and the embedding $(IC^*L, F) \simeq (\Omega^*L_2, F)$ defines a filtered acyclic resolution of $(j_{1*}L, F)$. Moreover, the natural pairing by wedge product and the $(-1)^{w'}$-symmetric product $S$ on $L$ extends to the complex

$$S : (\Omega^*L_2, F) \otimes (\Omega^*L_2, F) \xrightarrow{S \circ \Lambda} (\Omega_X^{2n}, F) \qquad (3.18)$$

and it is compatible with $F$ ([KaK 87], Theorem 6.4.2).

Let $\sigma$ denotes the symmetry on the tensor product $\sigma(\omega^p \otimes \omega^q) := (-1)^{pq} \omega^q \otimes \omega^p$, then $S \circ \sigma = (-1)^{w'}S$ (compatible with the convention of signs in [Sa 88] §5.2). We deduce the definitions below.

**Definition 3.10.** i) A bilinear $\pm$ symmetric pairing in the category $D_c^b(X, \mathbb{Q})$ is defined by a morphism $S: \mathcal{F} \otimes \mathcal{F} \to \mathbb{D}_X$ satisfying $S \circ \sigma = \pm S$.

$S$ is non degenerate if it induces a quasi-isomorphism: $\mathcal{F} \xrightarrow{\sim} \operatorname{Hom}^\bullet((\mathcal{F}, \mathbb{D}_X))$.

ii) A polarization of the intermediate extension of a polarized VHS $(j_{1*}^pL, F) := (IC^*L, F)$ of weight $pw$ on a complex algebraic variety $X$ of dimension $n$, is defined by a $pw$-symmetric morphism

$$S : (j_{1*}^pL, F) \otimes (j_{1*}^pL, F) \to \mathbb{D}_X(n - pw). \qquad (3.19)$$

Equivalently $S^* : (j_{1*}^pL, F) \xrightarrow{\sim} \operatorname{Hom}^\bullet((j_{1*}^pL, F), \mathbb{D}_X(n - pw))$ induced by $S$ is a filtered quasi-isomorphism.

iii) A graded polarized variation of MHS is defined with a family of polarizations $S_i$ on $(Gr_i^W L, F)$ inducing isomorphisms:

$$(Gr_i^W j_{1*}^pL, F) \xrightarrow{\sim} D_X^i (Gr_i^W j_{1*}^pL, F).$$

iv) If the weight is fixed, we still use the symbol of duality $D_X$ instead of $D_X^{pw}$. We write $D_X(j_{1*}^pL) \simeq j_{1*}(D_X^pL)$ as a consequence of the polarization.

Now we can state the following purity result after ([KaK 87], Theorem 6.4.2):

**Theorem 3.1 (Purity).** i) Hard Lefschetz: Let $X$ be a complex non singular projective variety. The cup product with the class of an hyperplane section induces isomorphisms of Hodge structures

$$\eta^i : \mathbb{H}^{-i}(X, (j_{1*}^pL, F)) \xrightarrow{\sim} \mathbb{H}^i(X, (j_{1*}^pL, F))(i).$$

ii) The following pairing induced by $^pS$ (Equations 3.18, 3.19) on hypercohomology is denoted by $(\cdot, (D_X^{pw}$ denoted $D_X$):

$$\mathbb{H}^{-i}(X, j_{1*}^pL) \otimes \mathbb{H}^i(X, j_{1*}^pL) \xrightarrow{\mathbb{H}^0(^pS)} \mathbb{H}^0(X, D_X) \xrightarrow{\text{trace}} \mathbb{C}$$

Set $P^{-i} := \ker \eta^{i+1} : \mathbb{H}^{-i}(X, j_{1*}^pL) \to \mathbb{H}^{i+2}(X, j_{1*}^pL)$, then the pairing defined by $(-1)^{i(i+1)/2}(a, \eta^i b)$ induces a polarization of the HS of weight $pw-i$ on $P^{-i}$. Hence, the filtration $F$ induces on $\mathbb{H}^i(X, j_{1*}^pL)$ a polarized HS of weight $pw + i$.

The purity theorem is also stated as follows:
Let $(\mathcal{P}\mathbf{L}, F)$ be a shifted polarized variation of HS of weight $p_w$ on a dense open subset of a compact variety $X$. We have a duality isomorphism with equal weights

$$ \mathbb{H}^i(X, j_*\mathcal{P}\mathbf{L})^* \xrightarrow{\sim} \mathbb{H}^{-i}(X, j_*\mathcal{P}\mathbf{L})(p_w) \quad (3.20) $$

where $w(\mathbb{H}^i(X, j_*\mathcal{P}\mathbf{L})^*) = -(p_w + i)$ and $w(\mathbb{H}^{-i}(X, j_*\mathcal{P}\mathbf{L})(p_w)) = p_w - i - 2p_w$.

The statement is deduced from the purity theorem, the polarization and Verdier duality with weights $j_*\mathcal{P}\mathbf{L} \xrightarrow{\sim} D(j_*\mathcal{P}\mathbf{L})$.

**Definition 3.11.** i) If the weight is fixed, we still use the symbol of duality $D_X$ instead of $D_X^{p_w}$. We write $D_X(j_*\mathcal{P}\mathbf{L}) \simeq j_* (D_X \mathcal{P}\mathbf{L})$ as a consequence of the polarization.

ii) A graded polarized variation of MHS $(\mathcal{P}\mathbf{L}, W, F)$ on a Zariski open subset is defined by a family of polarizations $S_i$ on $(Gr_i^{\mathcal{P}\mathbf{L}}, F)$. It induces isomorphisms:

$$ (Gr_i^W j_*\mathcal{P}\mathbf{L}, F) \xrightarrow{\sim} D^i(Gr_i^W j_*\mathcal{P}\mathbf{L}, F). $$

**Example 3.3.** *Duality of a shifted VHS.* As a complex of sheaves, $(\mathcal{P}\mathbf{L}, F) := (\mathbf{L}[n], F)$ has weight $p_w = w' + n$. Its dual in the filtered derived category

$$ D^{p_w}(\mathcal{P}\mathbf{L}) := \operatorname{Hom}^{\bullet}(\mathcal{P}\mathbf{L}, \mathbb{Q}[2n](n-p_w)) \xrightarrow{\sim} \mathbf{L}^*(n-p_w)[n] $$

has weight $-w' + n - 2n = p_w$.

*Polarization of a shifted VHS.* A pairing $S : \mathbf{L} \otimes \mathbf{L} \to \mathbb{C}$ translates into a pairing $^{pS} : \mathcal{P}\mathbf{L} \otimes \mathcal{P}\mathbf{L} \to \mathbb{C}[2n] : a \otimes b \mapsto S(a, b) \in \mathbb{C}[2n]$. If $S$ is $(-1)^{w'}$-symmetric, the pairing $^{pS}$ is $(-1)^{p_w}$-symmetric:

$\sigma : \mathcal{P}\mathbf{L} \otimes \mathcal{P}\mathbf{L} \to \mathcal{P}\mathbf{L} \otimes \mathcal{P}\mathbf{L} : a \otimes b \mapsto (-1)^{n^2} b \otimes a$, then $^{pS} \circ \sigma = (-1)^{p_w}^{pS}$ since: $^{pS} \circ \sigma(a, b) = (-1)^{n^2} S(b, a) = (-1)^{(n+w')} S(a, b) = (-1)^{p_w}^{pS}(a, b)$.

If $S$ is a polarization of a VHS of weight $w'$ then $^{pS}$ is a polarization of $\mathcal{P}\mathbf{L}$ in the sense that $\mathcal{H}^{-n}(\mathcal{P}\mathbf{L}) \simeq \mathbf{L}$ is polarized and equivalently $^{pS}$ induces an isomorphism: $\mathcal{P}\mathbf{L} \xrightarrow{\sim} D^{p_w}(\mathcal{P}\mathbf{L})$ ($\mathcal{P}\mathbf{L}$ is auto-dual).

*Graded polarized VMHS.* As a complex of sheaves, a shifted VMHS $(\mathcal{P}\mathbf{L} := \mathbf{L}[n], W, F)$ has its weight increased by $n$, such that $W^i(\mathcal{P}\mathbf{L}) := W^{i-n}(\mathbf{L})[n]$, and $F_i(\mathcal{P}\mathbf{L}) := F_i(\mathbf{L})[n]$. The filtration on its cohomology $\mathcal{H}^{-n}(\mathcal{P}\mathbf{L})$ satisfy: $(\mathcal{H}^{-n}(\mathcal{P}\mathbf{L}), W) = (\mathbf{L}, W)$.

**Remark 3.5.** i) The pairing and the polarization are transported to the direct image and its decomposition by duality applied to the proper morphism $f : X \to V$ (Equation 2.5)

$$ D_X^{w'}(\mathbf{L}, F) := \operatorname{Hom}^{\bullet}((\mathbf{L}, F), D_X^{w'}) \simeq \operatorname{Hom}^{\bullet}(f_*(\mathbf{L}, F), D_V^{w'}) := D_V^{w'} f_*(\mathbf{L}, F). \quad (3.21) $$

ii) *Mixed cone.* The mixed cone over a morphism $f : (K, W, F) \to (K', W', F')$ is the cone $(C(f), W, F)$ with the filtrations on $C(f) := K[1] \oplus K'$ defined by the convention $W_{j+1}(K[1]) := (W_j K)[1]$ and $F_j(K[1]) := (F_j K)[1]$.

The weight filtration on the mixed cone is not always naturally defined (see the second homotopy problem in §2.1), however we will frequently use this concept whenever it is naturally defined.
For example, the diagonal filtration defined by a simplicial covering in [De 75] is a repeated use of mixed cone. Simplicial coverings define natural weights on cohomology of algebraic varieties (see also [E 83]).

### 3.2.4 Perverse sheaves along a NCD

Let $Y \stackrel{i}{\hookrightarrow} X \stackrel{j}{\hookleftarrow} U$ denote the embeddings of a normal crossing divisor (NCD) $Y$ and of $U := X \setminus Y$. Let $\mathbf{L}$ be a local system on $U$, then $\mathbf{p}\mathbf{L} := \mathbf{L}[\dim X]$ is perverse on $U$.

Let $Z \subset Y \subset X$ be a NCD sub-divisor of $Y$, $i_Z : Z \to X$ and $j_Z : X \setminus Z \to X$. The previous constructions on the perverse sheaves $j_*\mathbf{p}\mathbf{L}$ and $j_{1*}\mathbf{p}\mathbf{L}$ extend to the following complexes:

$$i'_Z j_{1*} \mathbf{p}\mathbf{L}, i_Z^* j_{1*} \mathbf{p}\mathbf{L}, i'_Z j_{Z*} j_{Z*}^* j_{1*} \mathbf{p}\mathbf{L}, i_Z^* j_{Z*} j_{Z*}^* j_{1*} \mathbf{p}\mathbf{L}, j_{Z*} j_{Z*}^* j_{1*} \mathbf{p}\mathbf{L}, j_{Z*} j_{Z*}^* j_{1*} \mathbf{p}\mathbf{L}.$$

Let $\mathbf{L}$ be VHS of weight $w'$, then $\mathbf{p}\mathbf{L} := \mathbf{L}[\dim X]$ is perverse of weight $p w = w' + \dim X$.

Let $D$ denote Verdier duality. By definition of auto-duality of $\mathbf{p}\mathbf{L}$, we have an isomorphism $D\mathbf{p}\mathbf{L} \simeq \mathbf{p}\mathbf{L}(p w)$ where $(p w)$ is a twist of the weight.

Let $Z \subset Y$ be a NCD sub-divisor, we deduce the weight filtration on $i'_Z j_{1*} \mathbf{p}\mathbf{L}$ from $(IC^*\mathcal{L}(\log Z), W, F)$ and the weight filtration on $i_Z^* j_{1*} \mathbf{p}\mathbf{L}$ by duality with $i'_Z j_{1*} \mathbf{p}\mathbf{L}$:

$$D(j_{1*} \mathbf{p}\mathbf{L} \simeq j_{1*} \mathbf{p}\mathbf{L}(p w), D i'_Z j_{1*} \mathbf{p}\mathbf{L} \simeq i_Z^* D(j_{1*} \mathbf{p}\mathbf{L}), D j_{Z*} j_Z^* j_{1*} \mathbf{p}\mathbf{L} \simeq j_{Z*} j_Z^* D(j_{1*} \mathbf{p}\mathbf{L})$$

### 3.2.5 $i_Z^*i_Z^*j_{Z*}j_{Z*}^*\mathbf{p}\mathbf{L}$

The complex $i_{Z*} i_Z^* j_{Z*} j_Z^* \mathbf{p}\mathbf{L}$ occurs in the following triangle:

$$j_{Z*} j_Z^* j_{1*} \mathbf{p}\mathbf{L} \xrightarrow{\text{can}} j_{Z*} j_Z^* j_{1*} \mathbf{p}\mathbf{L} \to i_{Z*} i_Z^* j_{Z*} j_Z^* j_{1*} \mathbf{p}\mathbf{L} \xrightarrow{[1]} \quad (3.22)$$

The complexes $j_{Z*} j_Z^* j_{1*} \mathbf{p}\mathbf{L}$ and $j_{Z*} j_Z^* j_{1*} \mathbf{p}\mathbf{L}$ are perverse since $j_Z$ is affine (Stein in the analytic case). Let $\mathcal{K} := j_{Z*} j_{1*} \mathbf{p}\mathbf{L}$.

The associated perverse long exact sequence is reduced to

$$0 \to i_{Z*}{}^p \mathcal{H}^{-1} i_Z^* j_{Z*} \mathcal{K} \to j_{Z*} \mathcal{K} \xrightarrow{\text{can}} j_{Z*} \mathcal{K} \to i_{Z*}{}^p \mathcal{H}^0 i_Z^* j_{Z*} \mathcal{K} \to 0 \quad (3.23)$$

as $i_Z^* j_{Z*} \mathcal{K}$ has only two non trivial perverse cohomology sheaves ([BBD 83] Corollaire 4.1.10 ii). Since $j_{1*} \mathbf{p}\mathbf{L} = j_{Z*} \mathcal{K} := \text{Im} : j_{Z*} \mathcal{K} \xrightarrow{\text{can}} j_{Z*} \mathcal{K}$, we deduce (([BBD 83], Corollaire 4.1.12) and [Br 82], Proposition 2.3.8):
$$ \begin{aligned} i_{Z}^{*}j_{j!*}^{p}\mathbf{L}[-1] &\simeq {}^{p}\mathcal{H}^{-1}(i_{Z}^{*}j_{Z}^{*}j_{Zj!*}^{*}\mathbf{L}) \simeq \text{Ker can,} \\ i_{Z}^{!}j_{j!*}^{p}\mathbf{L}[1] &\simeq {}^{p}\mathcal{H}^{0}(i_{Z}^{*}j_{Z}^{*}j_{Zj!*}^{*}\mathbf{L}) \simeq \text{Coker can.} \end{aligned} \quad (3.24) $$

where Coker can $\simeq j_{Z*}j_{Zj!*}^{*}p\mathbf{L}/j_{j!*}^{*}p\mathbf{L}$.

In the text we describe the filtration $W$ on $\text{IC}^*\mathcal{L}(\text{Log}Z) \simeq j_{Z*}j_{Zj!*}^{*}p\mathbf{L}$. Then, we deduce the filtrations $W$ and $F$ on

$$ i_{Z}^{*}i_{Zj!*}^{!}p\mathbf{L}[1] \simeq (j_{Z*}j_{Zj!*}^{*}p\mathbf{L})/j_{j!*}^{*}p\mathbf{L} = \text{IC}^{*}\mathcal{L}(\text{Log}Z)/\text{IC}^{*}\mathcal{L} \quad (3.25) $$

and by duality the filtration $W$ on $i_{Z}^{*}j_{j!*}^{*}p\mathbf{L}[-1]$.

### 3.2.6 Combinatorial description

To deal with perverse sheaves including perverse sub-sheaves of the complex $\Omega^*\mathcal{L}$, for $\mathcal{L}$ locally constant on $U := X \setminus Y$, we recall the description of perverse sheaves locally constant on the non singular strata $Y_j^*$ defined the NCD $Y := V(z_1 \cdots z_n) := \{z_\bullet : z_1 \cdots z_n = 0\}$.

In [V 83], J. L. Verdier describes the extensions of a general perverse sheaf $\mathcal{P}$ along a divisor by reduction to a local divisor defined by an equation $f$ in terms of the nearby cycles $\Psi_f$ and the vanishing cycles $\varphi_f$. Since $\Psi_f \mathcal{P}$ and $\varphi_f \mathcal{P}$, are shifted perverse, Verdier's description can be repeated inductively in the case of NCD to obtain the combinatorial description below.

#### 3.2.6.1 Combinatorial description of perverse sheaves along a NCD $Y \subset \mathbb{C}^n$ as a de Rham family

The NCD $Y$ has a natural stratification. For each $M \subset [1, n]$, let $Y_M := \{z \in \mathbb{C}^n : z_j = 0 \text{ for } j \in M\}$ and $Y_M^* := \{z \in Y_M : z_j \neq 0 \text{ for } j \notin M\}$ (the non singular locus of $Y_M$). The perverse sheaves $\mathcal{P}$ such that for each $M \subset [1, n]$, the cohomology of $P|_{Y_M^*}$ is locally constant, are described by the following combinatorial construction ([Ka 86] §2.1, see also [GGM 85] following a description by Deligne).

The category $\mathcal{P}$ of perverse sheaves $P$ on $\mathbb{C}^n$, with respect to the NCD stratification $Y_M^*$ of $Y$ is equivalent to the abelian category defined by :

(i) A family of vector spaces $L_A$ for $A \subset [1, n]$,

(ii) A family of morphisms

$f_{AB}: L_B \to L_A$ and $h_{BA}: L_A \to L_B$ for $B \subset A \subset [1, n]$ such that :

$f_{AB} \circ f_{BC} = f_{AC}$, $h_{CB} \circ h_{BA} = h_{CA}$ for $C \subset B \subset A$

$f_{AA} = h_{AA} = \text{id}$, $h_{A,A \cup B} \circ f_{A \cup B, B} = f_{A,A \cap B} \circ h_{A \cap B, B}$ for all $A, B$

and if $A \supset B$, $|A| = |B| + 1$, then $1 - h_{BA}f_{AB}$ is invertible.
*Intermediate extensions.* For each $A \subset [1, n]$, the description of the category $IE_A$ of intermediate extensions of a locally constant sheaf $L$ on $Y_A^*$ in terms of the family of vector spaces $L_B$ for $B \subset [1, n]$ is as follows:

(i) $L_B = 0$ for $A \subset B, A \neq B$

(ii) $f_{BA}$ is surjective and $g_{AB}$ is injective for $A \subset B$.

