### 10.3.2.1 Hard Lefschetz

The duality between $^{p}\mathcal{H}^i(K)$ and $^{p}\mathcal{H}^{-i}(K)$ follows from the auto-duality of $j_*^{\bullet}\mathcal{L}$ by the general statement of Verdier duality for proper morphisms.

**Lemma 10.16 (Hard Lefschetz).** By induction, let the restriction of the iterated cup-product $\eta^i : {}^p\mathcal{H}^{-i}(K) \to {}^p\mathcal{H}^i(K)$ for $i \ge 0$ to $V \setminus v$ be an isomorphism and set

$$ {}^{p}\mathcal{L}_{v}^{i} := \mathrm{Im} (\mathbb{H}_{X_{v}}^{i}(X, j_{*}{}^{p}\mathcal{L}) \xrightarrow{I^{i}} \mathbb{H}^{i}(X_{v}, j_{*}{}^{p}\mathcal{L})). $$

i) The HS on ${}^{p}\mathcal{L}_v^i$ is Poincaré dual to ${}^{p}\mathcal{L}_v^{-i}$ for $i \ge 0$ and the iterated cup-product induces isomorphisms $\eta^i : {}^{p}\mathcal{L}_v^{-i} \to {}^{p}\mathcal{L}_v^i$ for $i > 0$.

ii) The morphism $\eta^i$ for $i \ge 0$ extends across $v$ to an isomorphism on $V$.

Since the restriction of $\eta^i$ on $k^*{}^p\mathcal{H}^{-i}(K)$ to $V \setminus v$ is an isomorphism, it remains an isomorphism on the intermediate extension $k_{\bullet*}k^*{}^p\mathcal{H}^i(K)$ across the point $v \in V_0$. Hence it remains to prove Hard Lefschetz on the family of components ${}^{p}\mathcal{L}_v^i$. i) The duality between ${}^{p}\mathcal{L}_v^i$ and ${}^{p}\mathcal{L}_v^{-i}$ is deduced from Verdier duality between the functors $i_{X_v}^*$ and $i_{X_v}^*$ in the definition of ${}^{p}\mathcal{L}_v^i$.

The proof is by induction on a general hyperplane section $H$ transversal to the NCD $Y$ and $X_v$. Let $i_H$ denote the closed embedding of $H$ in $X$; the morphism $\eta$ is equal to the composition of Gysin and restriction morphisms:

$$ j_{*}{}^{p}\mathcal{L} \xrightarrow{\rho} i_{H*}i_{H}^{*}{}^{p}\mathcal{L} \xrightarrow{G} j_{*}{}^{p}\mathcal{L}[2]. $$

We consider the induced morphisms on perverse cohomology. By (Cohol-ary 10.1) it remains to consider the case $i = 1$ in i). Let

$$ {}^{p}\mathcal{L}(H)^{0}_{v} := \mathrm{Im}(\mathbb{H}_{X_{v} \cap H}^{0}(X, j_{*}{}^{p}\mathcal{L}[-1])) \xrightarrow{I^{0}} \mathbb{H}^{0}(X_{v} \cap H, j_{*}{}^{p}\mathcal{L}[-1])) $$

be a polarized HS by induction on $H$ and set $K_H = f_*i_{H*}i_H^*j_*{}^p\mathcal{L}[-1]$. Recall the diagram

$$ {}^{p}\mathcal{H}^{-1}(K) \xrightarrow{\rho_{-1}} {}^{p}\mathcal{H}^{0}(K_H) \xrightarrow{G_1} {}^{p}\mathcal{H}^{1}(K)(1) \quad (10.40) $$

where $\rho_{-1}$ is a monomorphism, $G_1$ an epimorphism. We consider the sequence

$$ {}^{p}\mathcal{L}_{v}^{-1} \xrightarrow{\rho_{-1}} {}^{p}\mathcal{L}(H)^{0}_{v} \xrightarrow{G_{1}} {}^{p}\mathcal{L}_{v}^{1}(1). $$

By induction ${}^{p}\mathcal{L}(H)^{0}_{v}$ is a polarized HS, while ${}^{p}\mathcal{L}_{v}^{-1}$ and ${}^{p}\mathcal{L}_{v}^{1}(1)$ are HS. Hence, the image $\mathrm{Im}\rho_{-1}$ is a polarized sub-HS of ${}^{p}\mathcal{L}(H)^{0}_{v}$ and by duality ker $G_1$ is the sub-HS orthogonal to $\mathrm{Im}\rho_{-1}$.

It follows that $\eta$ is injective and by duality surjective. The lemma follows since the isomorphism extends at $v$ on the intermediate extension $k_{\bullet*}k^*{}^p\mathcal{H}^i(K)$. Recall that this concept of proof has been used for the first time by Deligne in (Weil II [De 80], théorème 4.1.1).

**Lemma 10.17 (Compatibility with MHS).** i) The filtration ${}^{p}\tau^*$ on $\mathbb{H}^j(X_v, j_*{}^{p}\mathcal{L})$ induced by the filtration ${}^{p}\tau$ on $f_*j_*{}^{p}\mathcal{L}$ is a filtration by mixed Hodge sub-structures.
ii) Equivalently the induced filtrations W and F on $i_v^* p\mathcal{H}^i(f_*j_*p\mathbf{L}), W, F)$ define a graded polarized MHS.

*Proof.* The filtration is defined as in (§10.1.0.1) on the link. The proof is also similar to (Lemma 10.2). Let $B^j := \mathbb{H}^j(B_{X_v}, j_*p\mathbf{L}, W, F)$ with the same definition of the filtration $p_\tau$ on $B := \oplus_j B^j$. Then $\eta: B \to B(1)$ defined by cup-product with the hyperplane class is a morphism of MHS. We define an increasing filtration $A$ on $B$ by the same (Equation 10.7). We deduce: $\eta^i: \mathrm{Gr}_i^A = \mathrm{Gr}_{-i}^{p_\tau}B^{-i} \simeq \mathrm{Gr}_i^{p_\tau}B^i = \mathrm{Gr}_{-i}^A$. Hence $A_i$ is the unique increasing filtration defined by the nilpotent endomorphism $\eta$ on the MHS on $B$. Then $A$ is necessarily a filtration by MHS.

**Corollary 10.4 (Splitting of F).** *The decomposition (Equation 10.28) is compatible with the induced filtrations F:*

$$ (i_{v,*} p\mathbf{L}_v^i, F) \oplus (k_{l_*} k^* p\mathcal{H}^i(K), F) \xrightarrow{\varphi} (p\mathcal{H}^i(K), F) \quad (10.41) $$

The induced filtration $F$ on $i_{v,*} p\mathbf{L}_v^i$ is a HS embedded in the MHS on $i_v^* p\mathcal{H}^i(f_*j_*p\mathbf{L}), W, F)$ and the decomposition is in the category of MHS.

*Example 10.3.* In the case of the desingularization $f: X \to V$ of an isolated singularity $v \in V$, the decomposition is described in (Example 10.2) in terms of $H_{X_v}^i(X, \mathbb{Q}[\dim X])$ for $i < 0$ and $H^i(X_v, \mathbb{Q}[\dim X])$ for $i \ge 0$, since the morphism $H_{X_v}^i(X, \mathbb{Q}[\dim X]) \to H^i(X_v, \mathbb{Q}[\dim X])$ is injective for $i < 0$, surjective for $i > 0$ and an isomorphism for $i = 0$.

**Corollary 10.5.** *The perverse filtration $p_\tau$ on $(\mathbb{H}^{i+j}(X, j_*p\mathbf{L}), F)$ is compatible with Hodge filtration and the following decomposition:*

$$ \mathrm{Gr}_i^{p_\tau} \mathbb{H}^{i+j}(X, j_* p\mathbf{L}) \simeq \mathbb{H}^j(V, p\mathcal{H}^i(f_* j_* p\mathbf{L})) \xrightarrow{\sim} \oplus_{S \subset V} \mathbb{H}^j(V, i_{S_*} p\mathbf{L}_S^i). $$

consists of a direct sum of polarized HS of weight $a + i + j$

*Proof.* The proof is similar to (Lemma 10.2). Let $B^j := \mathbb{H}^j(X, j_*p\mathbf{L}, F)$ with the same definition of the filtration $p_\tau$ on $B := \oplus_j B^j$.

Then $\eta: B \to B(1)$ defined by cup-product with the hyperplane class is a morphism of HS. We define an increasing filtration $A$ on $B$ by the same (Equation 10.7). We deduce: $\eta^i: \mathrm{Gr}_i^A = \mathrm{Gr}_{-i}^{p_\tau}B^{-i} \simeq \mathrm{Gr}_i^{p_\tau}B^i = \mathrm{Gr}_{-i}^A$.

Hence $A_i$ is the unique increasing filtration defined by the nilpotent endomorphism $\eta$ on the MHS on $B$. Then $A$ is necessarily a filtration by HS.

The HS on $\mathbb{H}^j(V, p\mathcal{H}^i(f_*j_*p\mathbf{L}))$ is isomorphic to $\mathrm{Gr}_i^{p_\tau} \mathbb{H}^{i+j}(X, j_*p\mathbf{L})$. Moreover, Since the decomposition of $p\mathcal{H}^i(f_*j_*p\mathbf{L})$ is naturally compatible with the filtration $F$, the decomposition of $\mathbb{H}^j(V, p\mathcal{H}^i(f_*j_*p\mathbf{L}))$ is also compatible with HS.

**Remark 10.5.** Let $V$ be projective. The decomposition into primitive subspaces may be carried in two steps:
1) We deduce a scalar product on $^{p}\mathcal{H}^{-i}(f_{*}j_{*}^{!}p\mathbf{L})$ from Lefschetz isomorphism $\eta^i$ and the duality $^{p}\mathcal{H}^{-i}(f_{*}j_{*}^{!}p\mathbf{L}) \simeq D^{p}\mathcal{H}^{i}(f_{*}j_{*}^{!}p\mathbf{L})$. The primitive subperverse sheaf is defined by $\text{Ker } \eta^{i+1} \subset {}^{p}\mathcal{H}^{i}(f_{*}j_{*}^{!}p\mathbf{L})$ [De 93].

2) Let $\eta_V$ denote the cup product with the class $c_1(V)$ of an hyperplane section $H$ of the projective variety $V$. The scalar product is defined by Hard Lefschetz isomorphisms on $V$ defined by iteration of $\eta_V$ and duality. The polarized primitive sub-HS is defined by $\text{Ker } \eta_V^{j+1} : \mathbb{H}^{-j}(V, \text{Ker } \eta^{i+1}) \to \mathbb{H}^{j+2}(V, \text{Ker } \eta^{i+1})$. The proof of polarization, by induction on the hyper-plane section $H_V$ of $V$ is similar to Deligne's original proof (Weil II [De 80], théorème 4.1.1) (see also [CaMi 5]).

### 10.3.2.2 Octahedron diagram

With the notations of (Theorem 10.1), set $K := f_*j_*!p\mathbf{L}$ and $I := \gamma \circ \alpha$ the composition map $I : i_{v_*}i_v!K \xrightarrow{\alpha} K \xrightarrow{\gamma} i_{v_*}i_v^*K$. We consider the following upper and lower 'cap' diagrams ([BBD 83], 1.1.6)

<figure><img src="image_16.png" /></figure>

where $i_v : v \to V$ and $k : V \setminus v \to V$, $i_{v_*}i_v^*k_*k^*K$ is isomorphic to the cone of $I : i_{v_*}i_v!K \to i_{v_*}i_v^*K: i_{v_*}i_v^*k_*k^*K \simeq C(I) \simeq C(\beta')$.

**Corollary 10.6.** i) The long perverse exact sequence of four terms defined by the common boundary of the diagrams is exact:

$$ \xrightarrow{\to} {}^{p}\mathcal{H}^{i}(i_{v_*}i_v!K) \xrightarrow{\to} {}^{pI_i}{}^{p}\mathcal{H}^{i}(i_{v_*}i_v^*K) \xrightarrow{\to} {}^{p\delta'}{}^{p}\mathcal{H}^{i+1}(k_!k^*K) \xrightarrow{\to} {}^{p\beta'}{}^{p}\mathcal{H}^{i+1}(k_*k^*K) \xrightarrow{\to} {}^{p\delta_{i+1}}{}^{p}\mathcal{H}^{i+1}(k_*k^*K) \xrightarrow{\to} \quad (10.43) $$

ii) The morphism ${}^p\gamma_i$ in (Equation 10.36) induces an isomorphism:

$$ \text{Im} \, {}^{p}\alpha^i \xrightarrow{\sim} \text{Im} \, {}^{p}I^i := \text{Im}(\mathbb{H}_{X_v}^i(X, j_*!p\mathbf{L})) \to \mathbb{H}^i(X_v, j_*!p\mathbf{L})) \simeq \mathbf{L}_v^i \quad (10.44) $$

*Proof.* i) In the case of the decomposition: ${}^{p}\mathcal{H}^{i}(K) \xrightarrow{\sim} \text{Im} \, {}^{p}\alpha^{i} \oplus \text{Ker} \, {}^{p}\gamma_{i}$ (Lemma 10.15) in the left commutative triangle of the upper cap, the exact sequence (Equation 10.43) follows by general properties of octahedron diagrams.

ii) The isomorphism in (Equation 10.44) is induced by ${}^{p}\gamma_i$.
### 10.3.3 Singular variety X

The decomposition theorem has been proved in the previous chapters in the case of non singular variety X.

We present here a short review of the case where X is projective singular by reduction to a desingularization of X.

1) The decomposition theorem of $f: X \to V$ is deduced directly by Deligne's argument on the composition of $f$ with the desingularization morphism $\pi: X' \to X$ ([De 68] Proposition 2.16). We refer to (§6.2.2.2) for this method.

2) However, the theorem may be deduced again by the previous induction $D(n)$ applied in this setting.

3) The relative local purity theorem apply for all projective morphisms $f$ (of singular varieties).

#### 10.3.3.1 Hodge structure on $\mathbb{H}^*(X, j_*^p\mathcal{L})$ for singular X

Let $X$ be a projective complex variety maybe singular, $j: U \to X$ the embedding of a smooth Zariski open subset of $X$ and $p\mathcal{L}$ defined on $U$.

**Corollary 10.7 (HS on $\mathbb{H}^*(X, j_*^p\mathcal{L})$).** There exists a HS on $\mathbb{H}^i(X, j_*^p\mathcal{L})$ independent of the choice of a desingularization.

*Proof.* Let $\pi: X' \to X$ be a non singular modification of $X$, $j': U \to X'$ the embedding of $U$ in $X'$ and $j'_*^p\mathcal{L}$ the intermediate extension in $X'$. Let $S$ be a Thom-Whitney stratification of $\pi$ compatible with $j'_*^p\mathcal{L}$.

There exists a decomposition on $X$ of $p\mathbb{H}^i(\pi_*j'_*^p\mathcal{L}) \simeq \bigoplus_{S\subset X} i_{S!*}^p\mathcal{L}_S^i$ into a direct sum consisting of intermediate extensions of polarized VHS. We deduce a decomposition into a direct sum of polarized HS of weight $a + i + j$

$$Gr_i^{p\tau} \mathbb{H}^{i+j}(X', j_*^p \mathcal{L}) \simeq \mathbb{H}^j(X, p\mathbb{H}^i(f_*j_*^p \mathcal{L})) \xrightarrow{\sim} \bigoplus_{S\subset X} \mathbb{H}^j(X, i_{S!*}^p \mathcal{L}_S^i).$$

where for each stratum $S$, the hypercohomology is a direct sum of polarized primitive subspaces of the hypercohomology on $X$ with coefficients into the primitive decomposition of $i_{S!*}^p\mathcal{L}_S^i$.

Since on the big strata $U$ of $X$, we have $j_*^p\mathcal{L} = i_{U!*}^p\mathcal{L}_U^0$, we deduce the HS on $\mathbb{H}^i(X, j_*^p\mathcal{L})$ as a sub-quotient HS of $\mathbb{H}^i(X', j'_*^p\mathcal{L})$, precisely a polarized sub-HS of $Gr_0^{p\tau} \mathbb{H}^i(X', j'_*^p\mathcal{L})$.

The uniqueness is deduced by the method of comparison of two non singular modifications $X'_1$ and $X'_2$ with a non singular modification $X'$ of the fiber product $X'_1 \times_X X'_2$ such that $X' \to X_1$ (resp. $X' \to X_2$) is a composition of non singular modifications, then the HS on $\mathbb{H}^i(X, j_*^p\mathcal{L})$ deduced from $X'_1$ or $X'_2$ coincide with the HS deduced from $X'$.
### 10.3.4 Decomposition of $^{p}\mathcal{H}^i(f_*j_*!^{p}\mathbf{L})$

Independently of Deligne's argument (Equation 6.22), a direct proof by induction $D(n)$ (§10.0.0.1) applies. The case $dim V = 0$ as initial step, is proved (§10.3.3.1).

The assumption $D(n, V^*)$ on the complement of isolated singularity follows by general normally embedded sections of $V$, which reduces the proof to isolated strata of $V$.

Let $\pi: X' \to X$ be a desingularization, $^{p}\tau'$ the perverse truncation of $K' := \pi_*j'_*!^{p}\mathbf{L}$ on $X$ and $^{p}\tau_V$ the perverse truncation of $K := (f \circ \pi)_*j'_*!^{p}\mathbf{L} = f_*K'$ on $V$. We deduce from $^{p}\tau'$ a filtration $^{p}\tau_X := f_*(^{p}\tau')$ on $K$

$$ K := (f \circ \pi)_*j'_*!^{p}\mathbf{L} = f_*K', \quad ^{p}\tau_V, \quad ^{p}\tau_X := f_*(^{p}\tau'), \quad F := (f \circ \pi)_*F \quad (10.45) $$

The filtrations $^{p}\tau_X$, $^{p}\tau_V$ and $F$ are strict on $K$ since the respective spectral sequence degenerates at rank one.

