact sequence of perverse sheaves :

$$ 0 \to \mathbb{R}^{i-1} k_* k^* (\mathcal{P}_{\tau \le i-1} K) \xrightarrow{\mathcal{P}^{\delta^{i-1}}} \mathbb{H}_{X_v}^i(X, j_* \mathbf{P}_L) \xrightarrow{\mathcal{P}^{\alpha^i}} \mathbb{P}^i \mathbb{H}^i(K) \xrightarrow{\mathcal{P}^{\rho^i}} \quad (10.33) $$

where the perverse cohomology $\mathbb{P}^i(i_v^! K)$ of the derived complex of vector spaces $i_v^!(K)$ coincides with its cohomology $\mathbb{H}_{X_v}^i(X, j_* \mathbf{P}_L)$ and $\mathcal{P}^{\delta^{i-1}}$ induces an isomorphism onto $\mathrm{Ker} \mathcal{P}^{\alpha^i}$. On the other hand, by definition, $\mathbf{P}_L^i$ is the image of $\mathcal{I}^i$ in the exact sequence

$$ \mathbb{H}^{i-1}(i_v^* k_* k^* K) \xrightarrow{\delta^{i-1}} \mathbb{H}_v^i(V, K) \xrightarrow{\mathcal{I}^i} \mathbb{H}^i(i_v^* K) \xrightarrow{\rho^i} \mathbb{H}^i(i_v^* k_* k^* K). \quad (10.34) $$

If we prove $\mathrm{Im} \mathcal{P}^{\delta^{i-1}} = \mathrm{Im} \mathcal{P}^{\rho^{i-1}}$ up to the isomorphism $\mathbb{H}_v^i(V, K) \simeq \mathbb{H}_{X_v}^i(X, j_* \mathbf{P}_L)$ in (10.34) and (10.33), we deduce the result in the form of an exact sequence

$$ 0 \to i_{v,*} \mathbf{P}_v^i \xrightarrow{\mathcal{P}^{\alpha^i}} \mathbb{P}^i \mathbb{H}^i(K) \xrightarrow{\mathcal{P}^{\rho^i}} \oplus_{S \subset V \setminus v} i_{S_*} \mathbf{P}_S^i \to 0. \quad (10.35) $$
Again, by the decomposition theorem on $V \setminus v$, we have:

$R^{i-1}k_*k^*p_{\tau \le i-1}K = \mathbb{H}^{i-1}(B_v \setminus v, p_{\tau \le i-1}K) = p_{\tau \le i-1} \mathbb{H}^{i-1}(B_{X_v} \setminus X_v, j_*!p_L)$ in (Equation 10.33) and since $H^{i-1}(i_v^*k_*k^*K) = \mathbb{H}^{i-1}(B_{X_v} \setminus X_v, j_*!p_L)$ in (Equation 10.34), we need to prove:

**Sub-lemma.** $\delta^{i-1}(p_{\tau \le i-1} \mathbb{H}^{i-1}(B_{X_v} \setminus X_v, j_*!p_L)) = \delta^{i-1}(\mathbb{H}^{i-1}(B_{X_v} \setminus X_v, j_*!p_L))$ The proof is based again on the relative local purity theorem since then, the quotient space $\mathbb{H}^{i-1}(B_{X_v} \setminus X_v, j_*!p_L)/p_{\tau \le i-1}$ has weight $w < pw + i$, while $\mathbb{H}_{X_v}^i(X, j_*!p_L)$ has weight $w \ge pw + i$ (ch. 9, Corollary 9.4), hence $p^\delta \delta^{i-1}$ and $\delta^{i-1}$ have the same image in $\mathbb{H}_{X_v}^i(X, j_*!p_L)$, which ends the proof of (Lemma 10.14).