We denote by $IE$ the objects isomorphic to a direct sum of objects in $\cup_A IE_A$.

### 3.2.6.2 Distinguished pair and Kashiwara's criteria of decomposition into direct sum of intermediate extensions

Let $P \in \mathcal{P}$ and for all $B \subset A \subset [1, n]$, $L_B \xrightarrow{f_{AB}} L_A \xrightarrow{h_{BA}} L_B$, then by

([Ka 86] Proposition 2.3.1): The perverse sheaf $P \in \mathcal{P}$ is a direct sum of intermediate extensions if and only if the following distinguished pair criteria is satisfied

$$ \forall B \subset A \subset [1, n], A, B \neq [1, n], \quad L_A \simeq \mathrm{Im} f_{AB} \oplus \mathrm{Ker} h_{BA} \quad (3.26) $$

Moreover, it is enough to consider $|A| = |B| + 1$.

Let $P_B(P) = \cap_{C \subset B, C \neq B} \mathrm{Ker} h_{CB}$. The above condition is equivalent to the isomorphism:

$$ \oplus_{B \subset A} f_{AB}(P_B(P)) \xrightarrow{\sim} L_A \quad (3.27) $$

The induced morphism $h_{BA} : f_{AB}(P_B(P) \to P_B(P)$ is injective for $B \subset A$.

**Remark 3.6.** Originally, the local description of $IC^*L$ started with Deligne's local purity conjecture (see [CaKSc 87]). It took sometime to set the notations and reach the above statement. The reader would need as much time to meditate on the various ways to apply this combinatorial description.

i) The first version of the description in [GGM 85] is based on the cohomology with supports in the strata of the stratification defined by the NCD. Since then, the description by Verdier's extension prevail.

ii) In the text, we extract from this data the concept of de Rham family which determines the fiber $i_0^*\Omega^*\mathcal{L} := \Omega^*L$ (resp. $L_A := N_A L$ in the case of the complex $IC^*L = i_0^*IC^*\mathcal{L}$ (Definition 3.2)).

By the tilde embedding (Lemma 3.1) $L_A$ above for $A = \{i_1, \dots, i_a\}$, corresponds to $Ldz_{i_1} \wedge \dots \wedge dz_{i_a}$ (see Definition 3.7 for the general case $IC^*L(\mathrm{Log}Z)$).

iii) In the case of $\psi_f L$ where $f = z_1 \cdots z_n$, set $L_A := \Psi_A L$. In this case $L_\emptyset = 0$ since $\psi_f L$ is supported by $Y$. The de Rham family is applied to describe the fiber of the nearby cycle complex $\Psi^*L$ (see ch. 4 and 5).

On the (Example 5.1) in the case of $\Psi^*L$, we could as well represents the local system by $(L, N_1, N_2)$ on the generic stratum. At a point $z_2 \in Y_1^*$, $(L_\emptyset = L, N_1, L_{\{1\}} = L)$ represents the complex $(j_{1*}L)_{z_2}$, if we add the action of $N_2$ it represents $i_{Y_1^*}^*j_{1*}L$ as a complex of two local system on $Y_1^*$.
iv) In the proof of the decomposition theorem of the direct image by a morphism $f: X \to V$, it is an advantage to introduce a map $g$ locally on $V$ to $\mathbb{C}$ since $\psi_g$ is exact perverse and satisfy $\psi_g f_* j_* p_L \simeq \psi_{g \circ f} j_* p_L$ (similarly for and $\varphi_g$), then if we add the canonical morphism can, we can deduce $i_{g^{-1}(0)}^* f_* j_* p_L$.

v) For a general perverse sheaf, Saito in ([Sa 90], §3) express the terms of a combinatorial description by repeated application of the functors $\psi_{z_i}$ and $\varphi_{z_j}$ with the canonical and variation morphisms which determine all the terms of a perverse sheaf on a small ball including the local monodromy. We can recover the perverse sheaf completely by such data (see [Sa 90], 3.1).

*Example 3.4.* To describe $j_*{}^p L$ on $\mathbb{C}^n$ with singularities along $Y$, we find $L = {}^p\psi_{z_n} \circ \dots {}^p\psi_{z_1} j_*{}^p L$ where ${}^p\psi_{z_i} = \psi_{z_i} [-1]$.

If $n=1$, $i_0^* j_*{}^p L = (L \xrightarrow{N} NL)[1]$, $\psi_{z_1} j_*{}^p L = L[1]$, ${}^p\psi_{z_1} j_*{}^p L = L$.

By definition $\varphi_{z_1} j_*{}^p L$ is the cone of the morphism: $i_0^* j_*{}^p L \to \psi_{z_1} j_*{}^p L$, hence $\varphi_{z_1} j_*{}^p L = NL[1]$ and ${}^p\varphi_{z_1} j_*{}^p L = NL$.

vi) In the case where $L$ is a VHS the Hodge filtration $F$ is induced on the elements of the de Rham family (recall that $F$ is not a filtration by perverse sub-sheaves). Then on $Y_1^*$ we add the limit structures on $Y_1^*$ and $Y_2^*$ which are different than the limit structure at 0.

vii) We will consider a decomposition on the graded level of the graded polarized nilpotent orbit defined by the VHS. Although the decomposition above (Equation 3.27) is topological, the decomposition will be compatible with Hodge filtration which leads to a direct decomposition of the Hodge filtration by the following fact:

Let $H$ be a MHS, $A$ and $B$ two sub-MHS, if $H$ is a direct sum of the sub-spaces $A$ and $B$, then it is a direct sum of MHS by a simple count of dimensions of the filtrations:

Let $\mathrm{Gr}_i^W H = \bigoplus_{p,q \in \mathbb{Z}: p+q=i} H^{p,q}$, and $h^{p,q} := \dim H^{p,q}$, then $h^{p,q} \ge 0$ and $\sum_i h^{p,q} = \dim H$.

By application to $A$ and $B$, since $H^{p,q}(A) \oplus H^{p,q}(B) \subset H^{p,q}$, we find $h^{p,q}(A) + h^{p,q}(B) = h^{p,q}(H)$ and $H^{p,q}(A) \oplus H^{p,q}(B) = H^{p,q}$.
Chapter 4

# The Nearby cycles complex $\Psi^*I\mathcal{L}$

Fouad El Zein, Joseph Steenbrink

In this new section consisting of the next chapters 4–8, we develop the proof of the local purity and the decomposition theorems announced in §1.2.1.2.

A natural structure of bifiltered mixed Hodge complex on the nearby and vanishing cycles complex with coefficients in an intermediate extension $j_{1*}p^L$ of a (shifted) polarized variation of Hodge structure $(L, F)$ is defined in the first two chapters 4 and 5.

The structure of mixed Hodge complex is proved inductively with the decomposition theorem in chapter 6. The main feature is the reduction of the proof to the case of a vanishing cycles complex supported by the isolated strata on $V$. Hence the decomposition occurs in the category of IMHS.

By using a general hyperplane section, a remarkable fact in this case is the splitting of the weight spectral sequence which is the main simplification of the proof compared with [Sa 88].

**4.1 Introduction to chapters 4 – 8 (section 2)**

In view of Verdier's decomposition criteria (Equation 2.22), the decomposition theorem (see ch. 1, Theorem 1.2) can be reduced to a decomposition of the vanishing cycle complex.

First in the case where $X$ is a non singular complex variety, let $f: X \to D$ be a projective morphism to a complex disc with a NCD as central fiber and $j_{1*}\mathbf{L}$ on $X$. We construct in this chapter a (unipotent) limit nearby cycles complex $\Psi^*I\mathcal{L} \simeq \psi_f^u j_{1*}\mathbf{L}$ with coefficients in an intermediate extension $j_{1*}\mathbf{L}$ of a polarized variation of Hodge structure (VHS) $\mathbf{L}$ (resp. shifted $p^\mathbf{L}$) with a limit Hodge filtration $F$ (see ch. 2, §2.2.2) and (§4.2.1).

<footnote>IMJ-PRG, Université de Paris, Sophie Germain, F-75013, Paris, France .
IMAPP, Radboud University Nijmegen, The Netherlands.</footnote>
Since the definition of the complex $\psi_f j_1 * \mathbf{L}$ is topological (§2.2.2) and the polarized VHS $\mathbf{L}$ is on $X \setminus Y$, the definition of the Hodge filtration on the limit complex is subtle (the filtration $\psi_f F^p j_1 * \mathbf{L}$ is not interesting and $F^p \mathcal{L}$ is not constructible).

The extension of $F$ appears as Schmid's limit Hodge filtration [Sc 73] in the one variable case. On Steenbrink's limit complex [St 76] the filtration $F$ is deduced from the limit filtration on the logarithmic complex with constant coefficients.

The structure of mixed Hodge complex on $\Psi^* \mathcal{I} \mathcal{L}$ is defined by induction in chapter 5 in order to carry the proof mentioned in §1.2.1.2.

The decomposition theorem is proved by reduction to isolated strata and general hyperplane sections in chapter 6.

When the vanishing cycles complex is supported by the isolated strata on $V$, the decomposition occurs in the category of IMHS. Since the hyperplane section is general, the weight special sequence splits into a direct sum of a spectral sequence concentrated on the isolated strata which degenerates at rank 2 in the category of mixed Hodge structures (MHS), and a spectral sequence which degenerates at rank 1 (see ch.6, Corollary 6.3).

The structure of mixed Hodge complex on $\psi_f^u j_1 * \mathbf{L}$ in the case of a singular variety is defined by application of the decomposition theorem to the desingularization $\pi : \tilde{V} \to V$ in chapter 6 as follows. Let $j : U \to V$ denote the open embedding of a non singular Zariski open subset $U$ of a complex projective variety $V$ on which $\mathbf{L}$ is defined, and $j' : U \to \tilde{V}$ the embedding into the desingularization $\pi : \tilde{V} \to V$.

By the derived push forward functor, we have a Hodge filtration on $\pi_*(j'_* p_\mathbf{L}, F)$ and by the decomposition theorem $j_* p_\mathbf{L}$ is a component of $p\mathcal{H}^0(\pi_* j'_* p_\mathbf{L})$ with natural induced Hodge filtration $F$. This construction is independent of the desingularization up to isomorphism (see ch. 7, §7.1).

The category of intermediate extension of a polarized VHS on algebraic varieties is abelian (see ch. 7, §7.1).

In chapter 8, we fulfill our aim to adapt faithfully to complex varieties the original local purity theorem in positive characteristic, although the proof may be simplified by combination with the decomposition (see ch. 10).

For convenience, we do not use the letter $R$ in general for a derived functor (see ch.1, §2.4.0.2 for conventions and terminology) and we still say (shifted) intermediate extension for $j_* \mathbf{L}$.

### 4.1.0.1 Weigt and Hodge filtrations on $\psi_f j_* p_\mathbf{L}$

Let $f : X \to D$ be the restriction over a disc $D$ of an algebraic morphism on a non singular complex variety to $\mathbb{C}$ and $\mathbf{L}$ a polarized VHS on an open subset $U$ complement of a NCD $Y$ of $X$ containing the central fiber $Z \subset Y$.

The Hodge filtration $F$ on $\mathbf{L}$ extends to Deligne's canonical extension $\mathcal{L}_X$ of $\mathbf{L} \otimes_Z \mathcal{O}_U$ and to the logarithmic complex $\Omega^* \mathcal{L} := \Omega_X^*(\log Y) \otimes_{\mathcal{O}_X} \mathcal{L}_X$
(see ch.3, §3). We suppose the fiber of $f$ at $0 \in D$ a NCD $Z := f^{-1}(0)$ contained as a sub-divisor of $Y$. Let $j : U \to X$ and $j_Z : (X \setminus Z) \to X$, then $j_*L \simeq \Omega^*L$ and there exists a sub-complex $IC^*L(\log Z)$ of $\Omega^*L$, with induced Hodge filtration $F$, such that $j_{Z*}j^*_Z(j_{1*}{}^P L) \simeq IC^{*P}L(\log Z)$ (see ch.3, Definition 3.8).

To define the Hodge filtration $F$ on the unipotent sub-sheaf of nearby cycles complex $\psi_f^u j_{1*}{}^P L$ (see ch. 2, §2.2.2), first in the case of a VHS $L$ locally unipotent along a NCD $Y$, we extend the construction of Steenbrink's limit complex defined in the case of constant sheaf [St 76].

We define in §4.2.1 the limit complex $(\Psi^*IL, F)$ using the logarithmic complex $(IC^*L(LogZ), F)$. The Hodge filtration $F$ on $\psi_f^u j_{1*}{}^P L$ is defined by the isomorphism with the complex $\Psi^*I{}^P L$ (see Proposition 4.1).

The weight filtration $W$ is deduced in chapter 5, from the monodromy weight filtration $W(N)$ shifted by the weight ${}^P w$ of ${}^P L$.

The filtrations $W$ and $F$ on $\Psi^*IL$ are used to define the structure of a mixed Hodge complex on the nearby and vanishing cycles complexes $\psi_f^u$ and $\phi_f^u$ with coefficients in a polarized VHS on a Zariski open subset complement of a NCD in a complex variety (see chapter 5).

The decomposition of the graded monodromy weight filtration into direct sums of intermediate extensions (see ch.5, Theorem 5.3) is based on combinatorial constructions of perverse sheaves along a NCD [Ka 86] (see ch.3, §3.2.6 and §3.2.6.1).

The definition of the filtrations $W$ and $F$ on $\psi_f j_{1*}{}^P L$ for $L$ not necessarily locally unipotent along $Y$ is deduced directly from the locally unipotent case see (ch.5, §5.2).

### 4.1.0.2 Splitting of the perverse weight spectral sequence

Let $f : X \to V$ be a projective morphism of complex algebraic varieties. A proof of the decomposition theorem (ch.1, Theorem 1.2) is given in chapter 6 by reduction to the case of an isolated stratum $v \in V$ and a general hyperplane section $H$ through $v$.

Since the proof may be reduced to a desingularization of $X$ with a NCD as inverse image of $v$, we suppose already $X$ non singular and $v$ an isolated stratum in $V$. By considering a general pencil of hyperplane sections of $V$ containing $H$ and blowing up the axis of the pencil, the proof is reduced to the case where there exists a morphism $g : V \to \mathbb{P}$ to the projective complex line such that $g^{-1}(g(v)) = H$ is general and $X_H := f^{-1}(H)$ is a NCD in $X$.

Let $h := g \circ f$. The perverse weight spectral sequence of the nearby cycles $(f_*\psi_h j_{1*}{}^P L, W)$ splits in this case as $H$ is general, into a direct sum of a spectral sequence concentrated on $v$ which degenerates at rank 2 in the category of mixed Hodge structures (MHS), and a spectral sequence which degenerates at rank 1 (see ch.6, Corollary 6.3).
Moreover, the complex of vanishing cycles $\phi_h f_*j_*p^L$ is concentrated on $v$ which means that we are reduced to the case of a decomposition in the category of IMHS (see ch. 2, §2.3.2). Recall that the induced Hodge filtration on perverse cohomology is defined by restriction (see ch.2, §2.4).

### 4.1.0.3 Decomposition theorem on singular varieties

After the proof of the decomposition theorem we extend Hodge theory to singular varieties in (§6.2.2.2) and chapter 7. The definition of the abelian category of intermediate extensions of VHS (ch.7, Definition 7.1) on a singular variety $V$ and the definition of perverse variation of MHS (ch.7, Definition 7.2) are deduced from the definition on a desingularization of $V$ by application of the decomposition theorem which is already proved.

This is a significant simplification with respect to the proof in the theory of Hodge $D$-modules [Sa 88, Sa 90].

### 4.1.1 Invariant local and global cycles theorems

*Mixed Hodge structure defined along divisors.* As a consequence of the decomposition theorem, we construct a MHS on the hypercohomology of open or closed algebraic subsets of a nonsingular variety $X$ with coefficients in an intermediate extension in chapter 7.

To state the local purity theorem, we construct a MHS on the hypercohomology of a small analytic tubular neighborhood $B_{X_v}^*$ of a fiber $X_v$ of $f: X \to V$ with the central fiber $X_v$ deleted.

*Invariant theorems.* Let $V$ be a complex algebraic variety, $(L, F)$ a polarized VHS on a nonsingular Zariski open subset $U \subset V$, $j: U \to V$ and $j_*p^L$ its intermediate extension. Let $g: U' \to D$ be a projective morphism defined on an open subset $U' \subset V$ onto a disc with center 0 and central fiber $H = g^{-1}(0)$. The unipotent nearby cycles complex $\psi_g^u j_*p^L[-1]$ is perverse (Proposition 2.6) and we have:

**Theorem 4.1 (Local invariant cycle theorem).** *There exists an exact sequence*

$$ \mathbb{H}^q(H, j_*p^L) \xrightarrow{sp} \mathbb{H}^q(H, \psi_g^u j_*p^L) \xrightarrow{N} \mathbb{H}^q(H, \psi_g^u j_*p^L)(-1) \quad (4.1) $$

where $N$ is the logarithm of the unipotent monodromy.

The proof is based on the existence of mixed Hodge structures on the various hypercohomology groups of the sequence (ch.7, Proposition 7.5). Let $B_0 \subset D$ be a small disc, the specialization morphism $sp$ is precisely the composition map (where $\rho$ is surjective):

$$ \mathbb{H}^q(H, j_*p^L) \xrightarrow{\sim} \mathbb{H}^q(g^{-1}B_0, j_*p^L) \xrightarrow{\rho} \mathbb{H}^0(B_0, R^q g_*j_*p^L) \xrightarrow{\sim} \mathbb{H}^q(H, \psi_g^u j_*p^L)^T. $$
**Theorem 4.2 (Global invariant cycle theorem).** Let $g: V \to T$ be a morphism of projective varieties where $V$ is pure dimensional, $L$ a polarized VHS on a nonsingular Zariski open subset $U \subset V$ and $j_{1*}^P L$ the intermediate extension of $P L$.

There exists a Zariski open dense subset $T' \subset T$ of parameters $t$ over which $R^q g_* j_{1*}^P L$ is a polarized VHS, and for each fiber $V_t := g^{-1}(t)$ at $t \in T'$, the image of the restriction map

$$ \mathbb{H}^q(V, j_{1*}^P L) \to \mathbb{H}^q(V_t, j_{1*}^P L) \qquad (4.2) $$

is the invariant subset $\mathbb{H}^q(V_t, j_{1*}^P L)^{\pi_1(T',t)}$ of fixed elements under the monodromy action $\pi_1(T',t) \to \mathrm{Aut} \, \mathbb{H}^q(V_t, j_{1*}^P L)$.