The induced filtration $^{p}\tau_X$ on $^{p}\mathcal{H}^i(K) := Gr_i^{^{p}\tau_V}K$ is defined by

$$ (^{p}\tau_X)_j^{^{p}\mathcal{H}^i(K)} := \mathrm{Im}(^{p}\mathcal{H}^i(f_*(^{p}\tau_X)_j K) \to ^{p}\mathcal{H}^i(K)) $$

such that $Gr_j^{^{p}\tau_X}^{^{p}\mathcal{H}^i(K)} \simeq ^{p}\mathcal{H}^i(f_*Gr_j^{^{p}\tau'}K') \simeq ^{p}\mathcal{H}^i(f_*^{^{p}\mathcal{H}^j(K')}))$.

In terms of the decomposition of $^{p}\mathcal{H}^0(K') \simeq j_*!^{p}\mathbf{L} \oplus_{S \subset X, \dim S < \dim X} i_{S!*}!^{p}\mathbf{L}_S^0$:

$$ Gr_0^{^{p}\tau_X}^{^{p}\mathcal{H}^i(K)} \simeq ^{p}\mathcal{H}^i(f_*^{^{p}\mathcal{H}^0(K')})) \simeq ^{p}\mathcal{H}^i(f_*j_*!^{^{p}\mathbf{L}}) \oplus_S ^{p}\mathcal{H}^i(f_*i_{S!*}!^{^{p}\mathbf{L}}_S^0) \quad (10.46) $$

where the sum is over all strata $S$ of $X: \dim S < \dim X$.

#### 10.3.4.1 Relative local purity for all projective morphisms $f$

Let $V_0$ denote the union of the strata of dimension zero of $V$ and $k: (V \setminus V_0) \to V$. We suppose $V_0$ reduced to a point $v$ to simplify the notations.

**Corollary 10.8.** The weight $w$ of the space $\mathcal{H}^i(i_v^*k_*!^{p}\mathcal{H}^j(f_*j_*!^{p}\mathbf{L}))$, isomorphic to the Intersection cohomology $\mathbb{H}^i(B_v \setminus \{v\}, !^{p}\mathcal{H}^j(f_*j_*!^{p}\mathbf{L}))$, satisfies the relations:

$$ w \leq {}^{p}w + j + i \text{ if } i \leq -1 \text{ and } w > {}^{p}w + j + i \text{ if } i \geq 0 \quad (10.47) $$

The inequalities on the weights $w$ of the space $\mathbb{H}^i(B_v - \{v\}, !^{p}\mathcal{H}^j(K))$ in the equation 10.47 are satisfied since $f \circ \pi$ is a stratification by NCD over the isolated strata as proved in (§10.6). Since $\mathbb{H}^i(B_v \setminus \{v\}, !^{p}\mathcal{H}^j(K))$ is a sub-quotient MHS of $\mathbb{H}^i(B_v \setminus \{v\}, !^{p}\mathcal{H}^j(K))$, the inequalities on the weight $w$ are also satisfied.
Finally the corollary follows since $^{p}\mathcal{H}^j(f_*j_*!^{p}\mathbf{L})$ is a summand of $Gr_0^{p\tau X} p\mathcal{H}^j(K)$ (Equation 10.46) and there exists necessarily a decomposition of $^{p}\mathcal{H}^i(f_*j_*!^{p}\mathbf{L})$ (Equation 6.23) into intermediate extensions of $(!^{p}\mathbf{L})_S^i$ as in theorem (ch.1, Theorem 1.2).

**Corollary 10.9.** Let $V_m$ denote the union of maximal strata of $V$. We have the decomposition

$$ {}^{p}\mathcal{H}^{i}(f_{*}j_{*}!^{p}\mathbf{L}) \simeq \bigoplus j_{V_{*}}! (R^{i}f_{*}j_{*}!^{p}\mathbf{L}|_{V_{m}}[\dim V]) \oplus_{S\subset S, \dim S < \dim V} i_{S!*}!^{p}\mathbf{L}_{S}^{i} \quad (10.48) $$

### 10.3.4.2 Duality and polarization

The duality between $!^{p}\mathbf{L}_S^{-k}$ and $!^{p}\mathbf{L}_S^k$ follows from the auto-duality of $j_*!^{p}\mathbf{L}$ by Verdier duality formula for the proper morphism $f$.

Since Hard Lefschetz isomorphisms $!^{p}\mathbf{L}_S^{-k} \simeq !^{p}\mathbf{L}_S^k$ are also satisfied, we deduce that the VHS $!^{p}\mathbf{L}_S^k$ are direct sum of polarized primitive sub-VHS.

For each strata $S$ of $V$, the local system $!^{p}\mathbf{L}_S^i$ defined by $f_*j_*!^{p}\mathbf{L}$ is a direct sum polarized variation of Hodge structure on the smooth variety $S$.

**Corollary 10.10.** Let $f: X \to V$ be a morphism of projective varieties where $X$ is non singular. The decomposition consists of a direct sum of polarized HS of weight $a + i + j$

$$ Gr_i^{p\tau} \mathbb{H}^{i+j}(X, j_*!^{p}\mathbf{L}) \simeq \mathbb{H}^j(V, {}^{p}\mathcal{H}^i(f_*j_*!^{p}\mathbf{L})) \xrightarrow{\sim} \bigoplus_{S\subset V} \mathbb{H}^j(V, i_{S!*}!^{p}\mathbf{L}_S^i). $$

*Remark 10.6.* The proof of the existence of HS on the hypercohomology of a singular complex variety $X$ with coefficients in intermediate extensions of polarized VHS is reduced to the case where $X$ is projective, by the decomposition theorem applied to a birational morphism $\pi: X' \to X$ where $X'$ is projective and non singular (§6.2.2.1).

## 10.4 Relative local purity and Intersection morphisms

The relative local purity theorem may be interpreted in terms of the intersection morphism.

In the case of an isolated singularity $v \in V$, the local purity theorem has consequences on the weights of the cohomology with support $\mathbb{H}_{X_v}^i(X, \mathbb{Q})$. A direct proof of some consequences of the relative local purity appears in the work of Navarro Aznar ([Na 85] Proposition 5.1), where the results are treated in terms of the exceptional NCD in a desingularization and the MHS of $\mathbb{H}_{X_v}^i(X, \mathbb{Q})$.
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. In fact, the statement is equivalent to the purity theorem.

### 10.4.1 Intersection morphism

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 $B_v \subset V$. With the notations of Lemma 10.8, the perverse filtration $p_\tau$ on $K := f_*j_*!_*p_\tau L$ on $V$ induces a filtration $p_\tau$ on $i_v^!K$ defined functorially by $i_v^! p_\tau K$ and computed by $i_v^!$-acyclic filtered representative of $(K, p_\tau)$.

Similarly, we define $p_\tau$ on $i_v^!K$ and $i_v^!k_*k^*K$, from which we deduce increasing filtrations $p_\tau$ on $\mathbb{H}_{X_v}^i(X, j_*!_*p_\tau L)$, $\mathbb{H}^i(X_v, j_*!_*p_\tau L)$ and $\mathbb{H}^i(B_{X_v}^*, j_*!_*p_\tau L)$ (Lemma 10.2). The following proposition is equivalent to the local purity theorem.

**Proposition 10.2 (Intersection morphism).** i) The morphisms induced by the intersection morphism $I$ (ch. 2, Equation 2.1.3.1)

$$ \mathrm{Gr}_j^{p_\tau} \mathbb{H}_{X_v}^i (X, j_*!_* p_\tau L) \xrightarrow{p_\tau I_j} \mathrm{Gr}_j^{p_\tau} \mathbb{H}^i (X_v, j_*!_* p_\tau L) \quad (10.49) $$

are isomorphisms for each $i = j$ and vanish for all indices $i \neq j$.

ii) Conversely, if the intersection morphism $p_\tau I_j^i$ is an isomorphism in the case $i = j$, the inequalities on the weights in the local purity theorem hold.

We start the proof with a general result

**Lemma 10.18.** The induced perverse filtrations $p_\tau$ have the properties:

$$ p_{\tau_i} \mathbb{H}_{X_v}^i (X, j_*!_* p_\tau L) = \mathbb{H}_{X_v}^i (X, j_*!_* p_\tau L) \quad \text{and} \quad p_{\tau_{i-1}} \mathbb{H}^i (X_v, j_*!_* p_\tau L) = 0. $$

*Proof.* Let $K := f_*j_*!_*p_\tau L$. We have convergent spectral sequences associated to the increasing induced filtrations $p_\tau$

$$ \begin{align} p_\tau E_{v1}^{pq} &:= \mathbb{H}_v^{2p+q}(V, p\mathcal{H}^{-p}(K)) \Rightarrow \mathrm{Gr}_{-p}^{p_\tau} \mathbb{H}_{X_v}^{p+q}(X, j_*!_*p_\tau L) \tag{10.50} \\ p_\tau E_1^{pq}(v) &:= \mathbb{H}^{2p+q}(v, p\mathcal{H}^{-p}(K)) \Rightarrow \mathrm{Gr}_{-p}^{p_\tau} \mathbb{H}^{p+q}(X_v, j_*!_*p_\tau L). \end{align} $$

Since $p\mathcal{H}^{-p}(K)$ is perverse, we have

$$ \mathbb{H}_v^i(V, p\mathcal{H}^{-p}(K)) = 0 \text{ for } i < 0 \text{ (resp. } \mathbb{H}^i(v, p\mathcal{H}^{-p}(K)) = 0 \text{ for } i > 0) $$

then: $p_\tau E_{v1}^{pq} = 0$ for $2p + q < 0$, (resp. $p_\tau E_1^{pq}(v) = 0$ for $2p + q > 0$), and

$$ \begin{align} \mathrm{Gr}_j^{p_\tau} \mathbb{H}_{X_v}^i (X, j_*!_* p_\tau L) &= 0 \text{ for } i < j : \stackrel{p_\tau}{\mathrm{Gr}}_{i} \stackrel{p_\tau}{\mathbb{H}}_{X_v} \stackrel{p_\tau}{(X, j_*!_* p_\tau L)} = \stackrel{p_\tau}{\mathbb{H}}_{X_v} \stackrel{p_\tau}{(X, j_*!_* p_\tau L)}, \nonumber \\ \mathrm{Gr}_j^{p_\tau} \mathbb{H}^i (X_v, j_*!_* p_\tau L) &= 0 \text{ for } i > j : \stackrel{p_\tau}{\mathrm{Gr}}_{i-1} \stackrel{p_\tau}{\mathbb{H}}^i (X_v, j_*!_* p_\tau L) = 0 \tag{10.51} \end{align} $$
*Proof of the proposition.* The long exact sequence

$$ \mathbb{H}^{i-1}(B_{X_v}^*, j_{1*}^* \mathbf{p}\mathbf{L}) \xrightarrow{\partial_i} \mathbb{H}_{X_v}^i(X, j_{1*}^* \mathbf{p}\mathbf{L}) \xrightarrow{I_i} \mathbb{H}^i(X_v, j_{1*}^* \mathbf{p}\mathbf{L}) \xrightarrow{\gamma_i} \mathbb{H}^i(B_{X_v}^*, j_{1*}^* \mathbf{p}\mathbf{L}) \xrightarrow{[1]} $$
is deduced from the distinguished triangle:

$Ri_{X_v}^i j_{1*}^* \mathbf{p}\mathbf{L} \xrightarrow{I} i_{X_v}^* j_{1*}^* \mathbf{p}\mathbf{L} \xrightarrow{\gamma_i} Rj_{X_v*} j_{1*}^* \mathbf{p}\mathbf{L}$. By the decomposition theorem, the morphisms of the exact sequence are strictly compatible with the induced $p_\tau$ filtrations. We deduce the exact sequence (on the next two lines)

$$ \begin{aligned} \mathrm{Gr}_j^{p_\tau} \mathbb{H}^{i-1}(X_v, j_{1*}^* \mathbf{p}\mathbf{L}) \xrightarrow{p_\tau \gamma_{i-1}^j} \mathrm{Gr}_j^{p_\tau} \mathbb{H}^{i-1}(B_{X_v}^*, j_{1*}^* \mathbf{p}\mathbf{L}) \xrightarrow{p_\tau \partial_i^j} \mathrm{Gr}_j^{p_\tau} \mathbb{H}_{X_v}^i(X, j_{1*}^* \mathbf{p}\mathbf{L}) \\ \xrightarrow{p_\tau I_i^j} \mathrm{Gr}_j^{p_\tau} \mathbb{H}^i(X_v, j_{1*}^* \mathbf{p}\mathbf{L}) \xrightarrow{p_\tau \gamma_i^j} \mathrm{Gr}_j^{p_\tau} \mathbb{H}^i(B_{X_v}^*, j_{1*}^* \mathbf{p}\mathbf{L}) \xrightarrow{p_\tau \partial_{i+1}^j} \mathrm{Gr}_j^{p_\tau} \mathbb{H}_{X_v}^{i+1}(X, j_{1*}^* \mathbf{p}\mathbf{L}) \end{aligned} $$

By the lemma, the morphism $^{p_\tau}I_i^j$ vanishes if $i \neq j$.

For $j=i$, by the relative local purity theorem, the weights $w$ of the term $\mathrm{Gr}_i^{p_\tau} \mathbb{H}^{i-1}(B_{X_v}^*, j_{1*}^* \mathbf{p}\mathbf{L})$ satisfy $w \le p w + i - 1$ and those of the term $\mathrm{Gr}_i^{p_\tau} \mathbb{H}^i(B_{X_v}^*, j_{1*}^* \mathbf{p}\mathbf{L})$ satisfy $w > p w + i$. Moreover the weights $w$ of the term $\mathbb{H}_{X_v}^i(X, j_{1*}^* \mathbf{p}\mathbf{L})$ satisfy: $w \ge p w + i$, hence: $^{p_\tau}\partial_i^i = 0$.

We have $^{p_\tau}\gamma_i^i = 0$ since the weights $w$ of the term $\mathrm{Gr}_i^{p_\tau} \mathbb{H}^i(X_v, j_{1*}^* \mathbf{p}\mathbf{L})$ satisfy $w \le p w + i$. Hence $^{p_\tau}I_i^i : \mathrm{Gr}_i^{p_\tau} \mathbb{H}_{X_v}^i(X, j_{1*}^* \mathbf{p}\mathbf{L}) \to \mathrm{Gr}_i^{p_\tau} \mathbb{H}^i(X_v, j_{1*}^* \mathbf{p}\mathbf{L})$ is an isomorphism. Remark that by the decomposition theorem:

$$ \mathrm{Gr}_i^{p_\tau} \mathbb{H}_{X_v}^i(X, j_{1*}^* \mathbf{p}\mathbf{L}) \simeq H^0(i_v^! p^\mathcal{H}(K)) \simeq \mathbf{L}_v^i \simeq \mathrm{Gr}_i^{p_\tau} \mathbb{H}^i(X_v, j_{1*}^* \mathbf{p}\mathbf{L}) \simeq H^0(i_v^* p^\mathcal{H}(K)). $$

Conversely, we prove the crucial step of the relative local purity theorem (ch. 11, Equation 11.5): $\mathrm{Gr}_{p w}^W \mathrm{Gr}_0^{p_\tau} \mathbb{H}^{-1}((B_{X_v}^*, j_{1*}^* \mathbf{p}\mathbf{L})) = 0$. We consider the exact sequence:

$$ \begin{aligned} \mathrm{Gr}_0^{p_\tau} \mathbb{H}_{X_v}^{-1}(X, j_{1*}^* \mathbf{p}\mathbf{L}) \xrightarrow{p_\tau I_{-1}^0} \mathrm{Gr}_0^{p_\tau} \mathbb{H}^{-1}(X_v, j_{1*}^* \mathbf{p}\mathbf{L}) \xrightarrow{p_\tau \gamma_{-1}^0} \mathrm{Gr}_0^{p_\tau} \mathbb{H}^{-1}(B_{X_v}^*, j_{1*}^* \mathbf{p}\mathbf{L}) \\ \xrightarrow{p_\tau \partial_0^0} \mathrm{Gr}_0^{p_\tau} \mathbb{H}_{X_v}^0(X, j_{1*}^* \mathbf{p}\mathbf{L}) \xrightarrow{p_\tau I_0^0} \mathrm{Gr}_0^{p_\tau} \mathbb{H}^0(X_v, j_{1*}^* \mathbf{p}\mathbf{L}) \end{aligned} $$

By the above lemma $\mathrm{Gr}_0^{p_\tau} \mathbb{H}_{X_v}^{-1}(X, j_{1*}^* \mathbf{p}\mathbf{L}) = 0$, hence $^{p_\tau}I_{-1}^0 = 0$ and $^{p_\tau}\gamma_{-1}^0$ is injective. If the morphism $^{p_\tau}I_0^0$ is an isomorphism, then $^{p_\tau}\partial_0^0 = 0$ and $^{p_\tau}\gamma_{-1}^0$ is surjective, hence an isomorphism. By compatibility of $^{p_\tau}$ with MHS, $^{p_\tau}\gamma_{-1}^0$ induces an isomorphism

$$ \mathrm{Gr}_w^W \mathrm{Gr}_0^{p_\tau} \mathbb{H}^{-1}(X_v, j_{1*}^* \mathbf{p}\mathbf{L}) \simeq \mathrm{Gr}_w^W \mathrm{Gr}_0^{p_\tau} \mathbb{H}^{-1}((B_{X_v}^*, j_{1*}^* \mathbf{p}\mathbf{L})) $$

Since $X_v$ is closed, the weights of $\mathbb{H}^{-1}(X_v, j_{1*}^* \mathbf{p}\mathbf{L})$ are $\le a - 1$, which proves

$$ \mathrm{Gr}_{p w}^W \mathrm{Gr}_0^{p_\tau} \mathbb{H}^{-1}(X_v, j_{1*}^* \mathbf{p}\mathbf{L}) = \mathrm{Gr}_{p w}^W \mathrm{Gr}_0^{p_\tau} \mathbb{H}^{-1}((B_{X_v}^*, j_{1*}^* \mathbf{p}\mathbf{L}) = 0. $$

**Remark 10.7.** We deduce from the intersection isomorphism above and the duality map $D: \mathrm{Gr}_{-r}^{p_\tau} \mathbb{H}_{X_v}^{-r}(X, j_{1*}^* \mathbf{p}\mathbf{L}) \xrightarrow{\sim} \mathrm{Gr}_r^{p_\tau} \mathbb{H}^r(X_v, j_{1*}^* \mathbf{p}\mathbf{L})^*$ a non degenerate intersection form

$$ \mathrm{Gr}_r^{p_\tau} \mathbb{H}_{X_v}^r(X, j_{1*}^* \mathbf{p}\mathbf{L}) \otimes \mathrm{Gr}_{-r}^{p_\tau} \mathbb{H}_{X_v}^{-r}(X, j_{1*}^* \mathbf{p}\mathbf{L}) \to \mathbb{Q}, (a, b) \mapsto D(b)(I(a)) $$
When $X_v$ is a NCD on a smooth surface and in the case of intersection cohomology with rational coefficients viewed as homology, Grauert’s contractibility theorem states that the intersection form defined on $H_2(X_v)$ is negative definite if and only if $X_v$ contracts to a point $v$ on a normal surface (see also [CaMi 5], [BPV 84] Theorem 2.1).
<figure><img src="image_17.png" /></figure>
Chapter 11

Related results

We present in this chapter, in the first two sections interesting results that
has been originally used in the text but were not strictly necessary. In the
third section a tentative combinatorial description of the weight filtration on
the logarithmic complex is given.