*Proof of the splitting in (Equation 10.28:* Dually, we introduce the distinguished triangle $k_!k^*K \xrightarrow{\beta} K \xrightarrow{\gamma} i_{v_*}i_v^*K$, its exact sequence

$$ p\mathcal{H}^i(k_!k^*K) \xrightarrow{p\beta_i} p\mathcal{H}^i(K) \xrightarrow{p\gamma_i} p\mathcal{H}^i(i_{v_*}i_v^*K) \xrightarrow{p\delta_i} p\mathcal{H}^{i+1}(k_!k^*K) \quad (10.36) $$

and the sequence of morphisms: $\mathbb{H}_v^i(V, K) \xrightarrow{p\alpha_i} p\mathcal{H}^i(K) \xrightarrow{p\gamma_i} \mathbb{H}^i(i_v^*K)$.

**Lemma 10.15.** The restriction $p_{\gamma_i} : \mathrm{Im} p\alpha^i \xrightarrow{\sim} \mathrm{Im} p\gamma_i$ is an isomorphism and the perverse cohomology $p\mathcal{H}^i(K)$ splits on $V$ into a direct sum:

$$ \begin{align} p\mathcal{H}^i(K) &\xrightarrow{\sim} \mathrm{Im} p\alpha^i \oplus \mathrm{Im} p\beta_i = p\mathbb{L}_v^i \oplus k_*! p\mathcal{H}^i(K) = \mathrm{Ker} p\rho^i \oplus \mathrm{Im} p\beta_i \nonumber \\ \mathrm{Ker} p\gamma_i &= \mathrm{Im} p\beta_i := \mathrm{Im} (p\beta_i : p\mathcal{H}^i(Rk_*!K) \to p\mathcal{H}^i(K)) \simeq k_*! p\mathcal{H}^i(K) \nonumber \\ \mathrm{Ker} p\rho^i &\simeq \mathrm{Im} p\alpha^i \simeq \mathrm{Im} p\gamma_i \circ p\alpha^i \simeq \mathrm{Im} I^i := i_{v_*} p\mathbb{L}_v^i \tag{10.37} \end{align} $$

By (Remark 10.2) and by an argument dual to (Lemma 10.13), we deduce that $p\beta_i$ factors through the morphism $s_i$ in the sequence:

$$ p\mathcal{H}^i(k_!k^*K) \to \bigoplus_{S \subset V \setminus v} i_{S_*} p\mathbb{L}_S^i \xrightarrow{s_i} p\mathcal{H}^i(K) \xrightarrow{p\rho^i} \mathrm{Im} p\rho^i \subset p\mathcal{H}^i(k_*!K) \quad (10.38) $$

such that $p\rho^i \circ s_i$ induces an isomorphism onto $\mathrm{Im} p\rho^i = \bigoplus_{S_l \subset V \setminus v} i_{S_*} p\mathbb{L}_S^i$. Hence $s_i$ defines a section of $p\rho^i$ in (Equation 10.38).

Since $\mathrm{Im} p\alpha^i = \mathrm{Ker} p\rho^i$ in the exact sequence (Equation 10.30), we deduce the splitting. Moreover, we deduce from the relation $\mathrm{Ker} p\gamma_i = \mathrm{Im} p\beta_i$ (Equation 10.36) that $p\gamma_i$ induces an isomorphism:

$$ \mathrm{Im} p\alpha^i \xrightarrow{\sim} \mathrm{Im} p\gamma_i \circ p\alpha^i := i_{v_*} p\mathbb{L}_v^i \subset H^i(i_v^*K). $$

*Remark 10.4.* We extract from the sequence (Equation 10.30) an exact subsequence

$$ \begin{align} 0 &\to \bigoplus_{S \subset V \setminus v, i-j>0} R^{i-j-1} k_*! k^* i_{S_*} p\mathbb{L}_S^j \xrightarrow{p^{\delta^{i-1}}} \mathbb{H}_v^i(X, K) \xrightarrow{p^{\alpha^i}} i_{v_*} p\mathbb{L}_v^i \to 0 \tag{10.39} \\ \mathrm{Im} p^{\delta^{i-1}} &\simeq \bigoplus_{S_l \subset V \setminus v, i-j>0} H_v^{i-j} (i_{S_*} p\mathbb{L}_S^j) \simeq \mathrm{Ker} p^{\alpha^i} \simeq \mathrm{Ker} I^i \simeq \mathrm{Im} \delta^{i-1} \end{align} $$