Precisely, we have the following factorization where $u$ is surjective:

$$ \mathbb{H}^q(V, j_{1*}^P L) \xrightarrow{u} \mathbb{H}^0(T', R^q g_* j_{1*}^P L) \xrightarrow{\sim} \mathbb{H}^q(V_t, j_{1*}^P L)^{\pi_1(T',t)}. $$

### 4.1.2 Purity and local purity theorems

The global purity result (Theorem 3.1) after [CaKSc 87] and [KaK 87], is deduced in [Sa 88] and [Sa 90] from the decomposition of the derived direct image $f_* j_{1*} L$ by a projective morphism onto a curve and Zucker's purity result in the curve case. The same method apply here to prove the purity first on a non singular variety and then deduce the purity on singular complex varieties. The HS obtained by various methods are forced to be isomorphic by their own duality properties [KaK 86].

**Theorem 4.3 (Purity Theorem).** Let $L$ be a polarized VHS of weight $Pw$ defined on the complement $X \setminus Y$ of a NCD $Y$ in a projective variety $X$. The limit Hodge filtration $F$ on the intermediate extension $j_{1*} L$ defines a HS of weight $Pw + i$ on $\mathbb{H}^i(X, j_{1*}^P L)$.

See (Corollary 6.2 ii) if $X$ is singular and not projective.

**Local purity** (ch. 1, Theorem 1.1). We give a faithful interpretation on complex varieties of the proof in positive characteristic ([DeG 81] using the local and global invariant cycle theorems for the perverse cohomology (ch.7, Proposition 7.5, Lemma 7.2).

A new simple proof of the local purity theorem related to the decomposition theorem is given in the last section (chapter 10).

## 4.2 The limit Hodge filtration on $\Psi^*IL \simeq \psi_f^u j_{1*}L$

Let $X$ be a complex algebraic variety, $Y \subset X$ a NCD, $L$ a local system on $X \setminus Y$ (for the analytic topology) and $j: (X \setminus Y) \to X$.
Let $f: X \to \mathbb{C}$ be an algebraic map. The analytic restriction of $f$ over the punctured disc $D^*$ with small radius is a topological fibration. We suppose the central fiber $Z = f^{-1}(0) \subset Y \subset X$ a NCD sub-divisor of $Y$.

A filtered logarithmic limit complex $(\Psi^*IL, F)$ is defined in §4.2.1 with the following isomorphism (see Proposition 4.1):

$$ \Psi^*IL \xrightarrow{\sim} \psi_f^u j_{f_*} L. $$

The Hodge filtration $F$ on $\psi_f^u j_{f_*} L$ is deduced by this isomorphism.

Since we go back to the original work by Deligne ([DeK 73], Exposé XIV) various intermediary complexes are mentioned such as $\tilde{\Psi}^*IL$ (Definition 4.1) and $\tilde{\Psi}^*IL$ (Equation 4.31). The construction extends Steenbrink's limit complex with constant coefficients [St 76] to the case of coefficients in an intermediate extension of a polarized VHS $j_{f_*}L$. We refer to [NaG 90] for a clear exposition in the case of constant coefficients.

Since the definition of the complex $\psi_f j_{f_*} L$ is topological, the problem is to define correctly the Hodge filtration on the limit complex. The main contribution to the definition of $F^p \psi_f j_{f_*} L$ started with Schmid's extension of $F$ to Deligne's canonical $\mathcal{L}_X$ of $L \otimes_Z \mathcal{O}_U$ [De 70], [Ma 87] and the extension of $F$ to the logarithmic complex $\Omega^*\mathcal{L} := \Omega_X^*(\log Y) \otimes_{\mathcal{O}_X} \mathcal{L}_X$. Already, $j_*F$ on $j_*L$ is different from $F$ on $\Omega^*\mathcal{L}$.

Let $j_Z : (X \setminus Z) \to X$, the sub-complex $IC^*\mathcal{L}(\log Z)$ of $\Omega^*\mathcal{L}$ with the induced Hodge filtration $F$ represents $j_{Z^*}(j_{f_*}L)|_{X\setminus Z}$ (ch. 3, Definition 3.8).

### 4.2.0.1 Definition of $\psi_f j_{f_*} L$

We resume the definition of the sheaf of nearby cycles (§2.2.2, [DeK 73]) and the following diagram:

$$ \tilde{X}^* \xrightarrow{\tilde{p}} X^* \xrightarrow{j_Z} X \xleftarrow{i_Z} Z \qquad (4.3) $$

where $Z := f^{-1}(0)$, $X^* = X \setminus Z$, $\tilde{X}^* := X \times_D \tilde{D}^*$ the fiber product with a universal covering $p: \tilde{D}^* \to D^*$ of the punctured disc $D^*$.

Let $K \in D_c^+(X^*, \mathbb{Q})$ be a complex of sheaves of $\mathbb{Q}$-vector spaces, bounded below, with cohomology sheaves locally constant on the strata of a Whitney stratification of $X$, then

$$ \psi_f K := i_Z^* \tilde{j}_{Z^*} \tilde{p}^* K \in D^b(Z, \mathbb{C}) \quad \text{where} \quad \tilde{j}_Z = j_Z \circ \tilde{p} \qquad (4.4) $$

(we write $\tilde{j}_{Z^*}$ for $R\tilde{j}_{Z^*}$). The universal covering $\tilde{D}^*$ of $D$ may be represented by $\{u \in \mathbb{C}; Re u < 0\}$ with the projection $z = p(u) = e^u$ (equivalently $p(u) = e^{2i\pi u}$ for $u$ in the Poincaré upper half plane). The action of the counter clockwise generator of the fundamental group of $D^*$ on the hypercohomology of the Milnor fiber is the mondromy automorphism $T$ of $\psi_f K$.
If the complex $\mathcal{K}$ is perverse with respect to the middle perversity structure (§2.2.1 and [BBD 83], §4.0), the shifted complex $\psi_f \mathcal{K}[-1]$ as well $\phi_f \mathcal{K}[-1]$ are also perverse on $Z$ (Equation 2.15, Proposition 2.6, [Br 82], [Br 86] Théorème 1.2, [GMacP 88] §6.13). We denote by $\psi_f^u \mathcal{K}$ the perverse sub-sheaf unipotent under the action of $T$.

In general, if $T = T^s T^u$ is the decomposition of $T$ into its semisimple and unipotent part, the nilpotent endomorphism $N := \log T^u$ determines the monodromy weight increasing filtration $W_*(N)$ of $\psi_f(\mathcal{K})$ (§2.3.0.1). Explicitly, the filtration is given by $W_j(N) = \sum_i \text{Im } N^i \cap \text{Ker } N^{i+j+1}$.

Let us suppose that the central fiber $Z = f^{-1}(0)$ of the morphism $f$ is a NCD in a nonsingular complex algebraic variety $X$. We suppose that $Z$ is contained as a sub-divisor in a NCD $Y \subset X$ and that a local system $\mathbf{L}$ is given on the open subset $U := X \setminus Y$. In the next paragraph 4.2.1 we construct the limit complex $\Psi^*IL$ in terms of the logarithmic complex $\text{IC}^*\mathcal{L}(\log Z) \subset \Omega^*\mathcal{L}$ (chapter 3, Definition 3.8).

Since $\mathbf{L}$ underlies a polarized VHS, there exists a limit filtration $F$ [Sc 73], [Gr 74] and (§2.3.1) (Equation 3.16) on the restriction $i_Y^*\mathcal{L}_X$ of Deligne's extension $\mathcal{L}_X$ [De 70].

### 4.2.1 The limit complex $\Psi^*IL$

Let $T$ denote the monodromy action defined by a generator of $\pi_1(D^*)$ on the sheaf $\psi_f(j_{l_*} \mathbf{L})$. We denote by $\psi_f^u(j_{l_*} \mathbf{L})$ the unipotent sub-sheaf and $\psi_f^\alpha(j_{l_*} \mathbf{L})$ the unipotent sub-sheaf of the action $T - \alpha \text{Id}$ (which makes sense in the abelian category of perverse sheaves after an adequate shift of the complex). We construct a complex $\Psi^*IL$ on $Z$ which is a representative of the perverse sheaf $\psi_f^u(j_{l_*} \mathbf{L})$ by a complex of analytic $\mathcal{O}_X$-modules with support on $Z$ and bounded cohomology $\Psi^*IL \simeq \psi_f^u(j_{l_*} \mathbf{L})$ (Proposition 4.1), convenient for the representation of the perverse sheaf $\psi_f^u(j_{l_*} \mathbf{L})[-1]$ by the associated de Rham family §4.2.2.2.

To define the limit complex, we introduce the logarithmic complex with coefficients in Deligne's extension, since the Hodge filtration $F$ is not defined by the filtration $\tilde{j}_{Z^*} \tilde{j}_Z^* F$, but it is the expression of a limiting process.

Technically, since the Hodge filtration $F$ is not defined by the filtration $\tilde{j}_{Z^*} \tilde{j}_Z^* F$, but it is the expression of a limiting process, we use $(\text{IC}^*\mathcal{L}(\log Z), F)$ to construct various intermediary complexes:

1. the complex $\tilde{\Psi}^*IL \simeq \Psi^*IL$ (4.2.1.4)

2. the complex $\tilde{\Psi}^*IL$ (4.2.2.2)

3. local constructions (4.2.2.2, 4.2.1.2, 4.2.1.3)

The main result is: $\psi_f^u(j_{l_*} \mathbf{L}) \simeq \Psi^*IL$ (§4.21, Proposition 4.1). Since $\psi_f^u(j_{l_*} \mathbf{L})$ is perverse, the proof and constructions are based on the local version $\Psi^*IL$.
(§4.2.1.3, Definition 4.2) of $\Psi^*IL$ and its associated de Rham family (Equation 4.11).

### 4.2.1.1 $\Psi^*IL$ and its associated de Rham family

Let $L$ be a finite dimensional $\mathbb{C}$-vector space endowed with a set of commuting nilpotent endomorphisms $N_i$ for $i \in M \simeq [1, n]$, $\mathbb{C}[N]$ the polynomial ring in one variable $N$, and $\mathbb{C}[N, N^{-1}]$ the algebra of Laurent polynomials. We denote by $N$ the endomorphism on $L[N, N^{-1}] := L \otimes \mathbb{C}[N, N^{-1}]$ defined by multiplication by $N$. It maps the subspaces $L[N]$ and $NL[N]$ to themselves and acts on $L[N^{-1}] = L \otimes \mathbb{C}[N^{-1}]$ when we consider this module as $L[N, N^{-1}]/NL[N]$ (so $N = 0$ on $LN^0$).

The action of the commuting nilpotent endomorphisms $N_i$ extends as $N_i \otimes \text{Id}$, still denoted by $N_i$, on $L \otimes \mathbb{C}[N, N^{-1}]$ (resp. on $L[N]$ and $L[N^{-1}$]). Given a subset $M_Z \subset M$ and a sequence of positive integers $m_i$ for $i \in M_Z$, set

$$A_i = m_i N - N_i \text{ if } i \in M_Z, \text{ and } A_i = -N_i \text{ if } i \notin M_Z. \quad (4.5)$$

It is a family of commuting endomorphisms acting on $L[N, N^{-1}]$, $L[N]$ and on $L[N^{-1}]$.

**Remark 4.1.** The sign $-N_i$ in $A_i$ is due to the definition $N_i := \log T_i$ such that $\text{Res}_{Y_i} \nabla = -\frac{1}{2i\pi} N_i$. In general there is a problem of signs in the equations which goes back to the fact that monodromy is taken in opposite directions in the two schools in the field by Deligne and Griffith.

We associate to $L[N^{-1}]$ endowed with $A_i$ the de Rham family of modules with index $J \subset M$:

$$(L[N^{-1}]_J = L[N^{-1}] \text{ for } J \subset M, A_i := L[N^{-1}]_J \to L[N^{-1}]_{J \cup i} \text{ for } J \cup i \subset M, i \notin J).$$

### 4.2.1.2 Local differential complex $\tilde{\Psi}^*IL$ defined by de Rham family

We give first a local description, of a global complex $\tilde{\Psi}^*IL$, at a point $x \in Y_M^*$ where $M \subset [1, n]$ is a set of indices of equations of $Y_M^*$ at $x$.

Let $L := \mathcal{L}_X(x)$, $M_Z \subset M \subset [1, n]$. For each $J \subset M$ set:

$$J_Z := J \cap M_Z \text{ and } J'_Z = J \setminus J_Z \text{ such that } J = J_Z \cup J'_Z \quad (4.6)$$

With the notations of (ch. 3, §3.1.2), we consider two de Rham families:

$$\begin{align}
L[N]_J &:= L[N], A_i : L[N]_{J-i} \to L[N]_J, J \subset M, i \in M \tag{4.7} \\
L[N]_J &:= N_{J'_Z} L[N] \subset L[N], (A_i)_i, J \subset M, i \in M \nonumber
\end{align}$$
where $N_\emptyset = \mathrm{Id}$ and with induced filtration by $F$ since $L$ is a VHS.

**Definition 4.1** ($\tilde{\Psi}^*L$ and $\tilde{\Psi}^*IL$). The filtered complexes

$$ \tilde{\Psi}^*L := s(L[N]_J = L[N], A_i)_{J\subset M, i\in M}, \quad F^p\tilde{\Psi}^*L := s((F^{p-|J|}L)[N], A_i)_{J\subset M, i\in M} $$

$$ \tilde{\Psi}^*IL := s(N_{J'_Z}L[N], A_i)_{J\subset M, i\in M} \subset \tilde{\Psi}^*L $$

are associated to the de Rham families in equation 4.7, with filtration $F$ on $\tilde{\Psi}^*IL$ induced by $F$ on $\tilde{\Psi}^*L$.

*Example 4.1.* On the 3-dimensional disc $D^3 \subset \mathbb{C}^3$, let $W = W_1 \cup W_2 \cup W_3$ be the NCD defined by the coordinates $z_1z_2z_3 = 0$. A unipotent local system $\mathcal{L}$ is defined by a vector space $L$ with the action of 3 nilpotent endomorphisms $(L, N_i, i \in [1, 3])$.

Let $M_Z = \{1, 2\}$ and $Z = Z_1 \cup Z_2$ be defined by $z_1z_2 = 0$, then $\tilde{\Psi}^*IL$ is the simple complex associated to the double complex with differentials defined by $A_i$ and $N_i$ up to sign ($A_3 = -N_3$).

$$ \begin{array}{ccc} L[N] & \xrightarrow{A_1+A_2} & L[N] \oplus L[N] \\ \downarrow N_3 & & \downarrow (N_3 \oplus N_3) \\ N_3L[N] & \xrightarrow{A_1+A_2} & N_3L[N] \oplus N_3L[N] \end{array} \xrightarrow{A_2-A_1} L[N] \xrightarrow{\downarrow N_3} N_3L[N] \quad (4.8) $$

### 4.2.1.3 The complex $\Psi^*IL$

The construction of a global bounded logarithmic limit complex $\Psi^*IL$ where $\mathcal{L}$ is locally unipotent, is easily reduced to the construction of the finite complex $\Psi^*IL$ in the local case below, fit for the computation of the monodromy filtration.

For each subset $J \subset M$, let $A_J = A_{i_1} \circ \cdots \circ A_{i_j}$ for $J = \{i_1, \dots, i_j\}$ and let $\mathrm{Im} A_J \subset L[N]$ denote the image of the morphism

$$ A_J : L[N] \to L[N], \quad \mathrm{Im} A_J \subset N_{J'_Z} L[N] \subset L[N]. \quad (4.9) $$

To check that $\mathrm{Im} A_J \subset N_{J'_Z} L[N]$, note that $A_J = A_{J_Z} \circ A_{J'_Z} = \pm A_{J_Z} \circ N_{J'_Z}$.

We remark that the family $\mathrm{Im} A_J$ defines a sub-complex

$$ \mathrm{Im}^*A := s(\mathrm{Im} A_J, A_i)_{J\subset M, i\in M} \subset \tilde{\Psi}^*IL $$

**Definition 4.2** ($\Psi^*IL$). With the notations of (Equation 4.6), we define the finite dimensional quotient complex

$$ \Psi^*IL := \tilde{\Psi}^*IL/\mathrm{Im}^*A. \quad (4.10) $$

The de Rham family associated to $\Psi^*IL$ is defined by the finite quotient vector spaces
$$ \Psi_J(\mathrm{IL}) := (N_{J'_Z} L)[N]/\mathrm{Im} A_J, A_i := \Psi_J(\mathrm{IL}) \to \Psi_{J \cup i}(\mathrm{IL}) \text{ for } i \notin J \quad (4.11) $$

We write $\Psi^*L$ when $M_Z = M$.

To satisfy the conditions in (§3.2.6.1), set for $J \subset K : f_{JK} := A_{K \setminus J} : \Psi_J(\mathrm{IL}) \to \Psi_K(\mathrm{IL})$ which is well defined since $A_{K \setminus J} : N_{J'_Z} L[N] \to N_{K'_Z} L[N]$ and $A_{K \setminus J}(\mathrm{Im} A_J) \subset (\mathrm{Im} A_K)$. Set $p_{KJ} := \Psi_K(\mathrm{IL}) \to \Psi_J(\mathrm{IL})$ be the natural projection since $N_{K'_Z} L[N] \subset N_{J'_Z} L[N]$ and $\mathrm{Im} A_K \subset \mathrm{Im} A_J$.

**Lemma 4.1.** *There exists a natural filtered quasi-isomorphism*

$$ (\tilde{\Psi}^*IL, F) \xrightarrow{\sim} (\Psi^*IL, F). \qquad (4.12) $$

*Proof.* We prove that the complex $\mathrm{Im}^*A$ is acyclic. We write the complex $\mathrm{Im}^*A$ as the cone over the morphism $A_i$. Let $i \in M_Z \neq \emptyset$ such that $m_i > 0$, and $Z' := Z \setminus \{i\}$. We write the complex $\mathrm{Im}^*A$ as a cone complex over the morphism

$$ A_i : \mathrm{Im}^*A(\hat{i}) \to \mathrm{Im}^*A(\hat{i}) $$

where $\mathrm{Im}^*A(\hat{i})$ is defined by deleting the index $i$. We prove that $A_i$ is an isomorphism of complexes. Since $N_i$ is nilpotent, the inverse of $A_i$ on $L[N, N^{-1}]$ is defined and equal to the finite sum

$$ A_i^{-1} = \sum_{j \ge 0} m_i^{-j-1} N_i^j N_i^{-j-1}. \qquad (4.13) $$

In particular, the endomorphism $A_i$ is injective on $L[N]$. Since $A_i : \mathrm{Im} A_J \to \mathrm{Im} A_{J \cup \{i\}}$ is clearly surjective for $i \notin J$, we deduce that the morphism $A_i$ is an isomorphism of complexes and its cone $\mathrm{Im}^*A$ is acyclic.