11.1 Hodge theory on perverse cohomology

Let $f: X \to V$ be a projective morphism on a non singular complex algebraic
variety $X$ of dimension $n+1$, $j_{!*}p\mathbf{L}$ on $X$, $K := f_*j_{!*}p\mathbf{L}$, $v \in V$ a closed point
such that $Z = X_v := f^{-1}(v)$ is a NCD in $X$ and $B_v^*$ a punctured small ball
with deleted center $v$.

An independent direct proof that $p_{\tau^*} \mathbb{H}^r(X_{B_v^*}, j_*! p \mathbf{L})$ and $p_{\tau^*} \mathbb{H}^r(X_{B_v}, j_*! p \mathbf{L})$ are filtrations by sub-MHS (Proposition 11.3 below) may be deduced from the description of the perverse filtration directly on $X$ ([CaMi 10]). In chapter 10, the filtration $p_{\tau^*}$ has been deduced after the proof of the decomposition theorem (Lemma 10.17 and 10.2).

11.1.1 Preliminaries

Let $V$ be a quasi-projective variety and $K \in D_c^b(V, \mathbb{Q})$ a complex with con-
structible cohomology sheaves. The topological middle perversity truncations
on $K$ on $V$ define an increasing *perverse filtration* $p_{\tau}$ on $K$.

A corresponding increasing filtration $p_{\tau}$ is defined on the hypercohomology
of $K$ on open subsets $U$ of $V$

$$
p_{\tau_i} \mathbb{H}^k(U, K) := \mathrm{Im} \{ \mathbb{H}^k(U, p_{\tau_i} K) \to \mathbb{H}^k(U, K) \}. \quad (11.1)
$$
This perverse filtration $^{p}_{\tau}$ on the hypercohomology groups $\mathbb{H}^k(V, K)$ is described by algebraic-geometric techniques in [CaMi 10] as follows.

Given an affine embedding $V \subset \mathbb{P}^N$ and $n := \dim V$, we consider two families of hyperplanes $\Lambda_* := \Lambda_i$ and $\Lambda'_i := \Lambda'_i$ for $1 \le i \le n$ in $\mathbb{P}^N$, defining on $V$ two families $H_*$ and $W_*$ of increasing closed sub-varieties of $V$:
$H_{-r} := \cap_{1 \le i \le r} \Lambda_i \cap V$ and $W_{-r} := \cap_{1 \le i \le r} \Lambda'_i \cap V$

$$
H_* : V = H_0 \supset H_{-1} \supset \dots \supset H_{-n}, \quad W_* : V = W_0 \supset W_{-1} \supset \dots \supset W_{-n} \tag{11.2}
$$

and $H_{-n-1} = \emptyset = W_{-n-1}$. Let $h_i : (V \setminus H_{i-1}) \to V$ denote the open embeddings with indices in $[-n, 0]$. The decreasing filtration $\delta$:

$$
\delta^p \mathbb{H}^*(V, K) := \mathrm{Im} \left( \bigoplus_{i+j=p} \mathbb{H}_{W_{-j}}^*(V, (h_i)_i h_i^* K) \to \mathbb{H}^*(V, K) \right) \quad (11.3)
$$

([CaMi 10], Remark 3.6.6) determines $^{p}_{\tau}$ and is well adapted to computation
in Hodge theory.

**Proposition 11.1 ([CaMi 10], Theorem 4.2.1).**

For an affine embedding of $V$ into a projective space and for a general
choice of both families $H_*$ and $W_*$ depending on $K$ and the embedding of $V$,
the filtration $\delta$ is equal to the perverse filtration $^{p}_{\tau}$ up to a change in indices.
Precisely

$$
\delta^i \mathbb{H}^*(V, K) = {}^{p}\tau_{-i+l} \mathbb{H}^l(V, K)
$$

11.1.2 Hodge structure on the perverse filtration

We consider two cases: $U := V \setminus W$ is quasi-projective, complement of a closed subvariety $W \subset V$ (resp. $U$ is a punctured small ball $B_v^*$ with deleted center $v$).

**Proposition 11.2 (MHS on perverse hypercohomology).**

Let $f: X \to V$ be a projective morphism where $X$ is smooth, $V$ projective,
$\mathcal{P}\mathbf{L}$ a shifted admissible variation of MHS on $X \setminus Y$ the complement of a
normal crossing divisor $Y$ and $j: X \setminus Y \to X$.

Let $W \subset V$ be a closed algebraic subset of $V$ such that $X_W := f^{-1}(W)$ is a sub-normal crossing divisor of $Y$. The perverse filtration $^{p}_{\tau}$ on the hypercohomology $\mathbb{H}^*(X \setminus X_W, j_* \mathcal{P}\mathbf{L})$ (resp. $\mathbb{H}^*_c(X \setminus X_W, j_* \mathcal{P}\mathbf{L})$ by duality) is a filtration by sub-MHS.

Proof. The proof is in terms of logarithmic complexes. Let $Z := f^{-1}W$ the inverse image of $W$ and $j_Z : (X - Z) \to X$. We apply (Proposition 11.1) to $K := f_*j_Z j_Z^* j_{Z'}^* j_{Z'} \mathcal{P}\mathbf{L}$ (including $Z = Y$ or $Z = \emptyset$) (the case $f_*j_Z j_Z^* j_{Z'}^* j_{Z'} \mathcal{P}\mathbf{L}$ is dual).
The perverse filtration $p_{\tau}$ on $\mathbb{H}^k(X - Z, j_{1*} p_L)$ is defined via the isomorphism $\mathbb{H}^k(X \setminus Z, j_{1*} p_L) \simeq \mathbb{H}^k(V \setminus W, f_* j_{1*} \mathcal{L})$:

$$p_{\tau_i} \mathbb{H}^k(X \setminus Z, j_{1*} \mathcal{L}) := p_{\tau_i} \mathbb{H}^k(V \setminus W, f_* j_{1*} \mathcal{L}).$$

We check that the filtration $p_{\tau}$ is a filtration by sub-MHS on $\mathbb{H}^k(X \setminus Z, j_{1*} p_L)$ using (Proposition 11.1). The system of truncations morphisms:

$\dots \to p_{\tau \le -i} f_* IC^* p\mathcal{L}(\mathrm{Log} Z) \to p_{\tau \le -i+1} f_* IC^* p\mathcal{L}(\mathrm{Log} Z) \dots$ is defined over $\mathbb{Q}$ and it is isomorphic in $D_c^b(V, \mathbb{C})$ to a system of inclusions maps:

$\dots \to P^i K \to P^{i-1} K \dots$ where $P^i K \simeq p_{\tau \le -i} f_* IC^* p\mathcal{L}(\mathrm{Log} Z)$ is decreasing and where we suppose $K$, all $P^i K$ and $Gr_i^P K$ injective complexes ([BBD 83], 3.1.2.7) defined over $\mathbb{Q}$.

The filtration $P$ on $K$ is defined up to unique isomorphism in the category of derived filtered complexes. Moreover, we represent the filtrations $W$ and $F$ by acyclic resolutions for the global sections functor $\Gamma(V, *)$ on the same complex $(K, P, W, F)$ representative of $(f_* IC^* p\mathcal{L}(\mathrm{Log} Z), P, W, F)$.

To prove that $R\Gamma(V, Gr_i^P K, W, F)$ is a MHC, hence $\mathbb{H}^r(V, Gr_i^P K, W, F)$ is a MHS, it is enough to check the following lemma:

**Lemma 11.1.** Let $f : X \to V$ be a fibration by NCD over the strata, and $W$ a closed subvariety of $V$ such that $Z := f^{-1}(W)$ is a NCD in $X$.

The induced filtration $\delta$ (11.3) on $\mathbb{H}^j(X \setminus Z, j_{1*} p_L)$ corresponds to a filtration $\delta'$ by sub-MHS on $\mathbb{H}^j(X, IC^* p\mathcal{L}(\mathrm{Log} Z))$. Consequently the perverse filtration $p_{\tau}$ on $\mathbb{H}^j(X \setminus Z, j_{1*} p_L)$ is compatible with the MHS.

*Proof.* Let $U := V \setminus W$. We reduce the proof to the case of an affine embedding of $U$. Let $\pi : \tilde{V} \to V$ be the blowing up of $W$ such that the embedding of $U \hookrightarrow \tilde{V}$ is affine. Since $f^{-1}(W)$ is a NCD in $X$, the morphism $f : X \to V$ factors as $f = \pi \circ g$ with $g : X \to \tilde{V}$.

We apply the Proposition 11.1 to the embedding of $U$ in $\tilde{V}$, to construct the families $H_*$ and $W_*$ in $\tilde{V}$ as intersection with two general families of hyperplanes $\Lambda_*$ and $\Lambda'_*$ in the ambient projective space with inverse image $H'_* := g^{-1}H_*$ and $W'_* := g^{-1}W_*$ in $X$, such that the various intersections $H'_i \cap W'_j$ are transversal and intersect transversally the various strata in $X$. Let $h'_i : (X \setminus H'_i) \to X$, $i_{H'_i} : H'_i \to X$ and $K' := IC^* p\mathcal{L}(\mathrm{Log} Z)$. The filtration in the formula 11.3 is written on $X$ as

$$\delta^p \mathbb{H}^*(X, K') := \mathrm{Im}(\oplus_{i+j=p} \mathbb{H}_{W'_{-j}}^*(X, (h'_i)_{!} h_i'^* K')) \to \mathbb{H}^*(X, K') \quad (11.4)$$

Since the inverse image $H'_*$ and $W'_*$ are transversal in $X$ and not critical for $L$, the construction of the intermediate extension $j_{1*} p_L$, commute with the restriction to each $H'_i$ for various indices $i$.

By (ch 9, §9.5), there exists a structure of bifiltered complex on $h'_i h_i'^* K'$ (resp. on $i'_{W'_{-j}} h'_i h_i'^* K'$) such that $\delta^p \mathbb{H}^*(X, K')$ is defined by sub-MHS.

**Remark 11.1.** The filtrations may be constructed as mixed cones over morphisms of logarithmic complexes.
In particular, $(h'_i)h_i^*K'$ is the mixed cone $C(\rho')$ over the canonical morphism $\rho' : K' \to i_{H'_i}i_{H'_i}^*K'$.

To construct $i_{W'_{-j}}^!h'_i h_i^*K'$, since $W'_{-j}$ is not an hypersurface, we introduce the blowing up $\pi_{-j} : \tilde{X} \to X$ of $W'_{-j}$ in $X$ such that $\tilde{W}'_{-j} := \pi_{-j}^{-1}(W'_{-j})$ is a smooth hypersurface. We consider the bifiltered logarithmic complex $K'' := IC^* p\mathcal{L}(\mathrm{Log}(\pi_{-j}^{-1}Z) \cup \tilde{W}'_{-j})$ on $\tilde{X}$ and the cone $C(\rho'')$ over the morphism $\rho'' : \pi_{-j}^*K'' \to i_{W'_{-j}}^*i_{W'_{-j}}^*K'$. Then, $i_{W'_{-j}}^!K' \simeq C(\rho'')$. This construction applies similarly to the term $i_{H'_i}^*i_{H'_i}^*K'$ instead of $K'$, hence it applies also to the cone $C(\rho')$ to define the structure of bifiltered logarithmic complex on $i_{W'_{-j}}^!h'_i h_i^*K'$.

### 11.1.3 Mixed Hodge structure (MHS) on the Link

Let $f : X \to V$, $X$ non singular, $v \in V$ a closed point such that $X_v := f^{-1}(v)$ and $X_v \cup Y$ are NCD in $X$ and $i : X_v \to X$. The bifiltered complex $(K', W, F) := (i^*j_{X_v^*}(j_{i!*}^p\mathbf{L})|_{(X-X_v)}, W, F)$ (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. The link at $v$ corresponds to the tubular neighborhood of $X_v := f^{-1}(v)$.

In terms of $X_{B_v^*} = X_{B_v} \setminus X_v$, there exists an isomorphism

$$ \mathbb{H}^r(X_{B_v^*}, j_{i!*}^p \mathbf{L}) \xrightarrow{\sim} \mathbb{H}^r((X_v, i^*(j_{X_v})_* j_{X_v}^* j_{i!*}^p \mathbf{L})) \quad (11.5) $$

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 $(f_{|X_v})_*K' \simeq i_{v*}k_*((f_{*j_{i!*}^p\mathbf{L}})|_{(V-v)})$, defines by transport of structure, a filtration $p_{\tau^*}$ on $(f_{|X_v})_*K'$ deduced from the perverse filtration $p_{\tau}$ on $(f_{*j_{i!*}^p\mathbf{L}})|_{(V\setminus v)}$

$$ p_{\tau^*}(f_{|X_v})_*K' \simeq p_{\tau^*}(i_v^*k_*(f_{*j_{i!*}^p\mathbf{L}})|_{(V\setminus v)}) := i_v^*k_*(p_{\tau}(f_{*j_{i!*}^p\mathbf{L}})|_{(V\setminus v)}) \quad (11.6) $$

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 $\mathbb{H}^j(X_v, i^*j_{X_v^*}(j_{i!*}^p\mathbf{L})|_{(X\setminus X_v)})$.

**Proposition 11.3 (Compatibility with MHS).** i) The induced filtration $p_{\tau^*}$ on $\mathbb{H}^j(X_v, K')$ is a filtration by mixed Hodge substructures.
ii) The induced filtrations W and F on $Gr_i^{p_T^*}(f_{|X_v})_*K'$ define a MHS compatible with the isomorphism:

$$ H^j(v, \mathrm{Gr}_i^{p_T^*}(f_{|X_v})_*K', W, F) \simeq \mathrm{Gr}_i^{p_T^*} \mathbb{H}^{i+j}(X_v, K', W, F) \quad (11.7) $$

The filtration $^{p_T^*}$ in the proposition corresponds to the perverse filtration $^{p_T}$ on the hypercohomology of $B_{X_v}^*$ (Equation 11.5)

**Corollary 11.1.** If the restriction of $f$ to $X_{B_v^*}$ satisfy the decomposition theorem, there exists a MHS on $\mathbb{H}^j(B_v^*, {}^p\mathcal{H}^i(f_*j_*!^p\mathbf{L}))$ with filtrations W and F satisfying the isomorphism

$$ \mathbb{H}^j(B_v^*, {}^p\mathcal{H}^i(f_*j_*!^p\mathbf{L}), W, F) \simeq \mathrm{Gr}_i^{p_T} \mathbb{H}^{i+j}(B_{X_v}^*, j_*!^p\mathbf{L}, W, F) \quad (11.8) $$

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

The proof of the proposition is reduced to the global result in [CaMi 10]. To construct $^{p_T^*} := i_{v_*}k_*{}^{p_T}$ on $(f_*K', p_T^*, W, F)$ $(i_v^* {}^p\mathcal{H}^i(K) \neq {}^p\mathcal{H}^i(i_v^*K)$ in general), we introduce the blowing up of $v$ in the diagram

$$ \begin{aligned} Z &:= X_v \xleftarrow{i} X \xrightarrow{j} X^* \\ g_v \downarrow &\quad \downarrow g \quad \downarrow g' \\ \tilde{Z} &\quad \tilde{i} \quad \tilde{V} \quad \tilde{k} \quad \tilde{V}^* \\ \pi_v \downarrow &\quad \downarrow \pi \quad \simeq \downarrow \pi' \\ v &\quad i_v \quad V \quad k \quad V^* \end{aligned} \quad (11.9) $$

where $V^* := V \setminus \{v\}$ and $X^* := X \setminus X_v$, $\pi: \tilde{V} \to V$ is the blowing up of $v \in V$ such that the embedding of $V^*$ in $\tilde{V}$ is affine.

Since $Z := f^{-1}(v)$ is a divisor, the morphism $f$ to $X$, factors through $g: X \to \tilde{V}$. Let $P := (j_*!^p\mathbf{L})|_{X^*}$ and $K' := i_*j_{Z,*}P$:

$$ \mathbb{H}^j(Z, K') \simeq \mathbb{H}^j(f_{|Z})_*K' \simeq \mathbb{H}^j(i_v^*k_*((f_*j_*!^p\mathbf{L})|_{V^*})). $$

Let $(f_{|Z}) = \pi_v \circ g_v$. The complex $R\Gamma(Z, K')$ is a MHC (see Definition 9.35), endowed with a weight filtration W. Let $^{p}\mathcal{H}^i(g'_*P)$ denote the perverse cohomology on $\tilde{V}^*$. Since $\pi'_*{}^p\mathcal{H}^i(g'_*P) \simeq k^*{}^p\mathcal{H}^i(f_*j_*!^p\mathbf{L})$, we have:

$$ i_v^*k_*{}^p\mathcal{H}^i(f_*j_*!^p\mathbf{L}) \simeq \pi_{v_*} \tilde{i}^* \tilde{k}_* {}^p\mathcal{H}^i(g'_*P) \simeq R\Gamma(\tilde{Z}, \tilde{k}_* {}^p\mathcal{H}^i(g'_*P)) \quad (11.10) $$

where $\tilde{k}_* {}^p\mathcal{H}^i(g'_*P)$ is perverse. We apply (ch. 3, Equation 3.24) to write $\tilde{i}^* \tilde{k}_* {}^p\mathcal{H}^i(g'_*P)$ as a cone over the morphism can: $\tilde{k}_* {}^p\mathcal{H}^i(g'_*P) \to \tilde{k}_* {}^p\mathcal{H}^i(g'_*P)$. Hence $R\Gamma(\tilde{Z}, \tilde{k}_* {}^p\mathcal{H}^i(g'_*P))$ carry the structure of a mixed cone of two MHC by (Proposition 11.2).

**Remark 11.2.** A covering of $B_v^* \subset \mathbb{C}^n$ by Stein open subsets complements of linear hyperplanes through 0 in $\mathbb{C}^n$ corresponds by blowing up to a covering of $\tilde{Z}$.
## 11.2 Fibration by normal crossing divisors

The proof of the decomposition by induction assume the reduction to a fibration by NCD over the strata $S$, locally on $S$. It is possible once for all to define a class of fibrations by NCD (Definition 11.1) over the strata. In this case we can rely on logarithmic complexes in all arguments based on Hodge theory.

**Definition 11.1 (topological fibration by NCD over the strata).**

A morphism $f: X \to V$ is a topological fibration by NCD over the strata of a stratification $S = (S_\alpha)$ of $V$ underlying a Thom-Whitney stratification of $f$, if $X$ is smooth and the spaces $V_l = \cup_{\dim S_\alpha \le l} S_\alpha$ satisfy the following properties:

1) The subspaces $X_{V_i} := f^{-1}(V_i)$ with indices $0 \le i \le \dim V$ are successive sub-NCD embedded in $X$.

2) The restriction of $f$ to $X_S := f^{-1}(S)$ over each stratum $S$ of $S$ is a topological fibration: $f_\mid: X_S \to S$.

3) For each point $v \in V_i \setminus V_{i+1}$ (hence $V_i$ is smooth at $v$) and a general normal section $N_v$ to $V_i$ at $v$, the subvariety $f^{-1}(N_v)$ is smooth in $X$ and intersects the NCD $X_{V_i}$ transversally.

Sometimes we need to have the fibration by NCD to be adapted to a subspace $Y$ of $X$ or to a subspace $Z$ of $V$, hence it is convenient to introduce:

**Definition 11.2.** i) The fibration is adapted to a NCD $Y$ in $X$, or to a local system $\mathcal{L}$ defined on a Zariski open algebraic set in the complement of $Y$ in $X$, if in addition:

$Y$ is a union of strata of the underlying whitney stratification $S$ of $X$, the union of the sub-spaces $X_{V_i} \cup Y$ are relative NCD over the strata of $V$ for each $1 \le i \le n$ and the intermediate extension $j_*\mathcal{L}$ of $\mathcal{L}$ is constructible with respect to the strata.

ii) The fibration is adapted to a subspace $Z$ of $V$, if $Z$ is a union of strata of the underlying whitney stratification $S$ of $V$.

### 11.2.1 Fibration by NCD over the strata

Let $X$ be smooth, and $v \in V_i \setminus V_{i-1} \ne \emptyset$ (hence $V_i$ is smooth at $v$); the inverse image of a normal section $N_v$ to $V_i$ at $v$ in general position, is a smooth subvariety $f^{-1}(N_v)$ of $X$ intersecting the NCD $X_{V_i}$ transversally, then $X_{V_i} \cap f^{-1}(N_v) = f^{-1}(v)$ is a NCD in $f^{-1}(N_v)$.

We say for simplification, that $X_{V_i \setminus V_{i-1}} := f^{-1}(V_i \setminus V_{i-1})$ is a relative NCD, and that the stratification $S$ is admissible for $f$.
**Proposition 11.4.** Let $f: X \to V$ be a projective morphism of complex algebraic varieties and $Y$ a closed algebraic strict subspace containing the singularities of $X$.

There exists a diagram $X \xleftarrow{\pi'} X' \xrightarrow{f'} V$ where $\pi'$ is a non-singular modification of $X$, and Thom-Whitney stratifications of $f$, $\pi'$ and $f' := f \circ \pi'$ simultaneously, satisfying the following properties.

i) The morphism $\pi'$ (resp. $f'$) is a fibration by relative NCD over the strata of the underlying Whitney stratification of $X$ (resp. $V$).

ii) The inverse image $Y' := \pi'^{-1}(Y) \subset X'$ is a NCD and there exists a non singular Zariski open subset $U \subset f(X)$ dense in the image $f(X)$ of $f$ such that $f'$ is smooth on $U' := f'^{-1}(U)$ and $\pi'$ induces an isomorphism:

$$U' \setminus U' \cap Y' \xrightarrow{\sim} \Omega := f^{-1}(U) \setminus f^{-1}(U) \cap Y.$$

iii) The fibrations are adapted to $Y'$ over $U: U' \cap Y'$ is a strict relative NCD over $U$ in the smooth fibers of $f'$ (called horizontal or strict in the fibers, eventually empty).

*Remark 11.3.* i) We need the open subset $\Omega$ to be embedded in $X$ and in $X'$ such that we can lift a local system on $\Omega \subset X$ to a dense open subset in $X'$.

ii) The NCD $f'^{-1}(U) \cap Y'$ is called horizontal when at each point $v$ in a maximal stratum of $U$ the intersection $f'^{-1}(v) \cap Y'$ is a NCD in the smooth fiber $X'_v$, while at each point $v$ in a stratum $S$ of $V - U$ the fiber itself is a NCD in the inverse image of a general normal section at $v$.

We prove the Proposition 11.4 in three steps to simplify the exposition: First we transform the morphism $f$ (Lemma 11.2), then the desingularization $\pi: X' \to X$ (Corollary 11.2), then $f$ and $\pi$ simultaneously.

### 11.2.2 Existence of fibrations by NCD over the strata

Let $S = (S_\alpha)$ an algebraic Whitney stratification by a family of strata $S_\alpha$ of an algebraic variety $V$. The subspaces $V_l = \cup_{\dim S_\alpha \le l} S_\alpha$ form an increasing family of closed algebraic subsets of $V$ of dimension $\le l$, with index $l \le n$, where $n$ is the dimension of $V$,

$$V_0 \subset V_1 \subset \cdots \subset V_{n-1} \subset V_n = V$$

Let $f: X \to V$ be an algebraic map. The inverse of a subspace $Z \subset V$, is denoted $X_Z := f^{-1}(Z)$. We are interested in the inverses $X_S = f^{-1}(S)$ of strata $S$ of the stratification $S$.

**Definition 11.3.** A Thom-Whitney stratification of an algebraic map $f: X \to V$ consists of two Whitney stratifications one for $V$ and one for $X$ such that the inverse $X_S$ of each stratum $S$ of $V$ is a union of connected strata of $X$, the restriction of $f$ to each stratum of $X_S$ has maximal rank $\dim S$ and the restriction $f_\lambda: X_S \to S$ is a locally trivial topological fibration.
Using Thom-Mather first isotopy theorem ([Mat 12]), one can prove the following property:

• (W) The link at any point of a stratum is a locally constant topological invariant of the stratum [Mat 12], [LeT 83].

If $f$ is endowed with a Thom-Whitney stratification, the restriction of $f$ to $X_S$ is a topological fibration. (See [GMacP 88], [CaMi 5] subsection 3.5 for some consequences and [DeK 73], subsection 1.3.5).

The following lemma is based on the simple idea that the intersection of a NCD by a general non-singular subspace gives a family of relative NCD. This lemma is important for the proof of Proposition 11.4.

**Lemma 11.2.** Let $f: X \to V$ be a projective morphism of algebraic varieties. We consider $Y$ a closed algebraic strict subset of $X$ containing the singularities of $X$ and eventually the singularities of a local system on $X$. There exists a commutative diagram:

$$ \begin{array}{ccc} X & \xleftarrow{\pi_1} & X_1 \\ f \downarrow & & f_1 \downarrow \\ V & \xleftarrow{id} & V \end{array} \quad \begin{array}{ccc} \dots & \xleftarrow{\pi_i} & X_i \\ f_i \downarrow & & f_i \downarrow \\ \dots & \xleftarrow{id} & V \end{array} \quad \begin{array}{ccc} \dots & \xleftarrow{\pi_k} & X_k \\ f_k \downarrow & & f_k \downarrow \\ \dots & \xleftarrow{id} & V \end{array} \quad \begin{array}{ccc} \pi_{k+1} & \leftarrow & X_{k+1} \\ & & f_{k+1} \downarrow \\ & & V \end{array} $$

where $\rho_i := \pi_1 \circ \dots \circ \pi_i$, for $1 \le i \le k+1$, is a desingularization of $X$, $\pi_1$ is given by modifications over $Y$ and $Y_i := \rho_i^{-1}(Y)$ is a NCD in $X_i$ for $i \ge 1$ such that:

1. there exists $V^1 \subset V$ and inductively a decreasing sequence $V^i$, for $0 < i \le k$, of closed algebraic subspaces of $V$ of dimension $d_i > 0$;

2. for $1 \le j \le i$, the inverse image $f_{i+1}^{-1}(V^j)$ is a NCD in the non-singular variety $X_{i+1}$,

3. $f_{i+1}^{-1}(V^j) \setminus f_{i+1}^{-1}(V^{j+1})$ is a relative NCD over $V^j \setminus V^{j+1}$.

The morphisms $\pi_{i+1}$ are modifications over $f_i^{-1}(V^i)$, for all $i \ge 1$, obtained by blowing-ups over $f_i^{-1}(V^i)$ in $X_i$ and the space $f_i^{-1}(V^i)$ is a NCD in $X_i$ such that $\pi_{i+1}$ induces an isomorphism:

$$ (\pi_{i+1})_\mid : (X_{i+1} \setminus f_{i+1}^{-1}(V^i)) \xrightarrow{\sim} X_i \setminus f_i^{-1}(V^i). $$

Since $\rho_i$ are desingularizations, the $X_i$ are smooth for $i \ge 1$ and the commutativity of the diagram above gives $f_{i+1} := f_i \circ \pi_{i+1}$. Moreover, there exists a Whitney stratification $S^i$ of $V$ adapted to $V^j$ for $j \le i$, satisfying the following property: the strata of $S^i$ and of $S^{i-1}$ coincide outside $V^i$.

The open subsets $\Omega_i := f_i^{-1}(V \setminus V^1) \subset X_i$ (resp. $\Omega := f^{-1}(V \setminus V^1) \subset X$) are dense in $X_i$ (resp. $X$) and we can suppose that, for $k+1 \ge i \ge 1$, $\rho_i$ induces an isomorphism $(\Omega_i \setminus Y_i) \simeq (\Omega \setminus Y)$, and that, for $k+1 \ge i \ge 1$, the morphisms $f_i$ induce on $\rho_i^{-1}(Y \cap \Omega)$ a fibration by relative NCD over the strata of $V \setminus V^1$.
**Definition 11.4.** An algebraic morphism $f : X \to V$ of complex varieties is an admissible fibration if $X$ is a smooth variety and there exists a Thom-Whitney stratification $\mathcal{S}$ of $f$ satisfying the following conditions

1. Let $V_i$ denote the union of the strata of $V$ of dimension $\le i$, then $f^{-1}(V_i)$ is a NCD in $X$.

2. $f$ induces a fibration by relative NCD of $f^{-1}(V_i) \setminus f^{-1}(V_{i-1})$ over $V_i \setminus V_{i-1}$.

Moreover, let $Y$ be a strict NCD in $X$. Let $S_0$ the strata of maximal dimension in $V$. We say that $f$ is an admissible fibration with respect to $Y$ if $Y \cap f^{-1}(S_0)$ is a relative NCD over $S_0$.

*Remark 11.4.* 1. The open subset $\Omega$ is embedded in all $X_i$ such that one can lift a local system $\mathcal{L}$ defined on $\Omega$. The intersection $\rho_i^{-1}(Y \cap \Omega)$ in $X_i$ is a horizontal relative NCD (remark 11.3).

2. Consider the stratification $S^k$ of $V$ and let $d_i := \dim V^i$, $V_{d_i}$ the union of strata of the stratification $S^k$ of dimension $\le d_i$. Then, $f_{k+1}^{-1}(V_{d_i} \setminus V_{d_{i+1}})$ is a NCD in $X_{k+1}$ fibred by NCD over $V^i \setminus V^{i+1} = V_{d_i} \setminus V_{d_{i+1}}$.

3. Moreover, we can suppose that the family of subspaces $V^i$ is maximal in the following sense: the dimension of $V^i$ is $n-i$ for $0 < i \le k = n$.

4. From Lemma 11.2 we obtain that the morphism $f_{k+1} : X_{k+1} \to V$ is an admissible fibration.

**Proof of Lemma 11.2.** Since $f$ is projective, we can always suppose $V = f(X)$. By Hironaka's resolution result [Hiro 64] we have a desingularization morphism $\pi_1 : X_1 \to X$ of $X$ with NCD as exceptional divisor such that $Y_1 := \pi_1^{-1}(Y)$, as well as the inverse images of the irreducible components of $Y$, are sub-NCD of $Y_1$ in $X_1$.

Let $f_1 := f \circ \pi_1$. Since $X_1$ is smooth, there exists an open subset $U \subset V$ such that the restriction of $f_1$ to $f_1^{-1}(U)$ is smooth.

If the dimension of $f(Y)$ is strictly smaller than $n := \dim V$, let $U \subset V \setminus f(Y)$ such that the restriction of $f_1$ to $f_1^{-1}(U)$ is smooth and let $V^1 := V \setminus U$.

In the case $f(Y) = V$, since $Y_1$ is a NCD, there exists $U$ such that the restriction of $f_1$ to $Y_1 \cap f_1^{-1}(U)$ is a fibration by relative NCD over $U$. Let $V^1 := V \setminus U$, since $V$ is irreducible $d_1 := \dim V^1$ is strictly smaller than $\dim V = n$. We can always choose $U$, hence $V^1$, such that $d_1 = n-1$.

We consider a Thom-Whitney stratification of the morphism $f_1 : X_1 \to V$; in particular the image by $f_1$ of a stratum of $X_1$ is a stratum of $V$. We suppose also that the stratification of $X_1$ is compatible with the divisor $Y_1$. Let $\mathcal{S}^0$ denotes the underlying stratification of $V$.

We shall construct the algebraic subspaces $V^i$ of $V$, the morphisms $f_i = X_i \to V$, and $\pi_i : X_i \to X_{i-1}$ in the lemma by descending induction on $\dim V^i$. Let $\mathcal{S}^1$ be a Whitney stratification of $V$ compatible with $V^1$ which is a refinement of $\mathcal{S}^0$. Let $D^1 := f_1^{-1}(V^1)$; we introduce the horizontal divisor (remark 11.3):

$$Y_h^1 := \bar{Y}_1 \cap f_1^{-1}(V \setminus V^1).$$
Notice that $Y_h^1 \setminus D^1$ is a NCD. We construct $\pi_2 : X_2 \to X_1$ by blowing-ups over $D^1$ such that the inverse image $D^2 := \pi_2^{-1}(D^1)$ of $D^1$ and the union $D^2 \cup Y_h^2$ with $Y_h^2 := \pi_2^{-1}(Y_h^1)$ are NCD in the smooth variety $X_2$. We can do this construction without modification of $X_1 \setminus D^1$ because $Y_h^1 \setminus D^1$ is already a NCD.

Let $f_2 := f_1 \circ \pi_2$, hence $D^2 := f_2^{-1}(V^1)$, the next argument allows us to construct a subspace $V^2$ such that $D^2 \cap f_2^{-1}(V^1 \setminus V^2) = f_2^{-1}(V^1 \setminus V^2)$ is a relative NCD over $V^1 - V^2$.

The next lemma provides the argument used in the inductive construction of the diagram in lemma 11.2

**Lemma 11.3.** Let $f : X \to Z$ be a projective morphism on a non-singular space $X$, $T$ an algebraic subspace of $Z$ such that $D := f^{-1}(T)$ is a NCD in $X$ as well the inverse image of each irreducible component of $T$.

Then, there exists a non singular algebraic subset $S_0$ in $T$, such that the dimension of $T \setminus S_0$ is strictly smaller than $\dim T$ and $f$ induces a relative NCD over $T \setminus S_0$.