**Example 4.2.** In Diagram 4.8, the complex $\mathrm{Im}^*A$

$$ \begin{tikzcd} L[N] \arrow[r, "A_1+A_2"] \arrow[d, "N_3"] & A_1L[N] \oplus A_2L[N] \arrow[r, "A_2-A_1"] \arrow[d, "(N_3 \oplus N_3)"'] & A_2A_1L[N] \arrow[d, "N_3"] \\ N_3L[N] \arrow[r, "A_1+A_2"] & N_3A_1L[N] \oplus N_3A_2L[N] \arrow[r, "A_2-A_1"] & N_3A_2A_1L[N] \end{tikzcd} \qquad (4.14) $$

is the cone over the isomorphism defined in terms of $A_2$:

$$ \begin{tikzcd} L[N] \arrow[r, "A_1+N_3"] \arrow[d, "A_2"'] & A_1L[N] \oplus N_3L[N] \arrow[r, "N_3-A_1"] \arrow[d, "(A_2+A_2)"'] & A_1N_3L[N] \arrow[d, "A_2"'] \\ A_2L[N] \arrow[r, "-A_1+N_3"'] & A_2A_1L[N] \oplus A_2N_3L[N] \arrow[r, "-N_3-A_1"'] & A_2A_1N_3L[N] \end{tikzcd} $$

**Lemma 4.2.** Let $M_Z \subset M$, $J \subset M$, $J_Z = J \cap M_Z$, $l = |J|$ and $l_z = |J_Z|$, then

$$ \Psi_J(\mathrm{IL}) := (N_{J'_Z} L)[N]/\mathrm{Im} A_J \simeq \Psi_{J_Z}(N_{J'_Z} L) \simeq \bigoplus_{j \le l_z - 1} N_{J'_Z} L \otimes N^j \quad (4.15) $$

In particular, for $M_Z = M$, we have $A_J L \simeq L[N]/\mathrm{Im} A_J \simeq \bigoplus_{j \le l-1} L \otimes N^j$.

Let $A_{J_Z} := A_{l_z} \cdots A_1$, by definition:
$$ \Psi_{J_Z}(N_{J'_Z}L) := (N_{J'_Z}L)[N]/\operatorname{Im}(A_{J_Z} : (N_{J'_Z}L)[N] \to (N_{J'_Z}L)[N]) $$

Since $A_i = -N_i$ for $i \in J'_Z$, $A_{J_Z}(N_{J'_Z}L) \subset \operatorname{Im} A_J$, the isomorphism follows.

Since $A_{J_Z}$ acts as a polynomial in the variable $N$ on $\Psi_{J_Z}(N_{J'_Z}L)$ and vanish, we develop $A_{J_Z}$ in terms of the elementary symmetric functions $\sigma_j$, of $\frac{1}{m_i}N_i$ for $i \in J_Z$ to deduce the relation on $\Psi_{J_Z}(N_{J'_Z}L)$:

$$ N^{l_z} = \sum_{1 \le j \le l_z} (-1)^{j+1} \sigma_j \left( \frac{1}{m_i} N_i, i \in J_Z \right) \otimes N^{l_z-j}. $$

The action of the endomorphism $N$ is the multiplication by $N$ on $N_{J'_Z}L \otimes N^j$ for $j < l_z - 1$ and acts on $a \in N_{J'_Z}L \otimes N^{l_z-1}$ by the relation:

$$ N(a \otimes N^{l_z-1}) = \sum_{1 \le j \le l_z} (-1)^{j+1} \sigma_j \left( \frac{1}{m_i} N_i, i \in J_Z \right) (a) \otimes N^{l_z-j}. \quad (4.16) $$

**Example 4.3.** On the 2-dimensional disc $D^2 \subset \mathbb{C}^2$, let $W = W_1 \cup W_2$ be the NCD defined by the equation $z_1z_2 = 0$. A unipotent local system $L$ is defined by a vector space $L$ with the action of 2 nilpotent endomorphisms $(L, N_i, i \in [1, 2])$.

Let $M_Z = \{1, 2\}$ and $Z = W$, then $\Omega^*L[N](\log Z)$ is defined by the diagram

$$ L[N] \xrightarrow{A_1 \oplus A_2} L[N] \oplus L[N] \xrightarrow{A_2 - A_1} L[N] $$

with differentials defined by $A_i$ and $N_i$ up to sign. Taking the quotient in $\Psi_J L$, we obtain the complex $\Psi^*IL$ in degree 1 and 2

$$ \begin{array}{c} \Psi^1 IL := \Psi_1 \oplus \Psi_2 = L \oplus L \\ \downarrow d \quad \downarrow d \quad \downarrow d \end{array} \qquad (4.17) $$

$$ \Psi^2 IL := \Psi_{12} = L \oplus LN $$

where for $(a, b) \in \Psi_1 \oplus \Psi_2 : d(a, b) = -N_1b - N_2a \oplus (m_2a + m_1b)N \in \Psi_{12}$.

The action of $N$ on $\Psi$ is a matrix with entries defined by $N_1$ and $N_2$ as follows:

1. $N|\Psi_1 \oplus \Psi_2 : Na = N_1a, Nb = N_2b,$

2. $N|\Psi_{12} : N(a+bN) = -\frac{1}{m_1m_2}N_1N_2b + ((\frac{N_1}{m_1} + \frac{N_2}{m_2})b + a)N$

**Remark 4.2 (The complex $\bar{\Psi}^*L$).** We introduce the following complex:

(i) $\bar{\Psi}^*L$ is the exterior differential algebra (or Koszul complex) defined as the simple complex associated to the de Rham family

$$ (L[N^{-1}]_J = L[N^{-1}], A_i, i \in M)_{J \subset M}; $$

(ii) $\bar{\Psi}^*IL \subset \bar{\Psi}^*L$ is the simple complex associated to the de Rham sub-family $(N_{J'_Z}L[N^{-1}] \subset L[N^{-1}], A_i, i \in M)_{J \subset M}$ where $N_\emptyset L = L$

$$ \bar{\Psi}^*IL := s((N_{J'_Z}L[N^{-1}] \subset L[N^{-1}], A_i)_{J \subset M, i \in M}. \quad (4.18) $$
(iii) The exterior complex $\Omega^*(IL[N, N^{-1}], A_i)_{i \in M_Z}$ is acyclic since $A_i$ is invertible for $i \in M_Z$ (Equation 4.13). Equivalently, the simple complex associated to the de Rham sub-family $(N_{J'_Z} L[N, N^{-1}])$ is acyclic and we have a quasi-isomorphism: $\tilde{\Psi}^*IL \simeq \tilde{\Psi}^*IL$.

### 4.2.1.4 The logarithmic limit complex $\Psi^*IL \simeq \psi_f^u(j_*L)$

We resume the global hypothesis with a morphism $f: X \to D$ and a NCD $Z := \sum_{i \in I_Z} m_i Y_i$ as central fiber and let $L$ be a local system defined on a NCD $Y \supset Z$.

The global limit logarithmic complex $\Psi^*IL$ with indices in the right half plane is constructed as follows. Consider the double complex with gradings $p \ge 0$ and $q$:

$$ \tilde{\Psi}^{p,q}IL := \mathbb{C}u^{-(p+1)} \otimes_C i_Z^*IC^{p+q+1}\mathcal{L}(\log Z) \quad (4.19) $$

with differentials

$$ d' : \tilde{\Psi}IL^{p,q} \to \tilde{\Psi}IL^{p+1,q}, \quad d'(u^{-(p+1)} \otimes \varphi) = -(p+1)u^{-(p+2)} \otimes \frac{df}{f} \wedge \varphi, $$

$$ d'' : \tilde{\Psi}IL^{p,q} \to \tilde{\Psi}IL^{p,q+1}, \quad d''(u^{-(p+1)} \otimes \varphi) = u^{-(p+1)} \otimes \nabla \varphi, $$

where $d''$ is deduced from the differential $\nabla$ on $i_Z^*IC^*\mathcal{L}(\log Z)$, hence the total differential is: $d(u \otimes \varphi) = u \otimes \frac{df}{f} \wedge \varphi \pm 1 \otimes \nabla \varphi$. The associated simple complex is denoted by $\tilde{\Psi}^*IL$.

**Definition 4.3** ($\Psi^*IL$). There exists a unique complex $\Psi^*IL := \tilde{\Psi}^*IL/\mathcal{I}m^*A$, defined as a quotient complex by a sub-complex $\mathcal{I}m^*A \subset \tilde{\Psi}^*IL$ such that at a point $x \in Y_M^*$, the tilde embedding (chapter 3, Lemma 3.1) is a quasi-isomorphism:

$$ \Psi^*IL[1] \xrightarrow{\sim} \Psi^*IL_x, \quad L := \mathcal{L}(x) \quad (4.20) $$

(We remark that the embedding of $\Omega^*LN^j$ is equal to the embedding of $\Omega^*L$ in (ch. 2, Lemma 2.8) with a coefficient $(-2i\pi)^j$ times).

The complex $\Psi^*IL$ coincides in the case $Y = Z$ and $\mathcal{L} = \mathbb{C}$, with the complex $A^*$ introduced in Steenbrink [St 76].

### 4.2.1.5 The morphism $\nu$ of bidegree $(1, -1)$ on $\Psi^*IL$

The morphism $\nu$ of bidegree $(1, -1)$ on $\Psi^*IL$ is induced by

$$ \tilde{\nu} : \tilde{\Psi}IL^{p,q} \to \tilde{\Psi}IL^{p+1,q-1} \quad (4.21) $$

defined by $\tilde{\nu}(\varphi) := u^{-p} \otimes \varphi \to u^{-p+1} \otimes \varphi$ for $\varphi \in i_Z^*\Omega^{p+q+1}\mathcal{L}(\log Z)$).
**Proposition 4.1.** The unipotent perverse sheaf $\psi_f^u(j_{f_*}L)$ is isomorphic to the $\Psi^*IL$ in the derived category $D^b(Z, \mathbb{C})$, precisely

$$ \psi_f^u(j_{f_*}L) \simeq \tilde{\Psi}^*IL \simeq \Psi^*IL \qquad (4.22) $$

The proof may be reduced to the original Lemma in ([DeK 73], Exposé XIV, Lemma 4.18.5). The main point is the comparison theorem between the topological definition of $\psi_f^u(j_{f_*}L)$ and the analytic definition of $\Psi^*IL$ which allows the later use of Hodge filtration on logarithmic differential forms. By construction, we need to construct an adequate representative of the complex $\psi_f j_{f_*} L := i_Z^* \tilde{j}_{Z*} \tilde{p}^* j_{f_*} L \in D^b(Z, \mathbb{C})$ (see Equation 2.14) in order to define later the correct filtration $F$ on it. For this purpose we introduce another complex.

### 4.2.2 Proof of proposition 4.1

#### 4.2.2.1 The complex $\tilde{\Psi}^*IL$

We suppose the central fiber of the morphism $f: X \to D$ a NCD $Z \subset Y$ included but eventually distinct from the singular set $Y$ of $\mathcal{L}$. The wedge product

$$ \sim \eta : i_Z^*IC^*\mathcal{L}(\log Z) \to i_Z^*IC^*\mathcal{L}(\log Z)[1] : \varphi \mapsto \varphi \wedge \frac{1}{2\pi i} \frac{df}{f} $$

defines a morphism of degree 1. Recall that $t = \pi(u) = \exp 2\pi i u$. Then $c := [\frac{1}{2\pi i} \frac{dt}{t}] \in H^1(D^*, \mathbb{Q})$ and $I \sim \eta$ corresponds to the cup product with the inverse image $\eta = f^*c \in H^1(X^*, \mathbb{Q})$:

$$ \sim \eta : H^1(X^*, \mathbb{Q}) \otimes H^i(X^*, j_{f_*}L) \to H^{i+1}(X^*, j_{f_*}L) $$

The associated simple complex with indices in the left half plane

$$ \tilde{\Psi}^*IL := \mathbb{C}[u] \otimes_{\mathbb{C}} i_Z^*IC^*\mathcal{L}(\log Z) = \Sigma_{p \le 0} \mathbb{C}u^{-p} \otimes_{\mathbb{C}} i_Z^*IC^*\mathcal{L}(\log Z)[p] \quad (4.23) $$

where $du = \frac{df}{f}$, is the simple complex associated to the double complex with indices $(p, q)$, $p \le 0$, $p+q \ge -\dim X$

$$ \bar{\Psi}I\mathcal{L}^{p,q} = \mathbb{C}u^{-p} \otimes_{\mathbb{C}} i_Z^*IC^{q+p}\mathcal{L}(\log Z), \quad p \le 0, $$

$$ d' : \bar{\Psi}I\mathcal{L}^{p,q} \to \bar{\Psi}I\mathcal{L}^{p+1,q} : \quad d'(u^{-p} \otimes \varphi) = -pu^{-p-1} \otimes \frac{df}{f} \wedge \varphi, \quad (4.24) $$

$$ d'' : \bar{\Psi}I\mathcal{L}^{p,q} \to \bar{\Psi}I\mathcal{L}^{p,q+1} : \quad d''(u^{-p} \otimes \varphi) = u^{-p} \otimes \nabla \varphi, $$
**Definition 4.4.** The complex $\bar{\Psi}^*IL$ is the simple complex associated to this double complex:

$$ \bar{\Psi}^*IL := s(\bar{\Psi}IL^{p,q}, d', d'')_{p \le 0, q \in \mathbb{Z}} = s(i_Z^*IC^*\mathcal{L}(\log Z)[p], \wedge \frac{1}{2\pi i} \frac{f}{df})_{p \le 0}. $$

In the case $Z = Y$, the complex $\bar{\Psi}^*IL = \bar{\Psi}^*\mathcal{L}$ has been introduced in [St 76], [NaG 90]. It embeds into $i_Z^*\tilde{j}_{Z*}\tilde{p}^*j_1*L$. This embedding extends easily to the complex $\bar{\Psi}^*IL$ in the case $Z \neq Y$.

### 4.2.2.2 The quasi-isomorphism $\bar{\Psi}^*IL \simeq \tilde{\Psi}^*IL$

The morphism $\mu : \bar{\Psi}^*IL \to \tilde{\Psi}^*IL$ defined by: $\mu(u^p \otimes \omega_p) = 0$ if $p \neq 0$ and $\mu(u^0 \otimes \omega_0) = (-1)^{|\omega_0|}u^{-1} \otimes \frac{1}{2\pi i} \frac{df}{f} \wedge \omega_0$ is a quasi-isomorphism (to compare with [NaG 90] Lemma 2.5).

The morphisms $\nu \circ \mu$ and $\mu \circ \nu$ are homotopic. The homotopy $h : \bar{\Psi}^*IL \to \tilde{\Psi}^*IL$ of bidegree (0, 1) is defined by

$$ h(u^p \otimes \omega_p) = 0 \text{ if } p \neq 0 \text{ and } h(u^p \otimes \omega_p) = (-1)^{|\omega_0|} u^{-1} \otimes \frac{df}{f} \wedge \omega_0 \text{ if } p = 0. $$

**Proposition 4.2.** Let $L$ be a polarized VHS on $X \setminus Y$. Then there exists a natural quasi-isomorphism $\bar{\Psi}^*IL \xrightarrow{\sim} \tilde{\Psi}_f^u j_1*L$.

The local proof below (Lemma 4.4) is based on an extension of the following comparison result by Deligne [De 70] between analytic and topological complexes:

**Lemma 4.3.** Let $Y = Z$. There exists a quasi-isomorphism $\bar{\Psi}^*\mathcal{L} \xrightarrow{\sim} \tilde{\Psi}_f^u L$.

The proof is based on a comparison with the meromorphic complex $\Psi_f^m(\Omega_{X^*}^*(\mathcal{L}))$ on a complex manifold $X^* := X \setminus Z$. We remark that the embedding $j$ is Stein, hence $j_*$ is acyclic on coherent sheaves on $X \setminus Z$. The following isomorphism is proved in ([DeK 73], Exposé XIV, Proposition 4.15)

$$ \Psi_f^m(\Omega_{X^*}^*(\mathcal{L})) \xrightarrow{\sim} \tilde{\Psi}_f^u L. \qquad (4.25) $$

The complex $\bar{\Psi}^*\mathcal{L}$ is embedded as a sub-algebra in $\tilde{j}_{Z*}\tilde{p}^*\Omega^*\mathcal{L}$ by the relation

$$ I(u^p \otimes \varphi) := \frac{1}{p!} u^p \otimes \varphi \text{ for } p \ge 0 \text{ where } du = \frac{df}{2\pi i f} \text{ since } 2\pi i u = \log f. $$

This embedding defines the quasi-isomorphism

$$ \bar{\Psi}^*\mathcal{L} \xrightarrow{\sim} \Psi_f^m(\Omega_{X^*}^*(L)) \xrightarrow{\sim} \tilde{\Psi}_f^u L \qquad (4.26) $$

as we check by the following local computation due to Deligne.

Let $Y = Z = \cup_{i \in I} Z_i$ be the decomposition into smooth irreducible components $Z_i$ with indices $i \in I$ and let $L$ denote a local system on $X \setminus Z$, $B_y$
a polydisc in $X$ with center $y \in Z$, $B_y^* = B_y \setminus Z$ and $\tilde{B}_y^* := \tilde{D}^* \times_{D^*} B_z$. The unipotent cohomology $H^i(\Psi_f^u \mathcal{L})_y \simeq H^i(\tilde{B}_y^*, \mathbf{L})^u$ is related to $H^i(B_y^*, \mathbf{L})$ as follows.

Let $U$ be a product of discs $D^e$ with center 0 embedded as a polydisc $B_y \subset X$ such that $f: U \to D$ is defined by $\prod_{i \in [0,d]} z_i^{m_i}$ on $U$, $U^* = U \cap f^{-1}D^*$ and $\tilde{U}^* := \tilde{D}^* \times_D U \simeq \tilde{B}_y^* \subset \tilde{X}$.

A generator $c$ of $H^1(\tilde{D}^*, \mathbb{Q})$ defines an element $\eta := f^*(c) \in H^1(U^*, \mathcal{L})$ and the cup-product map $\rfloor \eta : H^{i-1}(U^*, \mathcal{L}) \to H^i(U^*, \mathcal{L})$.