**Proof.** i) Let $D_{\beta_i}$ denote the components of the divisor $D$ and $D_{\beta_1, \dots, \beta_j}$ the intersection of $D_{\beta_1}, \dots, D_{\beta_j}$. Let $d := \dim T$ be the dimension of $T$; there exists an open subset $S_{\beta_1, \dots, \beta_j}$ complementary of an algebraic subspace of $T$ of dimension strictly smaller than $d$, such that $f^{-1}(S_{\beta_1, \dots, \beta_j}) \cap D_{\beta_1, \dots, \beta_j}$ is either empty, or a topological fibration over $S_{\beta_1, \dots, \beta_j}$. Over the open subset $S_0' := \cap_j(\cap_{\beta_1, \dots, \beta_j} S_{\beta_1, \dots, \beta_j})$, the divisor $D \cap f^{-1}(S_0')$ is a topological fibration.

This property is required to obtain the big stratum in $T$ of a Whitney stratification over which we have a fibration by NCD and which satisfies the assertion (2) of the definition 11.1 above.

ii) Still, we need to check the assertion (3) of the definition 11.1 on a dense open subset of $S_0'$. Let $T_1$ be an irreducible component of $T$ of dimension $d$. Since the inverse image of $T_1$ is also a NCD contained in $D$, there exists a dense open subset $S_1'$ in $T_1$ with inverse image a sub-NCD of $D$.

As the argument is local, we consider an open affine subset $\mathcal{U}$ of $Z$ with non empty intersection $S_1 := T_1 \cap \mathcal{U} \subset S_1'$ and a projection $q : \mathcal{U} \to \mathbb{C}^d$ whose restriction to $S_1$ is a finite projection. There exists an open affine dense subset $U_1$ of $\mathbb{C}^d$ such that $q \circ f$ induces a smooth morphism over $U_1$.

Considering $S_0'$ defined in i) above, we notice that $f$ induces over $q^{-1}(U_1) \cap S_0'$ a fibration by NCD: indeed, let $x$ be a point in $q^{-1}(U_1) \cap S_0' \cap S_1$. There exists an open neighborhood $\mathcal{U}_x$ of $x$ in $(q^{-1}(U_1) \cap S_0') \subset \mathcal{U}$, small enough such that the restriction of $q$ to $\mathcal{U}_x \cap S_1$ is not ramified on its image.

On the other hand, for all $y \in \mathcal{U}_x$, the space $f^{-1}(q^{-1}(q(y)) \cap \mathcal{U}_x)$ is smooth, since $q(y) \in U_1$. The dimension of $f^{-1}(q^{-1}(q(y) \cap \mathcal{U}_x))$ is $\dim X - d$. As $x$ is in $S_0'$ the dimension of $f^{-1}(y) \cap D$ is $\dim X - 1 - d$ and it is a divisor with normal crossings in $f^{-1}(q^{-1}(q(y) \cap \mathcal{U}_x))$ which has dimension: $\dim X - d$.

We remark that $q^{-1}(q(x)) \cap \mathcal{U}_x$ is a normal section of $q^{-1}(U_1) \cap S_0'$ at $x$ in $Z$. Hence $f$ induces on $q^{-1}(U_1) \cap S_0'$ a fibration by NCD lying in the normal sections.
The open subset $S_0$ is the union of open subsets $q^{-1}(U_1) \cap S'_0$ for the various irreducible components $T_1$ of $T$ of maximal dimension $d$. The strata which are the connected component of $S_0$ satisfy the property (3) of the definition 11.1 and this ends the proof of the Lemma 11.3.

**End of the proof of Lemma 11.2.** Let $d_1$ be the dimension of $V^1$. According to Lemma 11.3 there exists an open algebraic subset $S_0$ of $V^1$, over which the restriction of $f_2$ induces a relative NCD over $S_0$, moreover, the dimension $d_2$ of the algebraic set $V^2 := V^1 \setminus S_0$ is $d_2 < d_1$. If $d_1 = n-1$, we can always choose $S_0$ such that $d_2 = n-2$.

We define a new Whitney stratification $S^2$ of $V$, compatible with $V^2$ which coincides with $S^1$ outside $V^1$, by keeping the same strata outside $V^1$ and by adding Thom-Whitney strata adapted to $V^1$ and to the connected components of the open algebraic subset $S_0$ that we have just defined.

We complete the proof by repeating this argument for the closed algebraic subspace $V^2$ of $V^1$.

This process is the beginning of an inductive argument. The initial step is the construction of $V^1$ above.

*Induction hypothesis:* Given projective morphisms $f_j : X_j \to V$ for $1 \le j \le i$ as in the diagram above, and $\pi_j : X_j \to X_{j-1}$, $f_j := f_{j-1} \circ \pi_j$ where $X_j$ is non-singular, a Whitney stratification $S^i$ of $V$ compatible with a family of algebraic subspaces $V^j \subset \dots \subset V^1 \subset V$ for $j \le i$ such that there exists a Whitney stratification on $X_i$ stratifying $f_i$, and such that $f_{i-1}^{-1}(V^j)$ are NCD in $X_i$, as well as the inverse image of each irreducible component of $V^j$ for $1 \le j < i$, and moreover $f_i$ induces an admissible morphism over $V \setminus V^i$. Also, $Y_i := \rho_i^{-1}(Y)$ is a NCD divisor of $X_i$ such that $f_i^{-1}(V^1) \cup Y_i$ is a NCD and $Y_i \cap (X_i \setminus f_i^{-1}(V^1))$ is a horizontal relative NCD over $V \setminus V_1$.

*Inductive step:* A sequence of blowings-up centered over $f_i^{-1}(V^i)$ leads to the construction of $\pi_{i+1} : X_{i+1} \to X_i$ such that $X_{i+1}$ is non-singular and the inverse images of $V^j$, as well as its irreducible components, by $f_{i+1} := f_i \circ \pi_{i+1}$, for $1 \le j \le i$, and their union with $Y_{i+1} := \pi_{i+1}^{-1}(Y_i)$ are NCD in $X_{i+1}$. Moreover $Y_{i+1} \setminus f_{i+1}^{-1}(V^1)$ is a fibration by NCD over the strata of $V \setminus V^1$.

The stratification $S^i$ of $V$ underlies a Thom-Whitney stratification of $f_{i+1}$. It follows from Lemma 11.3 that in each maximal stratum $S$ of $V^i$ in $S^i$, there exists an open dense subset $S_0^i(S)$ over which $\pi_{i+1}$ induces a relative NCD. Let $V_0^i$ be the union of all $S_0^i(S)$. The complement $V^i \setminus V_0^i$ of $V_0^i$ is a closed algebraic strict subspace $V^{i+1}$ of $V^i$, and $f_{i+1}$ is admissible over $V \setminus V^{i+1}$ (see definition in Remark 11.4 (3)). We construct a Thom-Whitney stratification $S^{i+1}$ of $V$ compatible with $V^{i+1}$, keeping the same strata of the stratification $S^i$ outside $V^i$ and introducing as new strata subsets of the connected components of the open subset $V_0^i$ of $V^i$.

The inductive argument ends when $V^{k+1} = \emptyset$, which occurs after a finite number of steps since the family $V^i$ is strictly decreasing.
**Corollary 11.2.** i) Let $f: X \to V$ be an algebraic morphism and $Y$ a strict subvariety containing the singularities of $X$, there exists a desingularization $X' \xrightarrow{\pi} X$ of $X$ such that $Y' = \pi^{-1}Y$ is a NCD and $f \circ \pi$ is admissible with respect to $Y'$.

ii) Let $X$ be an algebraic variety and $Y$ a strict algebraic subspace, there exists a desingularization $X' \xrightarrow{\pi} X$ of $X$ such that $Y' = \pi^{-1}Y$ is a NCD and $\pi$ is admissible with respect to $Y'$.

Indeed by Lemma 11.2, the morphism $f_{k+1}$ is a fibration by NCD over the strata, moreover we can suppose $k = n$.

To construct $\pi$ as a fibration by NCD, we apply the lemma to the case $f = \mathrm{Id}: X \to X$.

By the corollary we can choose $\pi$ or $f \circ \pi$ to be admissible. To show that we can choose both to be admissible we shall need the following improved version of lemma 11.2

**Lemma 11.4 (Fibrations relative to a subspace).** With the notations of Lemma 11.2, let $Z$ be a strict algebraic subspace of $V$. There exists a modification $\pi'$ of $X$

$$X \xleftarrow{\pi'} X' \xrightarrow{f'} V, \quad f' := f \circ \pi'$$

and a Thom-Whitney stratification of $f'$ compatible with $Z$ such that $f' := f \circ \pi'$ is admissible and $f'^{-1}(Z) = \pi'^{-1}(f^{-1}(Z))$ is a relative NCD over the strata in $Z$.

**Proof.** The proof is a slight modification of the proof of Lemma 11.2 such that we construct stratifications compatible with $Z$.

Let $\ell$ be the dimension of $Z$. Since $V$ is a variety and $Z$ is a strict algebraic subspace, we have $\dim V > \ell$. In the proof of lemma 11.2, if the dimension $d_1$ of $V^1$ is $\ge \ell$ we may suppose that $V^1 \supset Z$. Inductively, as long as $d_j \ge \ell$, we may suppose that $V^j \supset Z$.

If we reach an index $s$ such that $\dim V^s = d_s = \ell$, we may suppose $Z \subset V^s$, then by blowing-ups over $f_s^{-1}(V^s)$ we can assume that $f_{s+1}^{-1}(V^s) \subset X_{s+1}$ and $f_{s+1}^{-1}(Z)$ are transformed into NCD. Thus $f_{s+1}^{-1}(V^s)$ (resp. $f_{s+1}^{-1}(Z)$) induces a fibration over a open subset $U$ of $V_s$ (resp. $Z$) such that $\dim(V^s - U) < \dim V^s$. Then we suppose the stratification $S^{s+1}$ of $V$ compatible with $Z$ and we continue similarly as in the above proof of the Lemma 11.2.

Otherwise, if we reach the situation where $d_s > \ell > d_{s+1}$, we consider the subspace $W^{s+1} := Z \cup V^{s+1}$ of dimension $l$ and define the space $X_{s+1}$ by blowing-ups over the subspace $f_s^{-1}(W^{s+1}) \subset X_s$ such that the inverse image of $f_{s+1}^{-1}(Z)$ is a NCD and we continue with Whitney stratifications of $V$ compatible with $Z$. Then, by proceeding as in the end of the proof of Lemma 11.2, we obtain our proof.

**Proof of the Proposition 11.4.** We need to construct a diagram:
$$X' \xrightarrow{\pi} X \xrightarrow{f} V$$

such that $\pi$ and $f \circ \pi$ are simultaneously fibrations by NCD over the strata.

Going back to the inductive argument for $f$ in the Lemma 11.2, we apply at each step of the induction the corollary 11.2 to construct $\pi$.

The initial step of the induction starts with the desingularization by an admissible modification.

*Induction hypothesis.* We suppose there exists:

1) A diagram $\mathcal{D}_i$ of morphisms where $\pi'_i$ is admissible:

$$X \xleftarrow{\pi'_i} X_i \xrightarrow{f_i} V, \quad f_i = f \circ \pi'_i$$

2) A decreasing family of algebraic subspaces $V^i \subset V^j$ for $j < i$ with inverse image $f_i^{-1}(V^j)$ consisting of NCD in $X_i$ for $j < i$, and a stratification $S^i$ of $V$ compatible with the family $V^j$, such that the restriction of $f_i$ to $X_i - f_i^{-1}(V^i)$ over $V - V^i$ is a fibration by relative NCD over the strata.

*Inductive step.* Let $d_i$ (resp. $n - i$) be the dimension of $V^i$. We want to define a sub-space $V^{i+1} \subset V^i$ of dimension strictly smaller $d_{i+1} < d_i$ (resp. $n - i - 1$) and to extend the diagram $\mathcal{D}_i$ over the open subset $V^i - V^{i+1}$, that is, to construct a diagram $\mathcal{D}_{i+1}$ of morphisms: $X \xleftarrow{\pi_{i+1}'} X_{i+1} \xrightarrow{f_{i+1}'} V$ where $f_{i+1} = f \circ \pi'_{i+1}$ such that:

1) $\pi'_{i+1}$ is admissible and defined as a composition map $\pi'_{i+1} : X_{i+1} \xrightarrow{\pi_{i+1}'} X_i \xrightarrow{\pi'_i} X$ of $\pi'_i : X_i \to X$ with a modification $\pi_{i+1}$ inducing an isomorphism:

$$X_{i+1} \setminus f_{i+1}^{-1}(V^i) \xrightarrow{\sim} X_i - f_i^{-1}(V^i).$$

2) $f_{i+1}^{-1}(V^i)$ is a relative NCD over the open subset $V^i \setminus V^{i+1}$.

To achieve this step, we remark that the variety $f_i^{-1}(V^i)$ in $X_i$ is over $f^{-1}(V^i)$ in $X$. Then, we apply the lemma 11.4 to the morphism $\pi'_i : X_i \to X$ (instead of $f : X \to V$ in the lemma), and $f^{-1}(V^i) \subset X$ (instead of $Z \subset V$ in the lemma) to construct $\pi_{i+1}$ such that $\pi'_{i+1} := \pi'_i \circ \pi_{i+1}$ is an admissible morphism:

$$f^{-1}(V^i) \subset X \xleftarrow{\pi'_i} X_i \xleftarrow{\pi_{i+1}} X_{i+1}, \quad \pi'_{i+1} := \pi'_i \circ \pi_{i+1} : X_{i+1} \to X$$

That is we develop the constructions of the lemma 11.2 to construct $\pi'_{i+1}$ over $X$ by modification only of subspaces over $f_i^{-1}(V^i)$ to transform $f_i^{-1}(V^i)$ into a NCD, hence $X_{i+1}$ in the diagram $\mathcal{D}_{i+1}$ differs from $X_i$ in the diagram $\mathcal{D}_i$ only over $f_i^{-1}(V^i)$. The stratification of $X$ is modified inside the subset $f^{-1}(V^i)$, hence we modify adequately the stratification of $V$ inside the subset $V^i$.

There exists an open algebraic subset $U \subset V_i$ in the big strata of $V_i$ over which $f_{i+1}$ induces a fibration by NCD (Lemma 11.3). Let $V^{i+1} := V_i \setminus U$ of dimension strictly smaller than $d_i$. Then, the Thom-Whitney stratifications of
$f$ and $\pi'_i$ outside $V_i$ extend over $U$ into Thom-whitney stratifications of $f$ and $\pi'_{i+1}$. Hence, we deduce a diagram $\mathcal{D}_{i+1}$ which extend the diagram $\mathcal{D}_i$ such that $f_{i+1}$ (resp. $\pi'_{i+1}$) is a fibration by NCD over the strata of $f_{i+1}^{-1}(V^i \setminus V^{i+1})$ (resp. $f^{-1}(V^i \setminus V^{i+1})$.

Thus we complete the construction of the diagram $\mathcal{D}_{i+1}$ and the inductive step. At the end we define $\pi'_m$ and $f_m$ both admissible for some index $m$.

**Corollary 11.3.** *The decomposition theorem for $f$ can be deduced from both cases $\pi'$ and $f'$ in the proposition 11.4.*

Let $\mathcal{L}$ be a local system on $X-Y$, there exists an open algebraic set $\Omega \subset X \setminus Y$ dense in $X$ such that $\Omega' := \pi'^{-1}(\Omega) \subset X'$ is isomorphic to $\Omega$, which carry the local system $\mathcal{L}$. Let $j' : \Omega' \to X'$, then the decomposition theorem for $j_*\mathcal{L}$ with respect to the orginal proper algebraic morphism $f$ follows from both cases of $f'$ and $\pi'$ ([De 68] Proposition 2.16).

## 11.3 Combinatorial description of the weight

We resume the notations of (§3.1). The intersection complex may be described by a combinatorial complex [KaK 86]. Such description leads to a definition of the weight filtration $W$ on a combinatorial complex quasi-isomorphic to the de Rham complex $\Omega^*\mathcal{L}$.

First we take some time to associate to the set $I$ of indices of the irreducible components of the NCD $Y$, a category $S(I)$ whose objects are indices of the combinatorial complex (still denoted $\Omega^*\mathcal{L}$).

The combinatorial weights $W_0$ or $W_{-1}$ on $\Omega^*\mathcal{L}$ describe a combinatorial sub-complex $\mathrm{IC}^*\mathcal{L}$ quasi-isomorphic to the intermediate extension $j_*\mathcal{L}$.

To study the local properties of the graded complexes $\mathrm{Gr}_r^W \Omega^*\mathcal{L}$ for $r \in \mathbb{Z}$, we introduce locally the complexes $C_r^{KM}L$ for $K \subset M \subset I$. In particular, the main task is to show that for $K = M$ the complex $C_r^K L$ has a unique non zero cohomology $H^{|K|}C_r^K L$ for $r > 0$ (resp. $H^{|K|-1}C_r^K L$ for $r > 0$) with a pure polarized HS.

The result is a computation of the decomposition of $\mathrm{Gr}_r^W \Omega^*\mathcal{L}$ into intermediate extensions of the VHS defined by $H^{|K|}C_r^K L$ on the non singular intersections of components $Y_K$ of the NCD.

### 11.3.0.1 Complexes with indices in the category $S(I)$

The objects of the category $S(I)$ attached to a set $I$, consist of sequences of decreasing subsets of $I$ of the following form:

$$ (s_.) = (I = s_1 \supsetneq s_2 \dots \supsetneq s_p \neq \emptyset), \quad (p > 0) $$
Subtracting a subset $s_i$ defines a morphism $\delta_i(s.) : (s. \setminus s_i) \to s.$ and more generally $\operatorname{Hom}(s.^{\prime}, s.)$ is equal to one element iff $(s.^{\prime}) \le (s.)$ that is $(s.^{\prime})$ is obtained from $(s.)$ by deleting some subsets. We write $s. \in S(I)$ and define its degree or length $|s.|$ as the number of subsets $s_i$ in $(s.)$.

*Correspondence with an open simplex.* If $I = \{1, \dots, n\}$ is finite, $S(I)$ can be realized as a barycentric subdivision of the open simplex $\Delta_{n-1}$ of dimension $n-1$. A subset $s_\lambda$ corresponds to the barycenter $b_{s_\lambda}$ of the vertices of $s_\lambda$ and a sequence of subsets $s_\lambda$ corresponds to the oriented sub-simplex of $\Delta_{n-1}$ defined by the vertices $b_{s_\lambda}$.

For example, for $I = \{1, 2\}$, the barycenter $\{3/2\}$ of $]1, 2[$ is defined by $\{1, 2\}$, and the open simplices $]1, 3/2[$, $]3/2, 2[$ are defined resp. by the sequences $\{1, 2\} \supset \{1\}$ and $\{1, 2\} \supset \{2\}$, plus the integer $\{3/2\}$ corresponding to $\{1, 2\}$.

Since all sequences contain $I$, all corresponding simplices must have the barycenter of $\Delta_{n-1}$ defined by $I$ as vertex and the sub-simplices form a partition of the open simplex $\Delta_{n-1}^* = \Delta_{n-1} \setminus \partial\Delta_{n-1}$. In this way we define an incidence relation $\varepsilon(s., s.^{\prime})$ between two adjacent sequences equal to +1 or -1.

*Combinatorial objects of an abelian category with indices in $S(I)$* are defined as functors on $S(I)$.

We need the following construction. An algebraic or analytic variety over a fixed variety $X$ with indices in $S(I)$ is a contravariant functor defined by $\Pi(s.) : X_{(s.)} \to X$ and denoted by $\Pi$ with morphisms $\Pi(s.^{\prime} \le s.) : X_{s.^{\prime}} \to X_{s.^{\prime}}$ over $X$.

A complex $\mathcal{F}$ of abelian sheaves $\mathcal{F}_s$. over $X_s$, is defined with functorial morphisms $\varphi(s.^{\prime} \le s.) : \mathcal{F}_s \to \mathcal{F}_{s.^{\prime}}$ for $(s.^{\prime}) \le (s.)$. For each $s_\lambda \ne I \in (s.)$ and the couple $\delta_{s_\lambda} := ((s.) \setminus s_\lambda < (s.))$ corresponds a morphism $\Pi_{\delta_{s_\lambda}} : X_s \to X_{(s.) \setminus s_\lambda}$ and a morphism $\varphi(\delta_{s_\lambda}) : \Pi_*\mathcal{F}_s \to \Pi_*\mathcal{F}_{(s.) \setminus s_\lambda}$ with a sign $\epsilon_{\delta_{s_\lambda}}$ defined by the incidence relation in the open simplex interpretation.

The direct image of a complex of abelian sheaves over $\Pi$ is the simple complex complex associated to a double complex on $X$:

$$s(\mathcal{F}_s)_{s. \in S} = \bigoplus_{s. \in S} (\Pi_* \mathcal{F}_s)[|s.| - |I|], \quad d = \sum_{\delta_{s_\lambda}} \varepsilon(\delta_{s_\lambda}) \varphi(\delta_{s_\lambda})$$

denoted by $\Pi_* \mathcal{F}$ or preferably $s(\mathcal{F}_s)_{s. \in S(I)}$.

*Example 11.1.* In the case of the constant variety $X_s = X$ defined by a variety $X$, the constant sheaf $\mathbb{Z}$ lifts to a sheaf on $X_s$, such that the diagonal morphism:

$$\mathbb{Z}_X \to \bigoplus_{|(s.^{\prime})|=|I|} \mathbb{Z}_{X_s} : n \in \mathbb{Z} \to (\dots, n_{s.} = n, \dots) \in \bigoplus_{|(s.^{\prime})|=|I|} \mathbb{Z}$$

defines a quasi-isomorphism $\mathbb{Z}_X \cong \Pi_* \Pi^*(\mathbb{Z}_X)$, as the barycentric subdivision of an open simplex of dimension $|I| - 1$ shows.
### 11.3.0.2 Local definition of the weight filtration

If $Y \subset X$ is a NCD with indices $i \in I$ and $M \subset I$, we consider the category $S(M)$ attached to $M$ whose objects consist of sequences of decreasing subsets of $M$ of the form $(s.) = (M = s_1 \supseteq s_2 \dots \supseteq s_p \neq \emptyset)$, $p > 0$. Geometrically $S(M)$ corresponds to the case of a normal section to $Y_M^* = \cap_{i \in M} Y_i$ in $X$.

Let $(L, (N_i)_{i \in M}, F, m)$ be a polarized nilpotent orbit of weight $m$ with monodromy weight filtrations $W^J := W(\sum_{i \in J} N_i)$ for $J \subset M$.

We introduce a local definition of the weight filtration on the family of de Rham complexes $(\Omega^*L)_{s. \in S(M)}$.

**Notations.** For each $s. \in S(M)$ let $W^{s_\lambda} = W(\sum_{i \in s_\lambda} N_i)$ be centered at 0. For $J \subset M$ and an integer $r$, we define $a_{s_\lambda}(J, r) = |s_\lambda| - 2|s_\lambda \cap J| + r$, and the functorial filtered vector space

$$W_r(J, s.)L := \bigcap_{s_\lambda \in s.} W_{a_{s_\lambda}(J, r)}^{s_\lambda} L, \quad F^r(J, s.): = F^{r-|J|}L \quad (11.11)$$

The following definition reduces for $r = -1$ to the construction of the intersection complex by Kashiwara and Kawai [KaK 86].

**Definition 11.5.** The weight $W$ (centered at zero) and Hodge $F$ filtrations on the combinatorial de Rham complex $\Omega^*L := s(\Omega^*L)_{s. \in S(M)}$ are defined by summing over $J$ and $s.$:

$$W_r(\Omega^*L) := s\left(\bigcap_{s_\lambda \in s.} W_{a_{s_\lambda}(J,r)}^{s_\lambda} L\right)_{(J \subset M, s. \in S(M))} \quad (11.12)$$

where $a_{s_\lambda}(J, r) = |s_\lambda| - 2|s_\lambda \cap J| + r$ and $F^r(\Omega^*L) := s(F^{r-|J|}L)_{(J \subset M, s. \in S(M))}$. We denote the bifiltered sum complex by

$$(\Omega^*L, W, F)$$

The filtrations can be constructed in two times, first by summing over $J$ to get the sub-complexes:

$$\text{weight: } W_r(s.) = s(W_r(J, s.))_{J \subset M} \text{ and Hodge: } F^r(s.): = s(F^r(J, s.))_{J \subset M}.$$

**Example 11.2.** in dimension 2.

Let $W^{1,2} = W(N_1 + N_2)$, $W^1 = W(N_1)$ and $W^2 = W(N_2)$, the weight $W_r$ is a double complex:

$$W_r(\{1, 2\} \supseteq 1) \oplus W_r(\{1, 2\} \supseteq 2) \to W_r(\{1, 2\})$$

where the first line is the direct sum of $W_r(\{1, 2\} \supseteq 1) =$

$$(W_{r+2}^{1,2} \cap W_{r+1}^{1,2}) \xrightarrow{(N_1, N_2)} W_r^{1,2} \cap W_{r-1}^{1,2} \oplus W_r^{1,2} \cap W_{r+1}^{1,2} \xrightarrow{(-N_2, N_1)} W_{r-2}^{1,2} \cap W_{r-1}^{1,2})$$

and $W_r(\{1, 2\} \supseteq 2) =$

$$(W_{r+2}^{1,2} \cap W_{r+1}^{2,2}) \xrightarrow{(N_1, N_2)} W_r^{1,2} \cap W_{r+1}^{2,2} \oplus W_r^{1,2} \cap W_{r-1}^{2,2} \xrightarrow{(-N_2, N_1)} W_{r-2}^{1,2} \cap W_{r-1}^{2,2})$$

The second line for $\{1, 2\}$ is $W_r(\{1, 2\}) =$
$$ (W_{r+2}^{1,2} \xrightarrow{(N_1, N_2)} W_r^{1,2} \oplus W_r^{1,2} \xrightarrow{(-N_2, N_1)} W_{r-2}^{1,2}). $$

**Lemma 11.5 (Kashiwara and Kawai [KaK 86]).** There exists a quasi-isomorphism $IC^*L \simeq W_{-1}(\Omega^*L)$.

### 11.3.0.3 The complexes $C_r^{KM}L$ and $C_r^K L$

To study the graded part of the weight, we need to introduce the following full sub-categories:

For each subset $K \subset M$, let $S_K(M) = \{s. \in S(M) : K \in s.\}$ (that is $\exists \lambda : K = s_\lambda$). We remark the isomorphism of categories:

$$ S(K) \times S(M-K) \xrightarrow{\sim} S_K(M), \quad (s., s') \to (K \cup s., s.) $$

We consider the vector spaces with indices $J \subset M, s. \subset S_K M$,

$C_r^{KM}L(J, s.): = \bigcap_{K \neq s_\lambda \in s.} W_{a_\lambda(J,r-1)}^{s_\lambda} Gr_{a_{K(J,r)}}^{W^K} L$ and for each $(s.)$ the associated complex obtained by summing over $J$ (resp. over $(s.):$

$$ C_r^{KM}L(s.): = s(C_rL(J, s.))_{J \in M^+}, \quad C_r^{KM}L: = s(C_r^{KM}L(s.))_{(s.) \in S_K M}. $$

We write $C_r^K L(J, s.), C_r^K L(s.)$ and $C_r^K L$ when $K = M$.

**Definition 11.6.** For $K \subseteq M$ the complex $C_r^{KM}L$ is defined by summing over $J$ and $(s.)$

$$ C_r^{KM}L: = s\left( \bigcap_{K \neq s_\lambda \in s.} W_{a_\lambda(J,r-1)}^{s_\lambda} Gr_{a_{K(J,r)}}^{W^K} L \right)_{J \subset M, s. \in S_K(M)} \quad (11.13) $$

In the case $K = M$ we write $C_r^K L$

$$ C_r^K L = s\left( \left( \bigcap_{\neq s_\lambda \in s.} W_{a_{s_\lambda}(J,r-1)}^{s_\lambda} Gr_{a_{K(J,r)}}^{W^K} L \right)_{J \subset K, s. \in S(K)} \right) \quad (11.14) $$

**Example 11.3.** For $n = 1$, $K$ and $M$ reduces to one element 1 and the theorem reduces to

$$ Gr_r^W (\Omega^* L) \simeq C_r^1 L: = Gr_{r+1}^{W^1} L \xrightarrow{N_1} Gr_{r-1}^{W^1} L $$

By the elementary properties of the weight filtration of $N_1$, it is quasi-isomorphic to $Gr_{r-1}^{W^1}(L/N_1L)[-1]$ if $r > 0$, $Gr_{r+1}^{W^1}(ker N_1 : L \to L)$ if $r < 0$ and $C_0^1 L \simeq 0$.

**Remark 11.5.** The above combinatorial filtration $W$ on the combinatorial de Rham complex $\Omega^*\mathcal{L} \simeq s(\Omega^*\mathcal{L})_{s. \in S(I)}$ where $I$ is the set of indices of the NCD $Y \subset X$, has been studied in [E 08].

The decomposition of $Gr_*^W \Omega^*\mathcal{L}$ depends on the fact that the cohomology of the complex $C_r^K L$ vanishes in all but the degree $|K|$ for $r > 0$ and $C_0^K L \simeq 0$. There is a mistake in the proof which occurs on ([E 08] Lemma A.12 iii).

We give below a correction in the case of surfaces.
**Proposition 11.5.** Let $(L, N_1, N_2)$ be an IMHS and let $K = M = \{1, 2\}$. The non zero cohomology of $C_r^K L$ is $H^2(C_r^K L)$ for $r \ge 0$ and $C_0^K L = 0$.

*Proof.* Let $L_1$ resp. $L_2, L_3$ denote the following complexes

$$0 \to \mathrm{Gr}_{r+2}^{W^{12}} W_{r+1}^{1} \xrightarrow{N_1, N_2} \mathrm{Gr}_r^{W^{12}} W_{r-1}^{1} \oplus \mathrm{Gr}_r^{W^{12}} W_{r+1}^{1} \xrightarrow{N_2, N_1} \mathrm{Gr}_{r-2}^{W^{12}} W_{r-1}^{1} \to 0$$

$$0 \to \mathrm{Gr}_{r+2}^{W^{12}} W_{r+1}^{2} \xrightarrow{N_1, N_2} \mathrm{Gr}_r^{W^{12}} W_{r+1}^{2} \oplus \mathrm{Gr}_r^{W^{12}} W_{r-1}^{2} \xrightarrow{N_2, N_1} \mathrm{Gr}_{r-2}^{W^{12}} W_{r-1}^{2} \to 0$$

$$0 \to \mathrm{Gr}_{r+2}^{W^{12}} L \xrightarrow{N_1, N_2} \mathrm{Gr}_r^{W^{12}} L \oplus \mathrm{Gr}_r^{W^{12}} L \xrightarrow{N_2, N_1} \mathrm{Gr}_{r-2}^{W^{12}} L \to 0$$

The complex $C_r^K L$ is the sum of the double complex:

$$C_r^K L := (0 \to L_1 \oplus L_2 \xrightarrow{i_1, i_2} L_3 \to 0)$$

with natural embeddings $(i_1, i_2)$.

The sum of $L_1 \xrightarrow{i_1} L_3$ is a complex quasi-isomorphic to $L_4[-1]$ where $L_4 := (L_3/L_1)$ is:

$$\mathrm{Gr}_{r+2}^{W^{12}} (L/W_{r+1}^{1}) \xrightarrow{N_1, N_2} \mathrm{Gr}_r^{W^{12}} (L/W_{r-1}^{1}) \oplus \mathrm{Gr}_r^{W^{12}} (L/W_{r+1}^{1}) \xrightarrow{N_2, N_1} \mathrm{Gr}_{r-2}^{W^{12}} (L/W_{r-1}^{1})$$

We compute first the cohomology of $L_2$ and $L_4$.

**Lemma 11.6.** i) The cohomology of $L_2$ for $r \ge 0$ is equal to:

$$H^0(L_2) = 0, H^1(L_2) = \mathrm{Gr}_r^{W^{12}} \mathrm{Ker} (N_2 : L/N_1 \to L/N_1 L) \text{ and}$$

$$H^2(L_2) = \mathrm{Gr}_{r-2}^{W^{12}} \mathrm{Coker} (N_2 : L/N_1 \to L/N_1 L).$$

ii) The cohomology of $L_4$ for $r \ge 0$ is equal to:

$$H^0(L_4) = H^2(L_2) = 0, H^1(L_4) = \mathrm{Gr}_r^{W^{12}} \mathrm{Ker} (N_2 : L/N_1 \to L/N_1 L).$$

The computation of the cohomology is based on Kashiwara's canonical decomposition [KaK 86]: $\mathrm{Gr}_r^{W^{12}} L \simeq \bigoplus_k \mathrm{Gr}_r^{W^{12}} \mathrm{Gr}_k^{W^i} L$ for $i = 1, 2$, applied to the subspace $\mathrm{Ker} N_j \subset L, j = 1, 2$.

*Sub-lemma.* Let $\mathrm{Ker} N_2 := \mathrm{Ker} N_2 : L \to L$.

i) $\mathrm{Gr}_r^{W^{12}} \mathrm{Ker} N_2 = \bigoplus_{k \le 0} \mathrm{Gr}_r^{W^{12}} \mathrm{Gr}_k^{W^2}$ (Ker $N_2$),

$$\mathrm{Gr}_r^{W^{12}} \mathrm{Coker} N_2 = \bigoplus_{k \ge 0} \mathrm{Gr}_r^{W^{12}} \mathrm{Gr}_k^{W^2} \text{ (Coker } N_2).$$

ii) $\mathrm{Gr}_r^{W^{12}} \mathrm{Ker} N_2 = \bigoplus_{k > r-1} \mathrm{Gr}_r^{W^{12}} \mathrm{Gr}_k^{W^1}$ (Ker $N_2$).

Since the morphism $N_j, j = 1, 2$ is strict for the filtration $W^i, i = 1, 2$, an element $a \in L$ satisfy: $N_j(a) = 0$ if and only if its class in $\mathrm{Gr}_k^{W^i} L$ is in the kernel of the restriction $\tilde{N}_j$ of $N_j$ to $\mathrm{Gr}_k^{W^i} L$ (an element $c \in \mathrm{Gr}_k^{W^i} L$ such that $N_j c = 0$ is the class of an element $a_i \in W_k^i \mathrm{Ker} N_j$).

For i) we remark that $\mathrm{Ker} N_2$ is of weights $W_k^2$ for $k \le 0$, while ii) follows from the isomorphism: $\mathrm{Gr}_r^{W^{12}} \mathrm{Gr}_k^{W^1} L \simeq \mathrm{Gr}_{r-k}^{\tilde{W}^2} \mathrm{Gr}_k^{W^1} L$ where $\tilde{W}^2 := W(\tilde{N}_2)$ is the monodromy filtration of the restriction $\tilde{N}_2$ to $\mathrm{Gr}_k^{W^1} L$.
In fact $\tilde{N}_2 : \mathrm{Gr}_{r-k}^{\tilde{W}^2} \mathrm{Gr}_k^{W^1} \to \mathrm{Gr}_{r-k-2}^{\tilde{W}^2} \mathrm{Gr}_k^{W^1}$ is injective for $r-k \ge 1$ and surjective for $r-k \le 1$ which proves the sub-lemma.