On the one hand, the cohomology of the fiber of $\Psi_f^u \mathcal{L}$ at $y$ is isomorphic to

$$H^i(\Psi_f^u \mathbf{L})_y = H^i(\tilde{U}^*, \mathbf{L})^u \simeq H^i(U^*, \mathbf{L})/\eta H^{i-1}(U^*, \mathbf{L}) \quad (4.27)$$

(see [DeK 73], Exposé XIV, Lemma 4.18.5).

On the other hand, since $Y = Z$, the morphism $\rfloor \eta$ is induced by $\wedge \frac{df}{2\pi i f}$, and a spectral sequence argument proves the isomorphism

$$H^i(\bar{\Psi}^* \mathcal{L}) \simeq H^i(U^*, \mathbf{L})/\eta H^{i-1}(U^*, \mathbf{L}).$$

### 4.2.2.3 The general case: $Y \neq Z$

Let $Y' := \cup_{i \in I - I_Z} Y_i$ denote the subdivisor of $Y$ complement to $Z$ such that $Y = Z \cup Y'$. Let $X' := X \setminus Y'$ and $f': X' \to D$ the restriction of $f$ to $X'$.

The natural quasi-isomorphism in Proposition 4.2 is an extension of the quasi-isomorphism $\bar{\Psi}^* \mathcal{L} \xrightarrow{\sim} \Psi_f^u \mathbf{L}$ in Lemma 4.3. It remains to prove the quasi-isomorphism locally, at a point $x \in Y_M^* \subset Z_{M_Z}^*$.

With the notations of §4.2.1.2, and (Definition 4.1), we introduce the complex:

$$\bar{\Psi}^*IL := s((N_{J'_Z} L[N^{-1}] \subset L[N^{-1}], A_i)_{J \subset M, i \in M}. \quad (4.28)$$

such that there exists an isomorphism: $\bar{\Psi}^*IL \simeq (\bar{\Psi}^*IL)_x$ defined by the tilde embedding (see chapter 1, Lemma 3.1). The proof of the proposition is reduced to the following

**Lemma 4.4.** $\bar{\Psi}^*IL \xrightarrow{\sim} (\Psi_f^u j_{l_*} \mathbf{L})_x$

The lemma follows from the isomorphism: $(j_{Z^*} j_Z^* j_{l_*} \mathbf{L})_x \simeq \mathrm{IC}^*L(\log Z)$ (chapter 3, Proposition 3.2). The proof is similar.

Let $U := D^p \times D^q$ with local coordinates $(z, t) := (z_1, \dots, z_p, t_1, \dots, t_q)$ centered at $x$, such that $\mathbf{L}$ induces on $U^* := (D^*)^p \times (D^*)^q$ a local system unipotent along $Y \cap U$.

Let $g: D^p \to D$ denote the morphism $g(z.) = z_1^{m_1} \cdots z_p^{m_p}$ and $p_1: D^p \times D^q \to D^p$ the first projection such that $f = g \circ p_1$. The central fiber $W \subset D^p$ of $g$ is defined by $g(z.) = 0$ and its pre-image $Z \cap U = p_1^{-1}(W)$ is the central fiber of $f$ in $U$.

We reduce the proof of the lemma to the case $\psi_g^u p_{1*}(j_{l_*} \mathbf{L}|_U)$, where the derived functor $p_{1*}(j_{l_*} \mathbf{L}|_U)$ has locally constant cohomology on $D^p - W$. We
consider the diagram

$$
\begin{array}{ccc}
U' := U \setminus Z = (D^p \setminus W) \times D^q & \xrightarrow{j_Z} & D^p \times D^q \\
& \cup & \cup \\
U \cap Z = p_1^{-1}(W) = U \cap f^{-1}(0) = W \times D^q & \xrightarrow{p_1} & W = g^{-1}(0) \xrightarrow{g} & 0
\end{array}
$$

The cohomology sheaves $\mathbf{L}^i := R^i(p_{1|U'})_*(j_{1*}\mathbf{L}_{|U'})$ are locally constant on $(D^*)^p$. Their fiber at a point $z. \in (D^*)^p$ is isomorphic to the intersection cohomology of a local system on $(D^*)^q$ at the center $0 \in D^q$.

Let $j_W : (D^p \setminus W) \to D^p$ be the inclusion. Since the restriction of $p_1$ to the family of spheres in the fibers of $p_1$ is proper, the derived functor $p_{1*}$ commutes with $j_{W*}$ and $j_{Z*}$ and similarly, the functors $\psi_f$ on $U$ and $\psi_g$ on $D^p$ commute with $p_{1*}$ ([DeK 73] §2.1.7):

$$
p_{1|*} \psi_f^u j_{1*} \mathbf{L} \xrightarrow{\sim} \psi_g^u (p_{1|U'})_*(j_{1*} \mathbf{L}_{|U'}).
$$

By constructibility, we deduce for $U$ small enough, the following isomorphisms

$$
\begin{align*}
(\psi_f^u j_{1*} \mathbf{L})_0 &\simeq R\Gamma(U, \psi_f^u j_{1*} \mathbf{L}) = R\Gamma(D^p, R(p_{1|*} \psi_f^u j_{1*} \mathbf{L})) \\
&\xrightarrow{(Rp_{1|*} \psi_f^u (j_{1*} \mathbf{L}_{|U'}))}_0 \xrightarrow{\sim} (\psi_g^u R(p_{1|U'})_*(j_{1*} \mathbf{L}_{|U'}))_0
\end{align*}
$$

By a spectral sequence argument, Deligne's comparison applies to the com-
plex $R(p_{1|U'})_{j_*} L$ since it has locally constant cohomology.

We check now that the induced morphism $\bar{\Psi}^*IL \to (\psi_g^u R(p_{1|U'})_*(j_*\mathbf{L}_{|U'}))_0$ is an isomorphism. We write $\bar{\Psi}^*IL$ as the associated single complex of a double complex as in (ch. 1, Remark 3.4). Let $M'_Z := M \setminus M_Z$, $J'_Z := J \setminus J_Z$. For each fixed subset $J_Z \subset M_Z$ we define the following Koszul complex :

$$
\mathrm{IC}^*(L[N^{-1}], N_i)_{i \in M'_Z} := \mathbf{s}(N_{J'_Z} L[N^{-1}], N_i)_{J'_Z \subset M'_Z, i \in M'_Z}
$$

by summing over $J'_Z \subset M'_Z$, on which $A_j$ acts for $j \in M_Z$. We define

$$
\Omega^*(\mathrm{IC}^*(L[N^{-1}]), N_i)_{i \in M'_Z}, A_j)_{j \in M_Z}
$$

as the sum over $J_Z \subset M_Z$ of these constant complexes

$$
\bar{\Psi}^*IL := s(s(N_{J'_Z} L[N^{-1}]), N_i)_{J'_Z \subset M'_Z, i \in M'_Z, A_j)_{j \in M_Z} \quad (4.29)
$$

We use this structure of double complex to define filtrations on both terms of the lemma with isomorphic spectral sequences.

Let $H^j$ denote the fiber of the local system $\mathcal{H}^j := R^j(p_{|U'})_*(j_*\mathcal{L}_{|U'})$, with an action of $A_i$ for $i \in M_Z$ and let $\bar{\Psi}^*H^j$ be defined by $H^j$ as in (Remark 4.2, i).

**Sub-lemma.**
(i) We have $H^j[N^{-1})[-j] := \text{Gr}_j^\tau \mathbf{s}(N_{J'_Z}L[N^{-1}], N_i)_{J'_Z \subset M'_Z, i \in M'_Z}$, where $\tau$ denotes the truncation filtration of the complex.

(ii) The filtration $\tau$ extends to a filtration $W'$ on $\bar{\Psi}^*IL$ satisfying

$$ \text{Gr}_j^{W'} \bar{\Psi}^*IL := \mathbf{s}(H^j[N^{-1})[-j], A_i)_{J_Z \subset M_Z, i \in M_Z} \simeq \bar{\Psi}^*H^j. \quad (4.30) $$

*Proof of the lemma.* Let $\tau'_j \Psi_g^u (R(p_{|U'}), (j_* \mathbf{L}_{|U'})) := \Psi_g^u \tau_j R(p_{|U'}), (j_* \mathbf{L}_{|U'}))$, then

$$ (\text{Gr}_j^{\tau'} \Psi_g^u (R(p_{|U'}), (j_* \mathbf{L}_{|U'})))_0 = (\Psi_g^u \mathcal{H}^j [-j])_0. $$

Since $\mathcal{H}^j$ is a local system we apply Proposition 4.3 to deduce

$$ \bar{\Psi}^* \text{Gr}_j^\tau (\mathbf{s}(N_{J'_Z}L, N_i)_{J'_Z \subset M'_Z, i \in M'_Z}) = \bar{\Psi}^* H^j \simeq (\Psi_g^u \mathcal{H}^j)_0 $$

from which we deduce an isomorphism of spectral sequences defined by the filtrations $(\bar{\Psi}^*IL, W')$ and $(\Psi_g^u(R(p_{|U'}), (j_* \mathbf{L}_{|U'})), \tau')_0$, hence an isomorphism of complexes. This proves Proposition 4.2 and completes the proof of Proposition 4.1 as we established the following isomorphisms

$$ \psi_f^u j_* \mathbf{L} \stackrel{\sim}{\hookrightarrow} \bar{\Psi}^*IL \stackrel{\sim}{\longrightarrow} \tilde{\Psi}^*IL \stackrel{\sim}{\longrightarrow} \Psi^*IL \quad (4.31) $$
<figure><img src="image_6.png" /></figure>
Chapter 5

Limit mixed Hodge structure on $\Psi^*IL$

*Hypothesis*. Let $f : X \to D$ be a projective morphism on a non singular complex variety $X$ to a complex disc and $\mathbf{L}$ a local system defined on an open subset $U := X \setminus Y$ complement of a NCD $Y = \bigcup_{i \in I} Y_i \subset X$. We suppose the central fiber equal to a NCD $Z = f^{-1}(0)$ contained in $Y$ as a sub-divisor. The divisor $Z$ is a union of components of $Y$ indexed by a subset $I_Z \subset I$. For $i \in I_Z$ let $m_i > 0$ denote the multiplicity of $Y_i$ in the divisor of $f$, so $(f) = \sum_{i \in I_Z} m_i Y_i$.

*Results*. Let $\mathbf{L}$ be a polarized VHS on $U$, the filtration $F$ is transported onto the nearby cycle complex by the isomorphism $\psi_f^u j_* \mathbf{L} \simeq \Psi^*IL$ (see chapter 4, Proposition 4.1). The monodromy filtration $W(N)$ shifted by the weight $w'$ of $\mathbf{L}$ define the weight filtration $W := W(N)[w']$ on $\psi_f^u j_* \mathbf{L}$ such that $F$ and $W$ form a structure of mixed Hodge complex (MHC) on $\psi_f^u j_* \mathbf{L}$.

The finite increasing filtration $W(\nu)$ defined by the nilpotent endomor-
phism $\nu$ on $\Psi^*IL$ corresponds to the monodromy filtration $W(N)$ on $\psi_f^u j_*\mathbf{L}$.

We define also the vanishing cycles complex $\varphi_f(j_*\mathbf{L})$ as a mixed cone over the specialization sp : $i_Z^*j_*\mathbf{L} \to \psi_f(j_*\mathbf{L})$ (§5.1.3.2) with filtrations $W$ and $F$, on which the action of the nilpotent endomorphism $N$ extends.

The main result of this chapter is the decomposition of $Gr_*^{W(N)} \Psi_f^u j_* \mathbf{L}$ and $Gr_*^{W(N)} \varphi_f^u j_* \mathbf{L}$ into a direct sum of intermediate extensions on the smooth and projective components of the NCD $Z$.

Remark 5.1 (Conventions and terminology). With the notations of (§2.4.0.2), we denote by ($^p\mathbf{L}$, $F$) := ($\mathbf{L}$[dim $X$], $F$) the perverse shifted polarized VHS. If $L := \mathcal{L}_X(x)$ is the fiber of Deligne’s extension, then $^pL := {}^p\mathcal{L}_X(x)$. If $\nu$ acts on a perverse sheaf $\mathcal{P}$ as a nilpotent endomorphism, the filtration $W(\nu)$ is well defined in the abelian category of perverse sheaves.

A shifted perverse sheaf $\mathcal{P}$ on an algebraic variety $X$ is a complex such that $\mathcal{P}[k]$ is perverse when shifted by an integer $k$.

Consequently, the notation $\Psi^* I^p \mathcal{L} = \Psi^* I \mathcal{L}[\dim X]$ means a corresponding shift in the degrees. It is not a perverse sheaf on $Z$ but $\Psi^* I^p \mathcal{L}[-1]$ is perverse, hence we introduce the notations:
$$ {}^{p}\psi_{f}^{u}j_{1*}{}^{p}\mathcal{L} := \psi_{f}^{u}j_{1*}{}^{p}\mathcal{L}[-1], \quad {}^{p}\varphi_{f}^{u}j_{1*}{}^{p}\mathcal{L} := \varphi_{f}^{u}j_{1*}{}^{p}\mathcal{L}[-1]. \qquad (5.1) $$

ii) Conventions on the weight filtration W. In the text, we define a weight filtration on $\psi_f^u j_1^* \mathbf{L}$ correspondind to a weight W on ${}^p \psi_f^u j_1^* \mathbf{L}$. We apply the following conventions:

1) Let W be an increasing filtration on an object of an abelian category, the filtration $W[k]$ is defined in [De 72] and [De 75] by: $W(k)_n := W_{n-k}$. For example, this convention apply to $\psi_f^u j_1^* \mathbf{L}$ where the weight W is related to the monodromy filtration by $W = W(N)[w']$.

2) In the case of a HS $(H, F)$, the weight of the twisted HS $(H, F)(k)$ drops by $-2k$.

3) The weight W on a complex K is shifted with the complex: $W_{i+1}(K[1]) = (W_i K)[1]$ but not $F(F^i(K[1]) = (F^i K)[1]$.

4) A bifiltered complex of sheaves $(K, W, F)$ on a topological space X was called for a while: cohomological MHC if its hypercohomology on X is a mixed Hodge complex. This terminology did not survive, so by extension we call this bifiltered complex a MHC.

After the introduction of perverse sheaves, we define in the text the more rich concept of perverse variation of MHS, which yields also a MHC, so for a while we use both terminology.

**Definition 5.1.** The morphism $N := \text{Log} T$ acts on $\psi_f K$, and extends to the cone $C(sp)$ to define a morphism $N$ on $\varphi_f^u K$. The morphism $N$ induces a morphism $\text{var} : \varphi_f^u K \to \psi_f^u K$ such that $\text{can} \circ \text{var} = N$ and $\text{var} \circ \text{can} = N$.

With the above conventions, we have:

$$ \begin{aligned} \mathrm{Gr}_{a+p_{w-1}}^{W} {}^{p}\psi_{f}^{u}j_{1*}{}^{p}\mathbf{L} &= (\mathrm{Gr}_{a}^{W(N)} \psi_{f}^{u}j_{1*}{}^{p}\mathbf{L})[n-1] = \mathrm{Gr}_{a+p_{w}}^{W} \psi_{f}^{u}j_{1*}{}^{p}\mathbf{L}[-1] \\ \mathrm{Gr}_{a+p_{w}}^{W} \psi_{f}^{u}j_{1*}{}^{p}\mathbf{L} &= (\mathrm{Gr}_{a+w'}^{W} \psi_{f}^{u}j_{1*}{}^{p}\mathbf{L})[n] \end{aligned} \qquad (5.2) $$

In the case of the vanishing cycle, the lemma 5.1 apply and adds 1 to the weight of the structure of nilpotent orbit

$$ \begin{aligned} \mathrm{Gr}_{a+p_w}^{W} \varphi_f^u j_1^* {}^p\mathbf{L} &= (\mathrm{Gr}_{a+p_{w+1}}^{W} \varphi_f^u j_1^* {}^p\mathbf{L})[-1] = (\mathrm{Gr}_{a+w'+1}^{W} \varphi_f^u j_1^* {}^p\mathbf{L})[n-1] \\ \mathrm{Gr}_{a+w'+1}^{W} \varphi_f^u j_1^* {}^p\mathbf{L} &= \mathrm{Gr}_{a}^{W(N)} \varphi_f^u j_1^* {}^p\mathbf{L} \end{aligned} \qquad (5.3) $$

**Theorem 5.1 (Decomposition of $ {}^p\psi_f^u j_1^* {}^p\mathbf{L} $ and $ {}^p\varphi_f^u j_1^* {}^p\mathbf{L} $).** The weight filtration W and the Hodge filtration F on $ {}^p\psi_f^u j_1^* {}^p\mathbf{L} $ (resp. $ {}^p\varphi_f^u j_1^* {}^p\mathbf{L} $) satisfy the following decompositions:

$$ \begin{aligned} \mathrm{Gr}_{i}^{W} {}^{p}\psi_{f}^{u}j_{1*}{}^{p}\mathbf{L} &\simeq \bigoplus_{Z,J \subset Z} ({}^{p}\mathcal{P}_{i}^{J}, F) \\ \mathrm{Gr}_{i}^{W} {}^{p}\varphi_{f}^{u}j_{1*}{}^{p}\mathbf{L} &\simeq \bigoplus_{Z,J \subset Z} ({}^{p}\mathcal{Q}_{i}^{J}, F) \end{aligned} \qquad (5.4) $$

into intermediate extensions of shifted polarized VHS on $Y_Z^*$: $( {}^{p}\mathcal{P}_i^J, F )$ (resp. $( {}^{p}\mathcal{Q}_i^J, F )$) (see Equation 5.31). Moreover:
$Gr_{a+p_w-1}^W p_{\psi_f^u j_!_*}^u p_L \simeq Gr_a^{W(N)} p_{\psi_f^u j_!_*}^u p_L$, $Gr_{a+p_w}^W p_{\varphi_f^u j_!_*}^u p_L \simeq Gr_a^{W(N)} p_{\varphi_f^u j_!_*}^u p_L$
and there exists a diagram of morphisms :

$$ (p_{\psi_f^u j_!_*}^u p_L, W, F) \xrightarrow{can} (p_{\varphi_f^u j_!_*}^u p_L, W, F) \xrightarrow{var} (p_{\psi_f^u j_!_*}^u p_L, W, F) (-1) $$

such that: $var \circ can = N$ and $can \circ var = N$.

The statement here is a decomposition in terms of intermediate extensions in opposition with classical statements on a decomposition of cohomology in terms of HS (before the introduction of perverse sheaves).

As a consequence of this decomposition and the existence of a pure HS on the hypercohomology of intermediate extensions of polarized VHS on the smooth projective components of $Z$ (proved by induction on $\dim Z$), we deduce the existence of a MHS on $\mathbb{H}^*(Z, \psi_f^u j_!_* \mathbf{L}, W, F)$ and $\mathbb{H}^*(Z, \varphi_f^u j_!_* \mathbf{L}, W, F)$.

*Organization of the chapter.* After a local study (§5.1), the theorem follows from (Proposition 5.3). The proof of the local version of the theorem (see Equation 5.23) relies on the local description of perverse sheaves by de Rham families (§3.2.6.1, Remark 3.6) and a criterion by Kashiwara on the decomposition into a direct sum of intermediate extensions, based on the property of distinguished pairs (§3.2.6.2).

For the global version (see §5.1.3.1) and for the results on vanishing cycles $(\varphi_f j_!_* \mathbf{L}, F)$ (see §5.1.3.2).

## 5.1 Local proof of theorem 5.1

The proof is local at a point $x \in Z$ (see §5.1.2 and Corollary 5.2). It is necessary to use the local description of a perverse sheaf by the associated canonical de Rham families (Remark 3.6, Equation 4.11), on which natural structures of IMHS are defined below (§5.1.1).

Since the decomposition occurs locally in the abelian category of MHS, it is necessarily accompanied by the decomposition of the filtration $F$.

The fiber of Deligne's extension $L := \mathcal{L}_X(x)$ at a point $x \in Z$ is endowed with a structure of nilpotent orbit and there exists a local decomposition of the complex $Gr_i^{W(N)} \Psi^*IL$ into a direct sum of shifted complexes $IC^*P_a^JL$ defined by nilpotent orbits $(P_a^JL, W, F, N_i, i \in M \setminus J)$ of weight $a+w'+1-|J|$ with induced filtrations $W$ and $F$ defined by the equation (5.5) below.

We refer to the nilpotent orbit structure on $\Psi_J(L)$ (see Equation 5.7) proved by Kashiwara ([Sa 90], Appendix), and to the decomposition for distinguished pairs following the proof by Saito ([Sa 88], Lemma 5.2.15) (see Equation 5.17 below).

When $x$ varies on $Z_J^*$, the $P_a^JL$ form a polarized VHS $\mathcal{P}_a^J$ (§5.1.2). We deduce from the purity theorem on the components of $Z$ that the complex $Gr_a^{W(\nu)} \Psi^*IL$ is a Hodge complex (see §5.1.3.1).

The purity theorem on the hypercohomology of the complex $Gr_i^{W(N)} \Psi^*IL$ may be deduced from (Theorem 3.1) or by induction on $\dim X$, since $\dim Z < 0$.
$\dim X$. The purity theorem is proved in the same time with the decomposition theorem. The main difference with the proof in [Sa 88] and [Sa 90], is the reduction to the use of the category of IMHS at isolated strata on $V$.

The main property of the limit MHS is the functoriality of the monodromy filtration with respect to the derived direct image $f_*(\mathcal{I})$ (§7.1.1.1). This is proved by induction on $\dim X$.

The theory of nilpotent orbit (resp. admissible variation of MHS) appears in [Sc 73, GrSc 73, CaKSc 86] (resp. [StZ 85]) but we adopt the notations and terminology of [Ka 86] as summarized in §2.3 of chapter 2.

### 5.1.1 Infinitesimal mixed Hodge structure on $\Psi_J(L)$

We consider again a polarized VHS $\mathbf{L}$ of weight $w'$, $x \in Z_M^* \subset Y$ and the nilpotent orbit $(L, F, P, N_i, i \in M)$ of weight $w'$ on $L := \mathcal{L}_X(x)$ with monodromy $N_i, i \in M$ and polarization $P$. The filtrations $F$ and $W = W(\sum_{i \in M} N_i)[w']$ form a limit MHS on $L$. These filtrations are used to construct a MHS on $\Psi_J(L)$ (see ch. 4, Definition 4.2).

First, two filtrations $W$ and $F$ are defined on $L[N]$ as follows:

$$W_k(L[N]) = \sum_i W_{k+2i} L \otimes N^i, \quad F^p(L[N]) = \sum_i F^{p+i} L \otimes N^i \quad (5.5)$$

such that

$$(\mathrm{Gr}_k^W L[N], F) = \oplus_i (\mathrm{Gr}_{k+2i}^W L, F[i]) \otimes N^i.$$

where $F[i]^p := F^{p+i}$ defines a HS of weight $k$ for each index $i$, since $(\mathrm{Gr}_{k+2i}^W L, F)$ is of weight $k+2i$. The morphisms $N$ and $N_i$ induce morphisms:

$$W_k(L[N]) \to W_{k-2}(L[N]) \text{ and } F^p(L[N]) \to F^{p-1}(L[N])$$

(see §4.2.1.3, Equation 5.5).

Since for each $j \in M$, the endomorphisms $A_j = m_j N - N_j$ shift $W$ by $-2$ and $F_J$ by $-1$, $\mathrm{Im}A_J$ is compatible with $F$ and $W$ on $L[N]$, hence the filtrations $W$ and $F$ induce a MHS on the quotient (see Definition 4.2)

$$\Psi_J L = (L[N]/\mathrm{Im}A_J, W^J, F_J). \quad (5.6)$$

The filtration $W^J$ in Equation (5.6) is in fact related to the monodromy filtration $W(N + \sum_{i \in M} N_i)$ ([Sa 90], Appendix):

**Theorem** (Kashiwara) Let $(L, F, P, N_i, i \in M)$ be a polarized nilpotent orbit of weight $w'$ and $m_i > 0$ for $i \in M$ a set of integers, then there exists on

$$\Psi_J(L) = (L[N]/\mathrm{Im}A_J, N, N_1, \dots, N_n; F, P_J) \quad (5.7)$$

a structure of polarized nilpotent orbit of weight $\delta = w' + 1 - |J|$ defined by the nilpotent endomorphism $N_J := (N + \sum_{i \in M} N_i)|_{\Psi_J(L)}$ and the polarization
$P_J$ (Equation 5.10). The monodromy filtration shifted by $\delta$ and the filtration $F_J$ define a MHS on $\Psi_J(L)$:

$$W^J = W(N_J)[\delta] : W_{k+\delta}^J = W(N_J)_k \qquad (5.8)$$

Precisely: $\forall k \ge 0$, $N_J^k : \mathrm{Gr}_{\delta+k}^{W^J} \Psi_J(L) \xrightarrow{\sim} \mathrm{Gr}_{\delta-k}^{W^J} \Psi_J(L)$.

We recall that polarized means that $\mathrm{Gr}_i^{W(N_J)} \Psi_J(L)$ is a direct sum of primitive polarized structure of weight $\delta + i$.

Let $\ell = |J|$ We recall the construction by Kashiwara of the polarization on $\Psi_J L$. Since the endomorphisms $A_J$ shift $W$ by $-2$ and $F$ by $-1$, the isomorphism in Equation (5.51) is an isomorphism of MHS

$$(\Psi_J L, W^J, F_J) \simeq (\oplus_{j \le \ell-1} L \otimes N^j, W, F) \qquad (5.9)$$

The action of $N$ on the left hand side is defined by Equation 4.16:

$$N(a \otimes N^{l-1}) = \sum_{j=1}^{\ell} (-1)^{j+1} \sigma_j((N_i/m_i), i \in J)(a) \otimes N^{\ell-j}.$$

In order to define a polarization we introduce a product $P_J$ on $\Psi_J L$ as follows

$$P_J(a \otimes N^i, b \otimes N^j) = P(a, (-1)^i \mathrm{Res}(A_J^{-1}(b \otimes N^{i+j}))) \qquad (5.10)$$

where $A_J^{-1}$ is defined on $L[N, N^{-1}]$ and the map $\mathrm{Res} : L[N, N^{-1}] \to L$ is defined by $\mathrm{Res}(\sum_i u_i N^i) := u_{-1}$. This relation shows directly that the product is well defined on Coker $A_J$: note that $P_J(a \otimes N^i, A_J(c \otimes N^j)) = P(a, (-1)^i \mathrm{Res}(c \otimes N^{i+j}) = 0$ as $\mathrm{Res}(c \otimes N^{i+j}) = 0$.

Using the explicit expression of $A_i^{-1}$ (see ch. 4, Equation 4.13), we find

$$\forall a, b \in L : P_J(a, b \otimes N^j) = P(a, b) \text{ if } j = \ell-1, \text{ and } 0 \le j < \ell-1. \qquad (5.11)$$

**Example 5.1.** Let $\mathbf{L}$ be a local system on $\mathbb{C}^2 \setminus Y$ where $Y = Y_1 \cup Y_2$ is defined by $z_1 z_2 = 0$. Then, $\Psi_i L$ at $x \in Y_i^*$ defines a VMHS for a varying $x$ on $Y_i^*$ and $W(N) = W(N_i)$, while $\Psi_{12} L$ is a nilpotent orbit for $L = \mathcal{L}(0)$ with limit MHS defined by $W^{\{12\}}$ a shifted weight filtration $W(N + N_1 + N_2)$.

<figure><img src="image_7.png" /></figure>

where $W^J$ is defined by (Equation 5.8) for $J = \{1\}, \{2\}, \{12\}$.
Remark that the decomposition (Equation 5.15) depends on the point $x$. For example, $(\Psi_1L, W^{\{1\}}, F)_{x \in Y_{12}^*} \neq (\Psi_1L, W^{\{1\}}, F)_{x \in Y_{1*}}$ in the decomposition of $(\mathrm{Gr}^{W(N)}\Psi_{12}L, W^{\{12\}}, F)_0$ into a direct sum:

$$(N_2\mathrm{Gr}^{W(N)}\Psi_1, W^{\{1\}}L, F)_0 \oplus (N_1\mathrm{Gr}^{W(N)}\Psi_2L, W^{\{2\}}, F)_0 \oplus \mathrm{HS}.$$

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

A diagram of modules $A \xrightarrow{\rho} B \xrightarrow{G} C$ satisfy the decomposition property if $B = \mathrm{Im}\rho \oplus \mathrm{Ker}G$. In this case, we say that the diagram forms a distinguished pair and $G$ induces an isomorphism $\mathrm{Im}\rho \simeq \mathrm{Im}G \circ \rho$. If $C = A$, we say that the diagram forms a distinguished pair ([Ka 86, §2.2]). Such decompositions appear in Deligne's proof of the Hard Lefschetz Theorem.

The decomposition of $\mathrm{Gr}_a^{W(N)}\Psi^*L$ is based on the concept of distinguished pairs. The following proposition is proved by an argument similar to ([Sa 88] Lemma 5.2.15):

**Proposition 5.1.** Let $(L, F, P, N_i, i \in M)$ be a polarized nilpotent orbit of weight $w'$, $m_i > 0$ a family of integers for $i \in M$ and $J \subset M$ ($Y = Z$). Let $W(N)$ denote the filtration defined on $\Psi_JL$ by the nilpotent endomorphism $N$ (§4.2.1.3, Lemma 4.2 in the case $M_Z = M$). For each $J \subset M$, $a \in \mathbb{Z}$ and $i \in J$, the diagram

$$\mathrm{Gr}_a^{W(N)}\Psi_{J \setminus i}L \xrightarrow{A_i} \mathrm{Gr}_a^{W(N)}\Psi_JL(-1) \xrightarrow{p_i} \mathrm{Gr}_a^{W(N)}\Psi_{J \setminus i}L(-1) \quad (5.12)$$

where $A_i = m_iN - N_i$ and $p_i$ is the canonical projection, is a distinguished pair:

$$\mathrm{Gr}_a^{W(N)}\Psi_JL(-1) \simeq \mathrm{Im}A_i \oplus \mathrm{Ker}p_i. \quad (5.13)$$

and $p_i$ induces an isomorphism of $\mathrm{Im}A_i \subset \mathrm{Gr}_a^{W(N)}\Psi_JL$ with $\mathrm{Im}(N_i : \mathrm{Gr}_a^{W(N)}\Psi_{J \setminus i}L \to \mathrm{Gr}_a^{W(N)}\Psi_{J \setminus i}L)$.

*Proof.* Here $L := \mathcal{L}_X(x)$ at $x \in Y_M^*$ is endowed with the limit MHS $W$ of an IMHS defined in (§5.1.1) at $x$. Let $i \in M$, the induced morphisms $N, N_i$ and $A_i$, shift the limit MHS $W$ (Equation 5.5) by $-2$ and $F$ by $-1$.

Let and $N_0 := \sum_{k \in M \setminus \{i\}} N_k$, since $W(N + N_0 + N_i)$ is the relative weight filtration with respect to $W(N + N_0)$ on $\Psi_J(L)$ (resp. $W(N + N_0)$ is relative to $W(N)$) ([CaKSc 86], [Ka 86] Theorem 3.2.9, [StZ 85] Definition 2.5), we have a HS of weight $a + j + r + \delta$ where $\delta = w' + 1 - |J|$ on

$$\mathrm{Gr}_{j+r+a}^{W(N+N_0+N_i)}\mathrm{Gr}_{r+a}^{W(N_0+N)}\mathrm{Gr}_a^{W(N)}\Psi_JL \simeq \mathrm{Gr}_j^{W(N_i)}\mathrm{Gr}_r^{W(N_0)}\mathrm{Gr}_a^{W(N)}\Psi_JL$$

For all $j, r \in \mathbb{N}$, we have a commutative diagram
**Proof.** By (§4.2.1.3, Lemma 4.2): $\Psi_J(IL) \simeq \Psi_{J_Z(N_{J'_Z}L)}$. Set $\Psi_{J_Z(N_{J'_Z}L)} := \Psi_{J_Z,J'}$. First let $i \in J_Z$ and consider the sequence

$$ \mathrm{Gr}_a^{W(N)} \Psi_{J_Z \setminus i, J'} \xrightarrow{A_i} \mathrm{Gr}_a^{W(N)} \Psi_{J_Z, J'}(-1) \xrightarrow{p_i} \mathrm{Gr}_a^{W(N)} \Psi_{J_Z \setminus i, J'}(-1) $$

where $A_i = m_i N - N_i$ and $p_i$ is the natural projection.

By repeated application of the lemma (5.1) below, $N_{J'}L$ is a nilpotent orbit. Hence the above proposition applies to $N_{J'_Z}L$ and we have the decomposition:

$$ \mathrm{Gr}_a^{W(N)} \Psi_{J_Z, J'} \simeq \mathrm{Im} A_i \oplus \mathrm{Ker} p_i \quad (5.16) $$

Second, for $i \in M'_Z$, hence $A_i = -N_i$, we check that the following sequence

$$ \mathrm{Gr}_a^{W(N)} \Psi_{J_Z, J' \setminus i} L(1) \xrightarrow{N_i} \mathrm{Gr}_a^{W(N)} \Psi_{J_Z, J'} L \xrightarrow{p_i} \mathrm{Gr}_a^{W(N)} \Psi_{J_Z, J' \setminus i} L(-1) $$

satisfies the decomposition property

$$ \mathrm{Gr}_a^{W(N)} \Psi_{J_Z, J'} L \simeq \mathrm{Im} N_i \oplus (\mathrm{Ker} p_i = 0). \quad (5.17) $$

which is clear via the isomorphism (§4.2.1.3, Lemma 4.2):

$$ (\Psi_{J_Z, J'} L, W, F) \simeq (\oplus_{j \le l_z - 1} N_{J'} L \otimes N^j, W, F) $$

compatible with the filtrations. The surjectivity of $N_i$ (resp. the injectivity of $p_i$) follows from the surjection $N_{J'_i} L \xrightarrow{N_i} N_{J'} L$ (resp. the injection $N_{J'} L \hookrightarrow N_{J'_i} L$). The proof is a consequence of:

**Lemma 5.1 (Vanishing cycle).** ([KaK 87] Theorem. 2.1.5, [CaKSc 87] Descent Lemma 1.16). Let $(L, F, P, N_1, \dots, N_n)$ be a nilpotent orbit of weight $w'$ polarized by $P$. Set $F^{k+1}(N_1 L) := N_1 F^{k+1} L = N_1 L \cap F^k L$, then

$$ (N_1 L, N_1 F, P_1, (N_i)_{i \in N_1 L}), \text{ where } P_1(N_1 a, N_1 b) = P(a, N_1 b), \forall a, b \in L $$

is a polarized nilpotent orbit of weight $w' + 1$.

i) Set $N_0 := \sum_{i \in [1, n]} N_i$. The monodromy filtrations centered at 0 satisfy:

$$ W_k(N_0|_{N_1L}) = N_1 W_{k+1}(N_0) = N_1 L \cap W_{k-1}(N_0) $$

$$ \text{and } W_{k+w'+1} N_1 L = W(N_0|_{N_1L})_k = N_1 W_{k+w'+1} L = N_1 W_{k+1}(N_0). $$

ii) The morphism $N_1$ is written as the composition of morphisms of MHS:

$$ N_1 : (L, W, F) \xrightarrow{\mathrm{can}} (N_1 L, F, W) \xrightarrow{N_1'} (L, W, F)(-1) $$

where can is surjective and called canonical such that $F^{k+1}(N_1 L) := N_1 F^{k+1} L = N_1 L \cap F^k L$ and $W_{k+1}(N_1 L) := N_1 W_{k+1} L = N_1 L \cap W_{k-1} L$.

The difficulty in the lemma is to prove the structure of nilpotent orbit on $N_1 L$ as in the following diagram. Let $\tilde{N}_0 := (N_0)|_{N_1 L}$ the restriction of $N_0$, then for all $k \ge 0$, the induced morphism by $N_0^k$ on the first line, is an isomorphism:
$$
\begin{array}{ccc}
N_1 \mathrm{Gr}_{k+1}^{W(N_0)} L & \xrightarrow{N_0^k} & N_1 \mathrm{Gr}_{-k+1}^{W(N_0)} L \\
\downarrow & = & \downarrow \\
\mathrm{Gr}_k^{W(\tilde{N}_0)} N_1 L & \xrightarrow{\tilde{N}_0^k} & \mathrm{Gr}_{-k}^{W(\tilde{N}_0)} N_1 L
\end{array}
$$

Remark 5.2. i) Later, the lemma is used to deduce the weight filtration on the perverse vanishing cycle sheaf $\mathcal{P}_{\varphi_f^u j_1^* P_L}$ from the weight on the perverse nearby cycle $\psi_f^u$ which explains the name of the result.

ii) In [CaKSc 87], the convention in the Descent lemma is: $F^k N_1 L = N_1 F^{k+1} L$, which explains the weight $w' - 1$ there, hence we have the factorization: $N_1 : (L, W, F) \to (N_1 L, F, W)(-1) \to (L, W, F)(-1)$.