*Proof continued.* We write $L_2$ as the sum of a double complex $L'_2 \xrightarrow{N_2} L''_2$:

$$L'_2 = (\mathrm{Gr}_{r+2}^{W^2} W_{r+1}^2 \xrightarrow{N_1} \mathrm{Gr}_r^{W^2} W_{r+1}^2), L''_2 := (\mathrm{Gr}_r^{W^2} W_{r-1}^2 \xrightarrow{N_1} \mathrm{Gr}_{r-2}^{W^2} W_{r-1}^2)$$

Let $\tilde{N}_1$ be induced by $N_1$ then:

$$L'_2 = \bigoplus_{k \le r+1} (\mathrm{Gr}_{r+2-k}^{W(\tilde{N}_1)} \mathrm{Gr}_k^{W^2} L \xrightarrow{\tilde{N}_1} \mathrm{Gr}_{r-k}^{W(\tilde{N}_1)} \mathrm{Gr}_k^{W^2} L)$$

$$L''_2 = \bigoplus_{k \le r+1} (\mathrm{Gr}_{r+2-k}^{W(\tilde{N}_1)} \mathrm{Gr}_{k-2}^{W^2} L \xrightarrow{\tilde{N}_1} \mathrm{Gr}_{r-k}^{W(\tilde{N}_1)} \mathrm{Gr}_{k-2}^{W^2} L),$$

hence $\tilde{N}_1$ is injective on $L'_2$ and $L''_2$ since $r-k \ge -1$. We deduce

$$L'_2 = \bigoplus_{k \le r} \mathrm{Prim} \ \mathrm{Gr}_{r-k}^{W(\tilde{N}_1)} \mathrm{Gr}_k^{W^2} L[-1] := \bigoplus_{k \le r} \mathrm{Prim}_{r-k}^1 \mathrm{Gr}_k^{W^2} L[-1]$$

$$L''_2 = \bigoplus_{k \le r} \mathrm{Prim} \ \mathrm{Gr}_{r-k-2}^{W(\tilde{N}_1)} \mathrm{Gr}_{k-2}^{W^2} L[-1] := \bigoplus_{k \le r} \mathrm{Prim}_{r-k-2}^1 \mathrm{Gr}_{k-2}^{W^2} L[-1]$$

In particular $H^0(L'_2) = H^0(L''_2) = 0$. Let

$$L_2(k) := (\mathrm{Prim}_{r-k}^1 \mathrm{Gr}_k^{W^2} L \xrightarrow{N_2'} \mathrm{Prim}_{r-k-2}^1 \mathrm{Gr}_{k-2}^{W^2} L)[-1] \quad (11.15)$$

where $N'_2$, induced by $N_2$, is injective for $k \ge 1$ and surjective for $k-2 \le -1$, hence we consider the cases $k \in [1, r]$ where $r \ge 0$, $k < 1$, $k = 1$ or $r+1$.

The complex $L_2$ splits into: $(\bigoplus_{k<1} L_2(k)) \oplus (\bigoplus_{k \in [1,r]} L_2(k))$ where:

$$\bigoplus_{k<1} L_2(k) \simeq \mathrm{Gr}_r^{W^{12}} \mathrm{Ker} \ N_2 : (L/N_1L) \to L/N_1L)[-1], \quad (11.16)$$

$$\bigoplus_{k \in [1, r]} L_2(k) \simeq \mathrm{Gr}_{r-2}^{W^{12}} \mathrm{Coker} \ N_2 : (L/N_1L) \to (L/N_1L)[-2]$$

Since $k \le r$, the formula does not cover all Coker $N_2$ but covers $\mathrm{Gr}_{r-2}^{W^{12}}$, and $L_2 \simeq 0$ for $k=1$ or $r+1$. In particular:

$$H^1(L_2) \simeq \mathrm{Gr}_r^{W^{12}} \mathrm{Ker} \ N_2 : (L/N_1L) \to L/N_1L),$$

$$H^2(L_2) \simeq \bigoplus_{k>1} \mathrm{Coker} \ N_2' : \mathrm{Gr}_{r-2}^{W^{12}} \mathrm{Coker} \ N_2 : (L/N_1L) \to (L/N_1L).$$

*Sub-lemma.* $H^1(L_4) \simeq \mathrm{Gr}_r^{W^{12}} \mathrm{Ker} \ N_2 : (L/N_1L) \to L/N_1L)$,

$$H^i(L_4) = 0 \text{ if } i \ne 1.$$

We apply a similar argument to $L_4$: $\tilde{N}_1 : \bigoplus_{k>r+1} \mathrm{Gr}_{r+2}^{W^{12}} \mathrm{Gr}_k^{W^1} \to \mathrm{Gr}_r^{W^{12}} \mathrm{Gr}_{k-2}^{W^1}$ is injective on $\mathrm{Gr}_k^{W^1}$ since $k > 1$ as $r \ge 0$, hence

$$L'_4[1] = \bigoplus_{k>r+1} \mathrm{Gr}_r^{W^{12}} \mathrm{Prim} \ \mathrm{Gr}_{k-2}^{W^1}, \quad L''_4[1] = \bigoplus_{k>r+1} \mathrm{Gr}_{r-2}^{W^{12}} \mathrm{Prim} \ \mathrm{Gr}_{k-2}^{W^1}$$

then $\tilde{N}_2 : \mathrm{Gr}_{r-k+2}^{W(\tilde{N}_2)} \mathrm{Prim} \ \mathrm{Gr}_{k-2}^{W^1} \to \mathrm{Gr}_{r-k}^{W(\tilde{N}_2)} \mathrm{Prim} \ \mathrm{Gr}_{k-2}^{W^1}$ is surjective since $r-k < -1$, hence $H^i(L_4) = 0$ for $i \ne 1$, while:

$$H^1(L_4) = \mathrm{Gr}_r^{W^{12}} \mathrm{Ker} \ (N_2 : L/N_1L \to L/N_1L)$$

**Corollary 11.4.** For $r=0$, $C_0^K$ is acyclic. For $r \ge 1$, $H^i(C_r^K L) = 0$ if $i \ne 2$ and $H^2(C_r^K L) \simeq \mathrm{Gr}_{r-2}^{W^{12}} (L/(N_1L + N_2L))$,

Indeed, $i_2 : H^1(L_2) \to H^1(L_4)$ is an isomorphism.

From now on, we assume:

$$L_r^K := H^{|K|}(C_r^K L), \quad C_r^K L \simeq L_r^K[-|K|], \quad C_0^K L \simeq 0 \quad (11.17)$$
The complex $C_r^{KM}L$ is related to $C_r^K L$ by the following formula

$$C_r^{KM}L \simeq IC^*(C_r^K L) \simeq IC^*(L_r^K)$$

where $IC^*(C_r^K L)$ is the simple complex defined by application of $IC^*$ to each term of $C_r^K L$.

**Lemma 11.7 (decomposition).** Let $(L, N_i, i \in M)$ be a nilpotent orbit. For all subsets $K \subset M$ there exist canonical embeddings of $C_r^{KM}L$ in $Gr_r^M \Omega^*L$ inducing a quasi-isomorphism (decomposition as a direct sum in the derived category)

$$Gr_r^W(\Omega^*L) \cong \bigoplus_{K \subset M} C_r^{KM}L, \quad Gr_0^W(\Omega^*L) \simeq 0 \quad (11.18)$$

The proof is based on the following elementary remark:

Let $W^i$ for $i = 1, 2$ be two increasing filtrations on an object $V$ of an abelian category and $a_i$ two integers, then we have an exact sequence:

$$0 \to W_{a_2-1}^2 Gr_{a_1}^{W^1} \oplus W_{a_1-1}^1 Gr_{a_2}^{W^2} \to W_{a_1}^1 \cap W_{a_2}^2 / W_{a_1-1}^1 \cap W_{a_2-1}^2 \to Gr_{a_1}^{W^1} Gr_{a_2}^{W^2} \to 0$$

The proof of the lemma is by induction by repeating such exact sequences on the terms of the combinatorial sum ([E 08], Theorem A.30).

### 11.3.0.4 Mixed case

In the case of a VMHS $(\mathcal{L}, W^0, F)$ on $X \setminus Y$ we define a structure of MHC on $\Omega^*\mathcal{L}$. The global construction is reduced to the following local case expressed in terms of an IMHS $(L, W^0, F, N_i, i \in M)$ attached to a point $x \in Y_M^*$.

In the definition (11.5) we consider now for each $s_\lambda \in S(M)$ the relative monodromy filtration on $L$ $M^{s_\lambda} := W(N_{s_\lambda}, W^0))$ where $N_{s_\lambda} := \sum_{i \in s_\lambda} N_i$ with respect to the constant filtration $W^0$ defined on $L$ by Deligne's extension of the locally constant filtration $W^0$ on $\mathcal{L}$.

Let $a_{s_\lambda}(J, r) = |s_\lambda| - 2|s_\lambda \cap J| + r$, we consider on the combinatorial de Rham complex $\Omega^*L \simeq s(\Omega^*L)_{s \in S(M)}$ various filtrations

$$ \begin{align} M_r(J, s.) &= \bigcap_{s_\lambda \in s.} M_{a_{s_\lambda}}^{s_\lambda}(J, r)L, \quad W_r(J, s.)L = W_r^0 \cap M_r(J, s.)L \notag \\ M_r(\Omega^*L) &= s(M_r(J, s.))_{J \subset M, s. \in S(M)}, \quad F^r(\Omega^*L) = s(F^{r-|J|}L)_{J \subset M, s. \in S(M)} \notag \\ W_r(\Omega^*L) &= s(W_r(J, s.))_{J \subset M, s. \in S(M)} \tag{11.19} \end{align} $$

**Definition 11.7.** The bifiltered complex $(\Omega^*L, W^0, W, F)$ sum of the combinatorial de Rham complex $\Omega^*L \simeq s(\Omega^*L)_{s \in S(M)}$ is endowed with

1. the constant filtration $W^0$ defined by Deligne's extension of the locally constant filtration $W^0$ on $\mathcal{L}$,
2. the weight filtration W (Equation 11.19) and Hodge filtration F .

Lemma 11.8 (decomposition) . let ( L, W $^{0}$, N$_{i}$, i ∈ M ) be an IMHS. There exist a canonical quasi-isomorphism (decomposition as a direct sum in terms of Gr W 0 i L in the pure case

$$
Gr_r^W (\Omega^* L) \cong IC^* (Gr_r^{W^0} L) \oplus_{k<r} Gr_{r-k}^W (\Omega^* (Gr_k^{W^0} L)) \quad (11.20)
$$

Proof. We use the exact sequence for each J ⊂ M, ( s. ) ⊂ S ( M )
0 → M ( J, s. )$_{r}$$_{−}$$_{1}$Gr W 0 r ⊕ W 0 r − $_{1}$Gr M ( J,s. ) r → Gr W ( J,s. ) r → Gr M ( J,s. ) r Gr W 0 r → 0
The first term may be expressed in terms of W ( J, s. ) of the pure case Gr W 0 r L

$$
M(J, s.)_{r-1} Gr_r^{W^0} L = W(J, s.)_{-1} (Gr_r^{W^0} L)
$$

(Equation 11.11). By (Lemma 11.5) the sum is the intersection complex

$$
s(M(J, s.)_{r-1} Gr_r^{W^0} L)_{J \subset M, s. \in S(M)} \simeq IC^*(Gr_r^{W^0} L)
$$

The second term decomposes into the sum of pure cases

$$
Gr_r^{M(J,s.)} W_{r-1}^0 \cong \oplus_{k<r} Gr_r^{M(J,s.)} Gr_k^{W^0} L = \oplus_{k<r} Gr_{r-k}^{W(J,s.)} (Gr_k^{W^0} L)
$$

The sum over J ⊂ M, s. ∈ S ( M ) decomposes as a sum of pure cases

$$
s(Gr_r^{M(J,s.)} W_{r-1}^0)_{J \subset M, s. \in S(M)} \cong \bigoplus_{k<r} Gr_{r-k}^W \Omega^*(Gr_k^{W^0} L)
$$

While the sum of the last term vanish by equation 11.18:

$$
s(Gr_r^{M(J,s.)} Gr_r^{W^0} L)_{J \subset M, s. \in S(M)} = \mathrm{Gr}_0^W \Omega^* (Gr_r^{W^0} L) \approx 0
$$

Example 11.4. For n = 1, x a singular point of L , K = M = x

$$
\begin{align*}
\Omega^* L &= (L \xrightarrow{N_1} L), W_r = (M_{r+1}^1 \cap W_r^0 L \xrightarrow{N_1} M_{r-1}^1 \cap W_r^0 L) \\
\operatorname{Gr}_r^W (\Omega^* L) &\simeq (\operatorname{Gr}_{r+1}^{M_1} W_{r-1}^0 \xrightarrow{N_1} \operatorname{Gr}_{r-1}^{M_1} W_{r-1}^0) \oplus (M_r^1 \operatorname{Gr}_r^{W^0} \xrightarrow{N_1} M_{r-2}^1 \operatorname{Gr}_r^{W^0}) \\
C_r^1 L &= (\operatorname{Gr}_{r+1}^{M_1} W_{r-1}^0 L \xrightarrow{N_1} \operatorname{Gr}_{r-1}^{M_1} W_{r-1}^0 L) \\
C_r^{\emptyset M} &= (M_r^1 \operatorname{Gr}_r^{W^0} \xrightarrow{N_1} M_{r-2}^1 \operatorname{Gr}_r^{W^0})
\end{align*}
$$

By the basic property of M $^{1}$: Gr M 1 r +1 W 0 r − $_{1}$L ≃ ⊕ k < $_{r}$Gr M 1 r Gr W 0 k L ,
the complex Gr W r C 1 r L ≃ ⊕ k ≤ r − $_{1}$C 1 r , $_{k}$L splits into a sum where

$$
\begin{align*}
C_{r,k}^{1} L &= (\mathrm{Gr}_{r+1}^{M_1} \mathrm{Gr}_k^{W^0} L \xrightarrow{N_1} \mathrm{Gr}_{r-1}^{M_1} \mathrm{Gr}_k^{W^0} L) \approx \\
&\quad (\mathrm{Gr}_{r-k+1}^{W(\tilde{N}_1)} \mathrm{Gr}_k^{W^0} L \xrightarrow{N_1} \mathrm{Gr}_{r-1-k}^{W(\tilde{N}_1)} \mathrm{Gr}_k^{W^0} L)
\end{align*}
$$

Since r − k − 1 ≥ 0 , C 1 r ,k L [1] ≃ Prim Gr W ( N 1 ) r − $_{1}$k Gr W 0 k L is a polarized HS of weight r − 1. Moreover C ⊢ M r ≃ Ker N$_{1}$ : M 1 r Gr W 0 r → M 1 r − $_{2}$Gr W 0 r ) coincides with IC $^{∗}$L .

The sheaf with restriction to the complement of x is Gr W 0 r L and with fiber
C ⊢ M r coincides with j ∗ $_{!}$L .
### 11.3.0.5 Global weight filtration on $\Omega^*\mathcal{L}$

Let $(\mathbf{L}, W^0, F)$ be a graded admissible polarized variation of MHS on $X^* := X \setminus Y$. There exists a filtration $W$ on $\Omega^*\mathcal{L}$, extending $W^0$ on $\mathcal{L}$, which coincides locally with the various local weight filtrations at points $x \in Y$ (Definition 11.7), by the isomorphism $\Omega^*L \simeq (\Omega^*\mathcal{L})_x$. We deduce:

**Theorem 11.1.** Let $(\mathbf{L}, W^0, F)$ be a graded polarized admissible VMHS on $X \setminus Y$. There exists a weight filtration $W$ by perverse sub-sheaves on the logarithmic complex with coefficients $\mathcal{L}_X$, and a Hodge filtration $F$ by complexes of analytic sub-sheaves such that the bi-filtered complex

$$ \Omega^*\mathcal{L} := (\Omega_X^*(\mathrm{Log} Y) \otimes \mathcal{L}_X, W, F) \qquad (11.21) $$

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

Let $\mathrm{Gr}_k^{W^0}\mathbf{L}$ denote the graded polarized VHS defined by $(\mathbf{L}, W^0)$ on $X \setminus Y$ and let $W$ be the combinatorial weight filtration defined on $\Omega^*(\mathrm{Gr}_k^{W^0}\mathcal{L})$ in (§11.3), we deduce from (Lemma 11.8):

$$ \mathrm{Gr}_r^W(\Omega^*\mathcal{L}) \cong \mathrm{IC}^*(\mathrm{Gr}_r^{W^0}\mathcal{L}) \oplus_{k<r} \mathrm{Gr}_{r-k}^W\Omega^*(\mathrm{Gr}_k^{W^0}\mathcal{L}) \qquad (11.22) $$

## References

[BBD 83] A.A. Beilinson, J. Bernstein, P. Deligne, Analyse et Topologie sur les espaces singuliers Vol.I, Astérisque, France 100, 1983.

[BPV 84] W. Barth, C. Peters, A. Van de Ven, Compact complex surfaces. Ergebnisse der Mathematik, Springer Verlag, 1984.

[Br 82] J.L. Brylinski, (Co)homologie d'intersection et faisceaux pervers. Séminaire Bourbaki 585, Astérisque, Soc. Math. France 92-93: 129-157,1982.

[Br 86] J.L. Brylinski, Transformations canoniques, dualité projective, théorie de Lefschetz, transformations de Fourier et sommes trigonométriques, Géométrie et analyse micro locales Astérisque 140-141, Soc. Math. France, 3-134, 1986.

[CaKSc 86] E. Cattani, A. Kaplan, W. Schmid, Degeneration of Hodge structures, Ann. of Math. 123: 457-535,1986.

[CaKSc 87] E. Cattani, A. Kaplan, W. Schmid, $L^2$ and intersection cohomologies for a polarizable variation of Hodge structure, Invent. Math. 87(2): 217-252, 1987.

[CEGL 14] E. Cattani, F. El Zein, W. Schmid, P. Griffiths, D.T. Lê, Hodge Theory, Princeton Univ. Press 49, 2014.