Example 5.2. We illustrate the vanishing cycle lemma as follows. Given $(L, N)$, let $L' := NL$, denote by $N' : L' \to L'$ the restriction of $N$ and by $\tilde{N} : L \to L'$ the morphism induced by $N$. We have:

$$
\begin{tikzcd}[column sep=2.8em, row sep=2.8em]
  P_2 = \text{Ker } N^3 \subset \text{Gr}_2^W L \arrow[r, "\tilde{N} \sim"] & \tilde{N} P_2 = \text{Ker } N'^2 \subset \text{Gr}_1^{W'} L' \\
  \bigvee_{\substack{N \\ \downarrow}} & \bigvee_{\substack{N' \\ \downarrow}} \\
  P_0 = \text{Ker } \tilde{N} \arrow[r, "\sim"] & P_0 \oplus N \text{Gr}_2^W L \subset \text{Gr}_0^W L \arrow[r, "\tilde{N}"] & \text{Gr}_{-1}^{W'} L' \simeq N^2 \text{Gr}_2^W L \\
  \bigvee_{\substack{N \\ \downarrow}} & \bigvee_{\substack{N' \\ \downarrow}} \\
  N^2 P_2 = \text{Ker } \tilde{N} \arrow[r, "\sim"] & N^2 \text{Gr}_2^W L \simeq \text{Gr}_{-2}^{W'} L \arrow[r, "\tilde{N}"] & \text{Gr}_{-3}^{W'} L' \simeq N \text{Gr}_{-2}^{W'} L
\end{tikzcd}
$$
$$
\begin{tikzcd}[column sep=2.8em, row sep=2.8em]
  P_1 \subset \text{Gr}_1^W L \arrow[r, "\tilde{N} \sim"] & \tilde{N} P_1 = \text{Ker } N' \subset \text{Gr}_0^{W'} L' \\
  \bigvee_{\substack{N \\ \downarrow}} & \bigvee_{\substack{N' \\ \downarrow}} \\
  NP_1 = \text{Ker } \tilde{N} \subset \text{Gr}_{-1}^W L \arrow[r, "\tilde{N}"] & \text{Gr}_{-2}^{W'} L'
\end{tikzcd}
$$

5.1.2 Local decomposition of Gr*<sub>*</sub>W(N)<sub>*</sub> Ψ*IL

The proof of Theorem 5.1 relies on the local description of perverse sheaves by de Rham families and a criterion by Kashiwara on the decomposition into a direct sum of intermediate extensions, based on the property of distinguished pairs (§3.2.6.2).

The restriction of the shifted perverse sheaf $\Psi^*IL$ to a small polydisc with center $x \in Y_M^*$ isomorphic to $D^{m+l}$ in the product $Y_M \times N_{Y_M}$ of $Y_M$ with the normal section to $Y_M$ at $x$, is described by a de Rham family attached to $L := L(x)$ ([Ka 86], §2):
$$DR(\Psi IL) := \{\Psi_{JIL}, p_J^K : \Psi_{JIL} \to \Psi_{KIL}, A_{J \setminus K} : \Psi_{KIL} \to \Psi_{JIL}\}_{K \subset J \subset M}$$ $$p_J^K := \prod_{i \in J \setminus K} p_i : \Psi_{JIL} \to \Psi_{KIL}, A_{J \setminus K} := \prod_{i \in J \setminus K} A_i : \Psi_{KIL} \to \Psi_{JIL} \quad (5.18)$$

where the morphisms $p_J^K$ and $A_{J \setminus K}$ satisfy compatibility relations ([Ka 86], (2.1.1) to (2.1.4)). The morphism $N$ acts on the de Rham family $DR(\Psi IL)$ and we consider the graded de Rham family

$$\mathrm{Gr}_a^{W(N)} DR(\Psi_{IL}) := \{\mathrm{Gr}_a^{W(N)} \Psi_{JIL}, p_J^K, N_{J \setminus K}\}_{K \subset J \subset M} \quad (5.19)$$

where we write $p_J^K$ for $\mathrm{Gr}_a^{W(N)} p_J^K$. Moreover, $\mathrm{Gr}_a^{W(N)} A_{J \setminus K}$ coincides with $N_{J \setminus K}$ up to a sign, since $N : \mathrm{Gr}_a^{W(N)} \Psi_{J \setminus i} IL \to \mathrm{Gr}_a^{W(N)} \Psi_{JIL}$ vanishes for all $i \in J$. So the distinguished pairs property is satisfied by Corollary 5.1 for all $K \subset J \subset M$:

$$\mathrm{Gr}_a^{W(N)} \Psi_{JIL} \simeq \mathrm{Im} \, N_{J \setminus K} \oplus \mathrm{Ker} \, p_J^K. \quad (5.20)$$

**Proposition 5.2.** Let $\mathbf{L}$ be defined on $X \setminus Y$. For each $J \subset M$, $x \in Z_J$, let $L := \mathcal{L}_X(x)$. We define $P_a^J(IL) \subset \mathrm{Gr}_a^{W(N)} \Psi_{JIL}$ by

$$P_a^J(IL) := \bigcap_{K \subset J, K \neq J} \mathrm{Ker}(p_J^K : \mathrm{Gr}_a^{W(N)} \Psi_{JIL} \to \mathrm{Gr}_a^{W(N)} \Psi_{KIL}) \quad (5.21)$$

then $P_a^J(IL) \neq 0$ if and only $J \subset J_Z$.

i) If $x \in Y_M^*$, then $x \in Y_J$ for all $J \subset M, J \neq M$ and we have the decomposition as a direct sum of MHS (hence including the decomposition of the Hodge filtration $F$)

$$\mathrm{Gr}_a^{W(N)} \Psi_{MIL} \xrightarrow{\sim} \bigoplus_{J \subset M_Z} N_{M \setminus J} P_a^J(L) \quad (5.22)$$

Let $\mathrm{Im} \, N_{M \setminus J} : P_a^J(L) \to P_a^J(L)$ denote the image of $N_{M \setminus J}$ acting on $P_a^J(L)$. Then $p_M^J$ induces an isomorphism:

$$\mathrm{Gr}_a^{WM} \Psi_{MIL} \supset N_{M \setminus J} P_a^J(L) \stackrel{p_M^J}{\cong} \mathrm{Im} \, N_{M \setminus J} \subset P_a^J(L).$$

ii) Let $J = M_Z$ and $x \in Y_J^*$, then $P_a^J(L)$ is a HS of weight $a+w'+1-|J|$. For varying $x \in Y_J^*$, $P_J\mathbf{L}$ defines a polarize VHS on $Y_J^*$. iii) Let $x \in Y_J \setminus Y_J^*$ and $x \in Y_M^*$, then $(P_a^J(L), F, N_i, i \in M \setminus J)$ with induced filtration $F$ and nilpotent endomorphisms $N_i$, is a nilpotent orbit of weight $a+w'+1-|J|$.

**Corollary 5.2.** When $x \in Y_M^*$ the decomposition in Equation 5.22 is interpreted as the following decomposition

$$(\mathrm{Gr}_a^{W(N)} \Psi^*_{IL}, F) \xrightarrow{\sim} \bigoplus_{K \subset M_Z} (\mathrm{IC}^* P_a^K L, F)[-|K|]. \quad (5.23)$$

The corollary follows by (Equation 5.22) since the filtered morphism

$$\bigoplus_{J \subset M_Z} (N_{M \setminus J} P_a^J L, W, F) \xrightarrow{\sim} (\mathrm{Gr}_a^{W(N)} \Psi_{MIL}, W, F) \quad (5.24)$$
is necessarily an isomorphism in the category of MHS by (Remark 3.6vii).

The local decomposition by (Equation 5.23) leads later to the global decomposition (Equation 5.31).

*Example 5.3.* The decomposition of Diagram 4.17 in chapter 4, where $Z = Y$, is as follows

$$ \begin{aligned} \mathrm{Gr}_a^{W(N)} \Psi^1 L &:= \mathrm{Gr}_a^{W(N)} \Psi_1 \oplus \mathrm{Gr}_a^{W(N)} \Psi_2 = \mathrm{Gr}_a^{W(N_1)} L \oplus \mathrm{Gr}_a^{W(N_2)} L \\ \downarrow & \downarrow \downarrow N_2 \downarrow N_1 \\ \mathrm{Gr}_a^{W(N)} \Psi^2 L &:= \mathrm{Gr}_a^{W(N)} \Psi_{12} = N_2 P_a^1(L) \oplus N_1 P_a^2(L) \oplus P_a^{12}(L) \end{aligned} $$

where the term $P_a^1(L) = \mathrm{Gr}_a^{W(N_1)} \Psi^1 L$, (resp. $P_a^2(L) = \mathrm{Gr}_a^{W(N_2)} \Psi^2 L$) is a direct sum of polarized HS for $L := \mathcal{L}_X(x)$, $x \in Y_1^*$ (resp. $x \in Y_2^*$). For $x \in Y_{12}^*$ the space $N_2 P_a^1 L \oplus N_1 P_a^2 L$ is endowed with the induced MHS $W(N_1 + N_2)$ on $\mathrm{Gr}_a^{W(N)} \Psi^2 L$. Hence $W(N_2)$ on $N_2 P_a^1 L$ is induced by the nilpotent orbit $(P_a^1 L, N_2)$, resp. $W(N_1)$ on $N_1 P_a^2 L$.

Thus, the two columns are fibres of intersection complexes on $Y_i$.

The term $P_a^{12}L$ is endowed with a sub-polarized HS defined by $W(N_1 + N_2)$.

We have a decomposition as a direct sum of fibres of complexes at 0:

$$ IC^* P_a^1 L[-1] \oplus IC^* P_a^2 L[-1] \oplus P_a^{12} L[-2] $$

including the decomposition of Hodge filtrations. In addition there exists natural morphisms: $p_{12}^1: \mathrm{Gr}_a^{W(N)} \Psi_{12} \to \mathrm{Gr}_a^{W(N)} \Psi_1$ vanishing on $N_1 P_a^1 L \oplus P_a^{12}L$ and injective on $N_2 P_a^2 L$ (resp. $p_{12}^2: \mathrm{Gr}_a^{W(N)} \Psi_{12} \to \mathrm{Gr}_a^{W(N)} \Psi_2$ vanishing on $N_2 P_a^2 L \oplus P_a^{12}L$ and injective on $N_1 P_a^2 L$).

*Proof of Proposition 5.2.* By (Equation 5.17), $\ker p_i = 0$ if $i \notin M_Z$.

i) The decomposition follows functorially from the property of distinguished pairs in the graded family 5.19 and properties of the morphisms $p_J^K$ and $N_{J \setminus K}$ ([Ka 86], Proposition 2.3.1, Equations (2.1.1) to 2.1.4)).

We suggest a direct proof by induction on the length of $M: n = |M|$ of the general formula

$$ \mathrm{Gr}_a^{W(N)} \Psi_M IL \xrightarrow{\sim} \oplus_{J \subset M} N_{M \setminus J} P_a^J (IL). $$

For $n=1$ the decomposition is reduced to the property of graded distinguished pair. Let $M' := \{1, \dots, n-1\}$. We consider two sub-families:

$$ (\mathrm{Gr}_a^{W(N)} \Psi_{J \setminus M}, p_J^K, N_{J \setminus K})_{K \subset J \subset M'}; $$

$$ (\mathrm{Gr}_a^{W(N)} \Psi_{J \cup n} IL, p_{J \cup n}^K, N_{J \setminus K})_{K \subset J \subset M'}; $$

By induction, we suppose the decomposition for the two families with indices $J \subset M'$, valid on the two rows of the following diagram

$$ \begin{aligned} \mathrm{Gr}_a^{W(N)} \Psi_{M'IL} &\xrightarrow{\sim} \oplus_{J \subset M'} N_{M' \setminus J} P_a^J (IL) \\ p_M^{M'} \uparrow \downarrow N_n &\xrightarrow{\sim} p_{J \cup n}^J \uparrow \downarrow N_n \qquad (5.25) \\ \mathrm{Gr}_a^{W(N)} \Psi_M IL &\xrightarrow{\sim} \oplus_{J \subset M'} N_{M' \setminus J} P_a^{J \cup n, M'} (IL) \end{aligned} $$
where $P_a^{J \cup n, M'}(IL)$ is defined with respect to the family $K \cup n$ for $K \subset M'$, while $P_a^{J \cup n}(IL)$ (for the family with index in $M$) is defined with respect to the extended family with index $K \subset J \cup n$, i.e. $K \subset J \subset M'$ and $K \cup n \subset J \cup n$. The columns are written in terms of the property of distinguished pairs, for each $J \subset M'$:

$$ P_a^{J \cup n, M'}(IL) \simeq N_n P_a^J(IL) \oplus \mathrm{Ker} p_{J \cup n}^J \quad (5.26) $$

$$ P_a^{J \cup n}(IL) = \mathrm{Ker} (p_{J \cup n}^J : P_a^{J \cup n, M'}(IL) \to P_a^J(IL)) $$

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

$$ \begin{aligned} \mathrm{Gr}_a^{W^{(N)}} \Psi_J(IL) &\simeq \oplus_{J \subset M'} (N_n N_{M' \setminus J} P_a^J(IL) \oplus N_{M' \setminus J} P_a^{J \cup n}(IL)) \\ &= \oplus_{J \subset M} N_{M \setminus J} P_a^J(IL) \end{aligned} \quad (5.27) $$

ii) For $i \in J \subset M_Z$ and $K = J \setminus i$, $N_i : \mathrm{Gr}_a^{W^{(N)}} \Psi_J L \to \mathrm{Gr}_a^{W^{(N)}} \Psi_J L$ is equal to the composition morphism $\mathrm{Gr}_a^{W^{(N)}} \Psi_J L \xrightarrow{p_J^K} \mathrm{Gr}_a^{W^{(N)}} \Psi_K L \xrightarrow{N_i} \mathrm{Gr}_a^{W^{(N)}} \Psi_J L$. Hence for all $i \in J$, $N_i$ vanishes on $P_a^J L \subset \mathrm{Ker} p_J^K$.

iii) For $x \in Y_J^*$, since $N_0 := \sum_{i \in J} N_i$ vanishes on $P_a^J L$ and $P_a^J L$ is a direct summand of $\mathrm{Gr}_a^{W^{(N)}} \Psi_J L$, we deduce that $P_a^J L = \mathrm{Gr}_0^{W^{(N_0)}} P_a^J L$ is a pure direct summand of weight $a + w' + 1 - |J|$ of $\mathrm{Gr}_0^{W^{(N_0)}} \mathrm{Gr}_a^{W^{(N)}} \Psi_J L$ with respect to the MHS induced on $\mathrm{Gr}_a^{W^{(N)}} \Psi_J L$ by $W^J$.

iii) When $x$ varies on $Z_J^*$, $P_a^J(L)$ defines a global polarized VHS, hence for $x \in Z_J$, $P_a^J(L)$ is a nilpotent orbit of weight $a + w' + 1 - |J|$.

**Remark 5.3.** Let $\mathbf{L}$ be a shifted local system on the open subset $U$ union of the big strata of a stratification $S$ of $X$, $j : U \to X$ and $f : X \to D$ a morphism to the disc $D \in \mathbb{C}$ with central fiber $Z$ a NCD in $X$ intersecting transversally the strata of $S$ and let $j' : Z \cap U \to Z$, then $\mathrm{Gr}_*^{W^{(N)}} \psi_f^u j_{j_*} \mathbf{L} \simeq j_{j_*}' \mathrm{Gr}_*^{W^{(N)}} \psi_{f|_U}^u \mathbf{L}$.

**Remark 5.4.** [Relation with the open embedding] Let $Z \subset X$ be the NCD defined by $f$ and $j : X \setminus Z \to X$. The filtration $W(N)$ on $\psi_f^u$ defines the weight filtration on $j_*$ in the general case of VMHS (see §9.7.1.1).

### 5.1.3 Perverse VMHS structure on $\psi_f^u j_{j_*} \mathbf{L}$ and $\varphi_f^u j_{j_*} \mathbf{L}$

With the notations of §4.2.1.4 in chapter 4, we give below the global definition of the weight and Hodge filtrations on $\Psi^*\mathcal{I}\mathcal{L}$ when $\mathbf{L}$ is a polarized VHS on the complement of $Y$. The action of the logarithm of the monodromy is defined by $\nu$ on $\Psi^*\mathcal{I}\mathcal{L}$ (Equation 4.21); the weight filtration is defined by $W(\nu)$.

By the preceding local results, the graded weight complex decomposes into a direct sum of intermediate extensions. In particular, when the NCD $Z$
is projective, the derived global sections functor is a mixed Hodge complex (MHC) inducing MHS on its cohomology. The same results apply to the vanishing cycle complex $\Phi^*IL$ (see §5.1.3.2 below).

### 5.1.3.1 Weight and Hodge filtrations on $\Psi^*IL$

The Hodge filtration $F$ on the quotient complex $\Psi^*IL$ is deduced from the Hodge filtration $F$ on $\tilde{\Psi}^*IL$:

$$ F^p \tilde{\Psi}^* I\mathcal{L} := \sum_{q \ge 0} \mathbb{C} u^{-(q+1)} \otimes_{\mathbb{C}} (i_Z^* IC^* \mathcal{L}(\log Z)[q+1], F^{q+p+1}) \quad (5.28) $$

The weight and the monodromy filtration $W(\nu)$ on $\Psi^*IL$.

The action of the nilpotent endomorphism $\nu$ defines a unique monodromy filtration $W(\nu)$ on $\Psi^*IL$. This filtration $W(\nu)$ on $\Psi^*IL$ corresponds locally at a point $x \in Y_M^* \subset Z$, to the filtration $W(N)$ on $\Psi^*IL[1] \simeq (\Psi^*IL)_x$ (Equation 4.20).

In terms of a set of coordinate equations $y_i$ for $i \in M$ at $x$ of $Y_M$, the subspace $W(\nu)_k$ is generated as an $\Omega_{X,x}^*$ sub-module by the sections $\tilde{v} \wedge_{j \in J} \frac{dy_j}{y_j} \in \Psi^*IL_x$ for all $v \in W(N)_k \Psi_JIL[1]$ and $J \subset M$ such that the quasi-isomorphism

$$ (\Psi^*IL_x, W(\nu), F) \simeq (\Psi^*IL[1], W(N), F) \quad (5.29) $$

(see Equation 4.20) is compatible with both filtrations. The theorem (5.1) follows in the unipotent case from the following remark that the components $P_a^K L$ in Equations 5.22 and 5.23 are local version of a global VHS:

**Definition 5.2** ($IC^*P_a^K L \subset \text{Gr}_a^{W(\nu)}\Psi^*IL$). There exists a unique complex $IC^*P_a^K L$, defined as a sub-complex of $\text{Gr}_a^{W(\nu)}\Psi^*IL[|K| - 1]$) such that at a point $x \in Y_M^*$, the tilde embedding by (Equation 4.20) induces a quasi-isomorphism:

$$ IC^*(P_a^K L) \xrightarrow{\sim} (IC^*P_a^K L)_x \subset (\text{Gr}_a^{W(\nu)}\Psi^*IL[|K| - 1])_x, \quad L := \mathcal{L}(x) \quad (5.30) $$

The corollary below follows from the local decomposition in (Equation 5.23).