[De 68] P. Deligne, Théorèmes de Lefschetz et critères de dégénérescence de suites spectrales, Publ. Math. IHES, 35, 107-126, 1968.

[De 70] P. Deligne, Equations différentielles à points singuliers réguliers. Lecture Notes in Math., Springer, Berlin 163, 1970.

[De 71] P. Deligne, Théorie de Hodge I, Actes, Congrès intern. math., 1, 425-430, 1971.

[De 72] P. Deligne, Théorie de Hodge II. Publ. Math. IHES, Paris 40:5-57, 1972.

[De 75] P. Deligne, Théorie de Hodge III. Publ. Math. IHES, Paris 44:6-77, 1975.

[De 80] P. Deligne, Conjecture de Weil II. Publ. Math. IHES, Paris 52:138-247, 1980.

[DeG 81] P. Deligne, O. Gabber, Théorème de pureté d'après Gabber. Note written by Deligne and distributed at IHES, Paris, 1981.
[DeK 73] P. Deligne, N. Katz, Groupes de monodromie en géométrie algébrique. *Lecture Notes in Math.* **340**, Springer, Berlin, 1973.

[De 93] P. Deligne, Décompositions dans la catégorie dérivée. *Motives*, Proc. Sympos. Pure Math. Amer. Math. Soc. Providence RI, 55, Part 1, 115–128, 1994.

[CaMi 5] M.A. de Cataldo, L. Migliorini: The Hodge theory of algebraic maps, *Ann. Sci. cole Norm. Sup.*, 38: 693-750, 2005.

[CaMi 10] M.A. de Cataldo, L. Migliorini: The perverse filtration and the Lefschetz hyperplane theorem, arXiv:0805.4634, *Annals of Math.*, 171, No. 3: 2089–2113, 2010.

[E 83] F. El Zein, Mixed Hodge structures. *Proceedings of Symposia in pure Maths* 40 Part I: 345 - 352, 1983.

[E 83] F. El Zein, Théorie de Hodge des cycles évanescents. *Ann. Scient. Ec. Norm. Sup.* 19:107–184, 1986. *Notes C.R. Acad. Sc., Paris, Série I* 295:669-672, 1982; 292:51–54, 1983; 296:199–202, 1983.

[E 08] F. El Zein, Topology of algebraic morphisms. *Contemporary Mathematics*, A.M.S 474:25–84, 2008.

[ELM 10] F. El Zein, Le D. T., L. Migliorini, A topological construction of the weight filtration. *Manuscripta Mathematica* 133(1–2): 173–182, 2010.

[EL 14] F. El Zein, D.T. Lê, Du théorème de décomposition à la pureté locale, *C.R.Acad.Sci. Paris, Ser. I*, 352: 1051–1055, 2014. De la pureté locale à la décomposition, *C.R.Acad.Sci. Paris, Ser. I*, 353: 75–80, 2015.

[EY 14] F. El Zein, X. Ye, Filtration perverse et théorie de Hodge, *C.R.Acad.Sci. Paris, Ser. I*, 352, 1045–1049, 2014.

[GGM 85] A. Galligo, M. Granger, P. Maisonobe, D-modules et faisceaux pervers dont le support singulier est un croisement normal. *Ann. Inst. Fourier* 35, 1985.

[G 58] R. Godement, Topologie algébrique et théorie des faisceaux. *Hermann Paris*, 1958.

[GMacP 80] M. Goresky, R. MacPherson, Intersection homology theory. *Topology* 19: 135–162, 1980.

[GMacP 83] M. Goresky, R. MacPherson, Intersection homology II. *Inv.Math.* 72: 77–129, 1983.

[GMacP 83-1] M. Goresky, R. MacPherson, Morse theory and intersection homology theory in Analyse et Topologie sur les espaces singuliers II, III. *Astérisque Soc. Math. France*, 101-102: 135-192, 1983.

[GMacP 88] M. Goresky, R. MacPherson, Stratified Morse theory. *Ergebnisse der Mathematik*, Springer Verlag, Berlin, Heidelberg, 3.folge. Band 2, 1988.

[Gr 74] P. Griffiths, Topics in Transcendental Algebraic Geometry. *Ann. of Math. studies*, Princeton Univ. Press.106, 1974.

[GrSc 73] P. Griffiths, W. Schmid, Recent developments in Hodge theory. Discrete subgroups of Lie Groups. *Bombay Colloquium, Oxford University Press* 1973.

[Gro 57] A. Grothendieck, Théorèmes de dualité pour les faisceaux algébriques cohérents séminaire Bourbaki, 149, Société Math. de France, 1957.

[Gro 58] A. Grothendieck, The cohomology theory of abstract algebraic varieties. *Int. Long. of Math. Edinburgh 1958, Cambridge UNiv. Press*: 103–118, 1960.

[Gro 66] A. Grothendieck, On the de Rham cohomology of Algebraic Varieties. *Publ. Math. IHES* 29, 1966.

[Ha 66] R. Hartshorne, Residues and duality. *Lecture Notes in Math.* 20, Springer, Berlin, 1966.

[Hiro 64] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero I, II. *Ann. of Math.* 79: 109 – 326, 1964.

[Ka 74] M. Kashiwara, On the maximally overdetermined system of linear differential equations I. *Publ. Res, Inst. Math. Kyoto Univ.* 10:563–579, 1974/75.

[Ka 86] M. Kashiwara, A study of variation of mixed Hodge structure. *Publ. RIMS, Kyoto Univ.* 22:991–1024, 1986.

[KaK 86] M. Kashiwara, T. Kawai, Hodge structure and holonomic systems. *Proc. Japan Acad. Ser. A* 62:1–4, 1986.
[Ka 87] M. Kashiwara, T. Kawai, Poincaré lemma for a variation of Hodge structure. *Publ. RIMS, Kyoto University* 23:345–407, 1987.

[KaS 90] M. Kashiwara, P. Schapira Sheaves on manifolds, *Grundl. math. Wissenschaft. Springer-Verlag, Berlin, Heidelberg* 292, 1990.

[KaOd 68] N. M. Katz, T. Oda, On the differentiation of de Rham cohomology classes with respect to parameters. *J. Math. Kyoto Univ.* 1: 199–213, 1968.

[LeT 83] D.T. Lê, B. Teissier, Cycles évanescents et conditions de Whitney II. *Proc. Symp. Pure Math.*, Amer. Math. Soc. 40, part 2, 65–103, 1983.

[Ma 87] B. Malgrange, Regular connections, after Deligne, in algebraic D-modules, by Borel A. *Perspective in Math. 2*, Academic Press, Boston, 151–173, 1987.

[Max 19] L. Maxim, Intersection Homology and Perverse sheaves. *Graduate Texts in Math.*, Springer International Publishing 281, 2019.

[Mat 12] J. Mather, Notes on topological stability. *Bull. Amer. Math. Soc.* 49(4):475–506, 2012.

[Me 80] Z. Mebkhout, Sur le problème de Hilbert-Riemann. *Proc. Int. Collq. centre Phys. Les Houches, 1979, Lecture notes in Physics*, Springer 126:90–110, 1980.

[Me 89] Z. Mebkhout, Le formalisme des six opérations de Grothendieck pour les DD–modules cohérents Hermann, Paris, 1989.

[Na 85] A. V. Navarro, Sur la théorie de Hodge des variétés algébriques à singularités isolées. *Astérisque* 130: 272–307, 1985.

[NaG 90] A. V. Navarro, F. Guillén, Sur le théorème local des cycles invariants. *Duke Math. J.* 61:133–155, 1990.

[PS 08] A. Peters, J. Steenbrink, Mixed Hodge Structures, *Ergebnisse der Mathematik*, Springer Verlag 2008.

[Sa 88] M. Saito, Modules de Hodge polarisables. *Publ. RIMS, Kyoto Univ.* 24: 849–995, 1988.

[Sa 90] M. Saito, Mixed Hodge modules. *Publ. RIMS, Kyoto Univ.* 26: 221–333, 1990.

[Sc 73] W. Schmid, Variation of Hodge structure: the singularities of the period mapping. *Invent. Math.* 22:211–319, 1973.

[Shi 93] Y. Shmizu, An introduction to Morihiko Saito’s theory of mixed Hodge modules, *Theory of mixed Hodge modules, Infinite analysis Lecture Note, Kyoto*, 1993.

[Sch 20] C. Schnell, An overview of Morihiko Saito’s theory of mixed Hodge modules, *Representation theory, Automorphic forms and Complex Geometry. A tribute to Wilfried Schmid. Int. Press*. 27-80, 2020.

[St 76] J. Steenbrink, Limits of Hodge structures. *Invent. Math.* 31:229–257, 1976.

[StZ 85] J. H. M. Steenbrink, S. Zucker, Variation of mixed Hodge structures I. *Invent. Math.* 80:489–542, 1985.

[Ve 65] J. L. Verdier, Dualité dans la cohomologie des espaces localement compacts Séminaire Bourbaki, 300, 1965–1966.

[Ve 76] J. L. Verdier, Stratifications de Whitney et théorème de Bertini–Sard. *Invent. Math.* 36:295–312, 1976.

[Ve 76b] J. L. Verdier, Classe d'homologie associée à un cycle. *Astérisque, Séminaire Bourbaki*, 36, 37–101, 1976.

[Ve 77] J. L. Verdier, Catégories dérivées Lecture Notes, Springer, Berlin 569: 262–312, 1977.

[V 83] J.L. Verdier, Spécialisation de faisceaux et monodromie modérée. Analyse et Topologie sur les espaces singuliers. *Astérisque*, 101–102: 332–364, 1983.

[V 85-1] J.L. Verdier, Extension of a perverse sheaf over a closed subspace. *Astérisque*, 130: 210–217, 1985.

[V 85-2] J.L. Verdier, Prolongement des faisceaux pervers monodromiques. *Astérisque*, 130: 218–236, 1985.

[Vo 2007] C. Voisin, Hodge theory and complex algebraic geometry I, II. *Cambridge University Press*, 76, 77, 2007.

[Zu 79] S. Zucker, Hodge theory with degenerating coefficients: $L_2$-cohomology in the Poincaré-metric, *Ann. Math.*, 109: 415–476, 1979.
# Index

## C

### Category

$\mathcal{I}\mathcal{H}_V$ of filtered intermediate extensions of VHS Lemma 7.1 p 136

$\mathcal{I}\mathcal{H}$ of Hodge intermediate extensions Def. 7.1 p 137

### Complex

constructible Definition 2.5 p 25

$\Omega^*L$ §3.1.2 p 54

$\psi_f^u j_* \mathbf{L}$ (Nearby cycle complex) §4.2, p 75,

$\Psi^*IL$ (Logarithmic limit complex) §4.2.1, p 77, §4.2.1.4 p 82

$\Psi^*IL$ §4.2.1.3 p 79

$\varphi_f(j_*\mathbf{L})$ (vanishing cycles) §5.2 p 104

### Cycles, Invariant

Nearby and Vanishing cycles on a variety $V$ §7.1.2 p 140

Local invariant cycle theorem 4.1 p 74, Proposition 7.5 p 143

Global invariant cycle theorem 4.2 p 75, Lemma 7.2 p 146

## D

Distinguished pairs of $\mathrm{Gr}_*^{W^{(N)}}\Psi^*IL$ §5.1.1.1 p 94

Diagram (octahedron) Equation 7.13 p 144, §10.3.2.2 p 211

### Decomposition theorem:

Theorem 1.2, p 10

Decomposition (local) of $\mathrm{Gr}_*^{W^{(N)}}\Psi^*IL$, §5.1.2 p 97

Decomposition of $\mathrm{Gr}_*^{W^{(N)}}\psi_f^u j_* \mathbf{L}$, Theorem 5.1 p 90

Decomposition Proposition 5.5 p 107

$D(n)$ Induction, ch 6 p 121, §10.0.0.1 p 191

$D(n, V^*)$, lemma 6.1 p 122, Lemma 10.1, p 192

Decomposition of $p_\varphi g^p \mathcal{H}^i(K)$ §6.2.0.4, p 129

Decomposition of $p^i \mathcal{H}^i(f_*j_*\mathbf{L})$ §6.2.1 p 130

Decomposition of $\mathrm{Gr}_*^W \Omega^*L$ §9.2.3 p 170

Decomposition theorem at $v$, §10.3 p 203

Poincaré - Verdier duality §2.1.1 p 22
Duality §2.1.1 p 22, §10.1.0.2 p 195

Duality and Polarization, §3.2.3 p 62, p 214

## F

Filtrations (Weight and Hodge) on $\Psi^*IL$, §5.1.1 p 92

Weight filtration on $\Omega^*L$, §9.1.0.1, Theorem 9.1 p 160

Weight filtration (local) on $\Omega^*L$, §9.2.1 p 164

Filtration $N * W$ §9.2.1.1 p 164, $W^J S$ §9.2.1.3 p 166

Fibration by normal crossing divisors §11.2 p 224

## H

Hard Lefschetz, §10.3.2.1 p 209

Hodge theory:

Along a NCD, §5.2.1.4 p 113, locally principal divisor, §7.1.1 p 139

Intermediate extension of VHS, §7.1.0.2 p 136

On $i^*j_Z^*j_Zj_*^P\mathbf{L}$ §5.2.2 p 117, §9.5.1 p 178

Hodge structure on $\mathbb{H}^*(X, j_*^P\mathbf{L})$ for singular X §10.3.3.1 p 212

## I

Intermediate extension §2.1.3.3 p 26, §2.2.1.3 p 33,

$$j_* \cdot L \simeq IC^*\mathcal{L} \text{ §3.1.4 p 57}$$

Intersection morphisms Definition 1.1 p 9, Proposition 12.1 p 173

Intersection morphisms (and Relative local purity) §10.4 p 2.1.4

## L

Limit Hodge filtration on $\Psi^*IL$ §4.2 p 75

Limit mixed Hodge structure on $\Psi^*IL$ chapter 5 p 89

Logarithmic complex:

$\Omega^*\mathcal{L}$, §3.1.3 p 56, $IC^*\mathcal{L}(\text{Log}Z)$ Theorem 9.3 p 174

$j_Z^*j_Zj_*^P\mathbf{L} \simeq IC^*\mathcal{L}(\text{log } Z)$ §3.2 p 58, Definition 3.8 p 59, §9.4 p 173

$$P_K^J(\mathcal{L}), \text{ §9.3.0.2 p 171}$$

Local Purity:

Theorem 1.1 p 7

Lemma (Induction) 8.1 p 149

Crucial case, §8.0.1 p 151, Proposition 10.1 p 195, §10.2 p 199

## M

Mixed Hodge structure (MHS):

Admissible VMHS Definition 2.16 p 47

Infinitesimal MHS §2.3.2 p 45, Definition 2.14 p 45, Proposition 2.7 p 46

Mixed Hodge structure on $\psi_f j_*^P\mathbf{L}$ and $\varphi_f j_*^P\mathbf{L}$ §5.1.3.3 p 102

Corollary 5.4 p 104 §5.20.2 p 107

MHS on $\mathbb{H}^j(B_v^*, j_*^P\mathbf{L})$ Proposition 7.4 p 140

MHS on the Link §10.1.0.1 p 193

Compatibility with MHS (Lemma 10.2) p 194

Monodromy:

Functoriality of the monodromy filtration §5.2.1 p 108

Monodromy filtration §2.3.0.1 p 43, Theorem 5.2 p 109,
N

Nilpotent orbit §2.3.1 p 44, Definition 2.12 p 44

P

Perverse:

sheaves §2.2 p 28, Definition 2.9 p 30,

$P_V$ Definition 2.10 p 31

$\psi_f j_*! \mathbf{L}$ and extensions §2.2.2 p 34

sheaves along a NCD §3.2.4 p 67

extension of a polarized VHS §1.2.1.3 p 14

cohomology §2.2.0.2 p 29

cohomology along a locally principal divisor §5.2.1.3 p 112

sheaves along a NCD §3.2.4 p 67

logarithmec sub-complexes of $j_* \mathcal{L} \simeq \Omega^* \mathcal{L}$ §3.1 p 52

VMHS on $p\psi_f j_*! p\mathbf{L}$ and $p\varphi_f j_*! p\mathbf{L}$, §5.1.3, Corollary 5.4 p 104

variation of mixed Hodge structures §7.1 p 135, Definition 7.2 p 138,

Purity Theorem 3.1 p 65

R

Relative Hard Lefschetz Equation (1.2) p 8

Relative complexes $IC_f^* \mathcal{L} \subset \Omega_f^* \mathcal{L}$ §9.6.1 p 180

Relative local purity §9.1.0.3 p 161, Theorem 9.2 p 161, Prop. 10.1 p 195

Relative local purity and Intersection morphisms §10.4.1 p 223

T

Thom-Whitney stratification $\mathcal{S}$ of $f$ §2.1.3.1 p 24, Definition 2.3 p 24

Thom and Gysin isomorphism §5.2.2.1 p 118,

Tilde correspondence Definition 3.1 p 53, embedding Lemma 3.1 p 55

Truncation functor $p_{\tau \le 0}$ §2.2.0.1 p 28

V

Verdier's classification of extensions §2.2.3 p 38

W

Weight filtration

On $p\psi_f j_*! p\mathbf{L}$ §5.2.0.2 p 107

Splitting of the spectral sequence Lemma 6.3 p 126

Weight filtration §9.2 p 163, Definition 9.1 p 167,

Weights of MHS §9.5.0.1 p 176

On $\Omega^* \mathcal{L}$ §9.1.0.1, p 160, Theorem 9.1 p 160,

On $\Omega^* \mathcal{L}$ §9.1.1.4, p 167, Theorem 9.1 p 160, Definition 9.1 p 167

On $IC^* \mathcal{L}$ §9.2.3 p 170