**Corollary 5.3.** There exists a polarized VHS: $P_a^J L$ of weight $a+w'+1-|J|$ on $Z_j^*$, such that $(IC^*P_a^J L, F)$ is a resolution of the intermediate extension of $P_a^J L$ by $j_Z: Z_j^* \to X:(IC^*P_a^J L, F) \simeq (j_{Z!*}P_a^J L, F)$.

**Proposition 5.3 (Decomposition of Gr$_*^{W(N)}$** $\psi_f^u j_*! L$). Let $L$ be of weight $w'$ on $X \setminus Y$ locally unipotent along $Y$ and let $W(\nu)$ be the monodromy filtration on $\Psi^*IL$. For each $a \in \mathbb{Z}$, there exists a decomposition into a direct sum over all $J \subset I_Z$ of intermediate extensions by $j_Z: Z_j^* \to X$ of weight $a+w'$:

$$ (\text{Gr}_a^{W(\nu)}\Psi^*IL, F) \simeq \bigoplus_{J \subset I_Z, J \neq \emptyset} (IC^*P_a^J L, F)[1-|J|](1-|J|) \quad (5.31) $$
**Proof.** By definition of $\mathrm{IC}^*P_a^J\mathcal{L}$, the decomposition in (Equation 5.31) corresponds to the local version (Equation 5.23) and (Equation 4.20):

$$ (\mathrm{Gr}_a^{W(\nu)}\Psi^*\mathcal{I}\mathcal{L}_x, F) \xrightarrow{\sim} \bigoplus_{K \subset M_Z} (\mathrm{IC}^*P_a^K L, F)[1 - |K|](1 - |K|) \quad (5.32) $$

Remark the twist of the weight by $-2(1 - |J|)$ and the shift by $[1 - |J|]$, since $P_a^J L$ is a nilpotent orbit of weight $a + w' + 1 - |J|$, hence: $\mathrm{IC}^*(P_a^J L)[1 - |J|](1 - |J|)$ is of weight $a + w' + 2(1 - |J|) - 2(1 - |J|) = a + w'$.

**Remark 5.5.** i) The decomposition is stated in the category of filtered derived complex and follows from the local quasi-isomorphism with the mixed Hodge complex (Equation 5.23) and the isomorphism of (Equation 5.24).

ii) The notations are compatible with [St 76] in the case of a constant system $\mathbf{L}$ of weight $w'$ on the complement of a NCD as central fiber, where the hypercohomology $\mathbb{H}^i(Z, \psi_f \mathbf{L})$ is of weight $w' + i$.

### 5.1.3.2 Unipotent vanishing cycles: $\varphi_f^u(j_{!*}\mathbf{L})$

The complex of sheaves of vanishing cycles $\varphi_f(j_{!*}\mathbf{L})$ (resp. $\varphi_f^u(j_{!*}\mathbf{L})$) is defined as the cone over the specialization morphism $\mathrm{sp} : i_Z^*j_{!*}\mathbf{L} \to \psi_f(j_{!*}\mathbf{L})$ (resp. $\mathrm{sp} : i_Z^*j_{!*}\mathbf{L} \to \psi_f^u(j_{!*}\mathbf{L})$) such that we have a triangle:

$$ \begin{align} i_Z^*j_{!*}\mathbf{L} \xrightarrow{\mathrm{sp}} \psi_f(j_{!*}\mathbf{L}) &\xrightarrow{\mathrm{can}} \varphi_f(j_{!*}\mathbf{L}) \xrightarrow{+1} \nonumber \\ i_Z^*j_{!*}\mathbf{L} \xrightarrow{\mathrm{sp}} \psi_f^u(j_{!*}\mathbf{L}) &\xrightarrow{\mathrm{can}} \varphi_f^u(j_{!*}\mathbf{L}) \xrightarrow{+1} \tag{5.33} \end{align} $$

The monodromy $T$ extends to $\varphi_f(j_{!*}\mathbf{L})$ since $T$ on $\psi_f(j_{!*}\mathbf{L})$ satisfy $T \circ \mathrm{sp} = \mathrm{sp}$. There exists a variation morphism $\mathrm{var} : \varphi_f^u(j_{!*}\mathbf{L}) \to \psi_f^u j_{!*}\mathbf{L}(-1)$ such that $\mathrm{var} \circ \mathrm{can} = N : \psi_f^u j_{!*}\mathbf{L} \to \psi_f^u j_{!*}\mathbf{L}(-1)$ (§2.2.3, Remark 2.7).

### 5.1.3.3 Weight and Hodge filtrations on $P_f\psi_f^u j_{!*}\mathbf{P}\mathbf{L}$ and $P_f\varphi_f^u j_{!*}\mathbf{P}\mathbf{L}$

The following relations in terms of $\mathrm{Ker}$ and $\mathrm{Im}$ are well defined in the abelian category of perverse sheaves (see Equation 5.49 and 5.50):

$$ i_Z^*j_{!*}\mathbf{P}\mathbf{L}[-1] \simeq \mathrm{Ker}(N : P_f\psi_f^u j_{!*}\mathbf{P}\mathbf{L} \to P_f\psi_f^u j_{!*}\mathbf{P}\mathbf{L}(-1)), \quad P_f\varphi_f^u j_{!*}\mathbf{P}\mathbf{L} \simeq \mathrm{Im} N \quad (5.34) $$

We also write by abuse of notations: $i_Z^*j_{!*}\mathbf{L} \simeq \mathrm{Ker}(N : \psi_f^u j_{!*}\mathbf{L} \to \psi_f^u j_{!*}\mathbf{L})$ and $\varphi_f^u j_{!*}\mathbf{L} \simeq \mathrm{Im} N$.

**Definitions of $F$ and $W$.** The filtration $W_k$ on $i_Z^*j_{!*}\mathbf{P}\mathbf{L}[-1]$ (resp. $P_f\varphi_f^u j_{!*}\mathbf{P}\mathbf{L}$) is defined as $\ker N|_{W_k} P_f\psi_f^u j_{!*}\mathbf{P}\mathbf{L}$ (resp. $\mathrm{Im} N$).

We may suppose $\ker N \xrightarrow{\sigma} P_f\psi_f^u j_{!*}\mathbf{P}\mathbf{L}$ embedded as a sub complex of $P_f\psi_f^u j_{!*}\mathbf{P}\mathbf{L}$, then the filtration $F$ is induced on $\ker N$ (resp. $\mathrm{Im} N$ as quotient
complex). The filtrations $W$ and $F$ on $p\varphi_f^u j_1^* p\mathbf{L}$ may be defined also on the mixed cone $C_M(\sigma)$.

This is easily checked on the de Rham family associated to $\psi_f^u j_1^* \mathbf{L}$. We suppose $\mathbf{L}$ locally unipotent. At each point $x \in Z$, set $L := \mathcal{L}(x)$ and with the notations of the de Rham family of nilpotent orbits:

$$K_{JIL} := \ker(N : \Psi_{JIL} \to \Psi_{JIL}(-1), \quad \phi_{JIL} := \mathrm{Im}(N : \Psi_{JIL} \to \Psi_{JIL}(-1))$$

The morphisms of MHS $A_i$ on $\Psi_{J-i}IL \to \Psi_{J}IL$ for each $i \in J$, induce corresponding morphisms on $K_{J-i}IL$ (resp. $\phi_{J-i}IL$). The total complex:

$$s(K_{JIL}, A_i)_{J \subset M_Z, i \in M_Z}, \quad (\text{resp. } s(\phi_{JIL}, A_i))$$

is isomorphic to the complex $K^*IL := \ker(N : \Psi^*IL \to \Psi^*IL) \simeq \ker N$ (resp. $\Phi^*IL := \mathrm{Im}(N : \Psi^*IL \to \Psi^*IL) \simeq \mathrm{Im} N$ in the category of perverse sheaves. Moreover the induced filtrations define a structure of MHC (if $Z$ is projective) and more precisely a structure of perverse VMHS to be made precise later (§, 9.7.1.3).

**Remark 5.6.** We write the endomorphism $\nu$ as the composition of a canonical morphism can and a variation morphism var:

$$\nu : \Psi^* I p \mathcal{L}[-1] \xrightarrow{can} \Phi^* I p \mathcal{L}[-1] \xrightarrow{var} \Psi^* I p \mathcal{L}[-1](-1).$$

with short exact sequences of perverse sheaves of MHC

$$0 \to i_Z^* j_1^* p \mathbf{L}[-1] \xrightarrow{sp} \Psi^* I p \mathcal{L}[-1] \xrightarrow{can} \Phi^* I p \mathcal{L}[-1] \to 0$$

and we have an isomorphism with the mixed cone: $C_M(sp) \simeq (\Phi^* I p \mathcal{L}[-1], W, F))$.

Let $N'$ be the nilpotent endomorphism on $\Phi^*IL$ induced by $N$. The vanishing cycle lemma 5.1 relate the weight filtration on $p\varphi_f^u j_1^* p\mathbf{L}$ to the monodromy filtration defined by the nilpotent endomorphism $N'$.

By construction, we have morphisms

$$(p\psi_f^u j_1^* p\mathbf{L}, W, F) \xrightarrow{can} (p\varphi_f^u j_1^* p\mathbf{L}, W, F) \xrightarrow{var} (p\psi_f^u j_1^* p\mathbf{L}, W, F)(-1).$$

where in this case can is an epimorphism and var is a monomorphism.

The graded complex $\mathrm{Gr}_*^{W(N')}\Phi^*IL$. As in the case $\Psi^*IL$ (5.1.2), the graded complex $\mathrm{Gr}_*^{W(N')}\Phi^*IL$ decomposes. We deduce a decomposition similar to (Equation 5.22)

$$\mathrm{Gr}_a^{W(N)} \Phi_{MIL} \xrightarrow{\sim} \bigoplus_{J \subset M} N_{M \setminus J} Q_a^J (IL) \quad (5.35)$$

**Definition 5.3.** The sub-complex $\Phi^*IL$ of $\Psi^*IL$ is generated locally at $x \in Z$ by the image of the tilde embedding of the complex $\Phi^*IL$ for $L := \mathcal{L}_X(x)$.

We denote by $\nu'$ the nilpotent endomorphism on $\Phi^*IL$ induced by $\nu$.

The endomorphism $\nu'$ corresponds locally to the nilpotent endomorphism $N'$ (up to a constant) on $\Phi^*IL$ induced by $N$.

The shifted perverse filtration $W(\nu')$ is defined locally by $\mathrm{Gr}_i^{W(N')}\Phi^*IL$.
**Proposition 5.4.** i) The complex $\Phi^*IL$ coincides with the image of $\nu$:

$$ \Phi^*IL \xrightarrow{\sim} \mathrm{Im}(\nu : \Psi^*IL \to \Psi^*IL). $$

It is endowed with filtrations W and F.

ii) Let $W(\nu')$ denote the monodromy weight filtration on $\Phi^*IL$ and $W := W(\nu')[w' + 1]$, then $(\mathrm{Gr}_{a+w'+1}^W \Phi^*IL, F) := (\mathrm{Gr}_a^{W(\nu')} \Phi^*IL, F)$ decomposes into the direct sum of shifted and twisted intermediate extensions of polarized VHS of weight $a + w' + 1$ on the various components of the NCD.

The shift of $w' + 1$ on $W(\nu')$ is compatible with the vanishing cycle Lemma. The proof is similar to (Proposition 5.3) or may be deduced by the same lemma.

**Corollary 5.4.** There exists a weight filtration W and a Hodge filtration F on $p\psi_f^u j_*!^p L$ (resp. $p\varphi_f^u j_*!^p L$) such that:

$$ \begin{aligned} \mathrm{Gr}_a^W p\psi_f^u j_*!^p L &\simeq \bigoplus_{Z,J \subset Z} (p\mathcal{P}_a^J, F) \\ \mathrm{Gr}_a^W p\varphi_f^u j_*!^p L &\simeq \bigoplus_{Z,J \subset Z} (p\mathcal{Q}_a^J, F) \end{aligned} \qquad (5.36) $$

where $(p\mathcal{P}_a^J, F)$ (resp. $(p\mathcal{Q}_a^J, F)$) is an intermediate extension of weight a of a shifted polarized VHS on $Y_Z^*$ (see Equation 5.31) such that:

$$ \mathrm{Gr}_{a+p_w-1}^W p\psi_f^u j_*!^p L \simeq \mathrm{Gr}_a^{W(N)} p\psi_f^u j_*!^p L, \quad \mathrm{Gr}_{a+p_w}^W p\varphi_f^u j_*!^p L \simeq \mathrm{Gr}_a^{W(N)} p\varphi_f^u j_*!^p L. $$

This result is used later to put a structure of mixed Hodge complex on $\psi_f^u(j_*!^p L)$ and $\varphi_f^u(j_*!^p L)$ by induction on dim $X$ as long as the purity of the hypercohomology of intermediate extension of VHS is proved.

## 5.2 Perverse VMHS on $p\psi_f j_*!^p L$ and $p\varphi_f j_*!^p L$

We extend Hodge theory to the perverse sheaves $\psi_f(j_*!^p L)$ and $\varphi_f(j_*!^p L)$ by reduction to the case where $L$ is not necessarily locally unipotent along $Y$.

### 5.2.0.1 Reduction to VHS unipotent along $Y$

In the case of a VHS $!^p L$ quasi-unipotent along $Y$, we introduce a ramified covering locally at a point $x \in Y$, to reduce the proof to the locally-unipotent case. Let $U \simeq D^n$ be a neighborhood of $x$ such that $Y$ is defined by a product of $q \le n$ local coordinates $t_1 \cdots t_q$. We can suppose $q = n$ such that $U \setminus Y \cap U \simeq (D^*)^n$.

Since the local monodromy $T_i$ on the limit vector space $L := \mathcal{L}(x)$ is quasi-unipotent, there exists $\nu_i$ such that $T_i^{\nu_i} - \mathrm{Id}$ is nilpotent.
Let $Z \subset X$ be the central fiber of $f: X \to D$. We suppose $Z \subset Y$ be a sub-NCD with irreducible components indexed by $I_Z \subset I$. Let $g$ denote the local ramified covering of a product of complex balls

$$ (Z \subset Y \subset D^n, t_.) \leftarrow_{g} (Z' \subset Y' \subset D_1^n, u_.) \qquad (5.37) $$

defined by $g: D_1^n \to D^n: (u_1, \dots, u_n) \to (t_1 = u_1^{\nu_1}, \dots, t_n = u_n^{\nu_n})$.

The inverse image $f^*(p\mathbf{L}|_{D^n})$ is a polarized VHS on $D_1^n \setminus Y'$ with locally unipotent monodromy $T'_i := T_i^{\nu_i}$ around the component $Y'_i \subset Y'$.

Let $j': D_1^n \setminus Y' \to D_1^n$. The finite group of covering transformations $G_i$ generated by $\xi_i := \exp(2i\pi/\nu_i)$, $i \in [1, n]$ acts on $D_1^n$ and on $p\mathbf{L}'$ by $T'_i$ as well on its intermediate extension $j'_i p\mathbf{L}'$.

**Lemma 5.2.** The perverse sheaf $j_* p\mathbf{L}$ (resp. $j'_i p\mathbf{L}$) is the invariant subspace of $g_* j'_i p\mathbf{L}$ (resp. $g_* j'_i p\mathbf{L}'$) by the action of the group $G = G_1 \times \dots \times G_n$.

*Proof.* Since the subgroups $G_i$ act independently and commutatively, the proof may be reduced to the case $n=1$.

The case $n=1$. Let $\nu_i := \nu$ be fixed. Since the monodromy $T$ is quasi-unipotent, let $L_b \subset L$ denote the subspace on which $T^s$ acts with eigenvalue $a_b = \exp(-2i\pi\alpha_b)$ where $\alpha_b = b/\nu$ for $b \in [0, \nu - 1]$ and let $N_b := \text{Log} T^u$ on $L_b$. Let $z = (1/2i\pi)\text{log} t$ denote the coordinate on $\tilde{D}$.

Recall the holomorphic section on $D$

$$ \tilde{v}(t) = \exp(\text{log} t(b/\nu - (1/2i\pi)N_b))v(z) = t^{b/\nu} \exp(-(1/2i\pi)(\text{log} t)N_b) v(z) $$

satisfying $\tilde{v}(te^{2i\pi}) = \tilde{v}(t)T^{-1}.T v(z) = \tilde{v}(t)$, where $v(z) \in L_b$ is a multiform section.

If $g: (D_1, u) \to (D, t)$ is of order $\nu$, $T' = T^\nu$ acts as $\exp(\nu N_b)$ on $L_b$. The action of the transformation group $G$ over $D$, generated by $\xi = e^{2i\pi/\nu}$, extends to the logarithmic complex $\Omega^* \mathcal{L}'_{D_1}$ as follows.

Let $z = (1/2i\pi)\log u$ and $v(z) \in L'_b$, then we have a section of $p\mathcal{L}'_{D_1}$:

$$ \tilde{v}^1 := \exp(-(1/2i\pi)(\log u)\nu N_b) v(z) $$

satisfying $\tilde{v}^1(ue^{2i\pi}) = \tilde{v}^1(u)(T')^{-1}T'v(z) = \tilde{v}^1(z)$.

The generator $\xi = e^{2i\pi/\nu}$ acts on the inverse image $g^*f$ of a holomorphic section $f(t) \in \Gamma(D, \mathcal{O}_D)$. We have: $g^*f(\xi u) = g^*f(u)$ since $f(\xi^\nu u^\nu) = f(u^\nu)$.

Reciprocally, let $h \in \Gamma(D_1, \mathcal{O}_{D_1})$ be a section on $D_1$. Set $(\xi.h)(u) := h(\xi u)$. If $\xi.h = h$, then $h(\xi u) = h(u)$ and $h$ is the inverse image of a holomorphic section $f(t)$ on $D$. This action extends to sections of $g_* p\mathcal{L}'_{D_1}$ and $g_* \Omega^* p\mathcal{L}'_{D_1}$.

**Sub-lemma.** Let $G$ acts on the linear combination of sections: $h(u)\tilde{v}^1(u)$ of Deligne's extension $p\mathcal{L}'_{D_1}$ as follows:

$$ \xi(h(u)\tilde{v}^1(u)) := h(\xi u)\tilde{v}^1(\xi u) \qquad (5.38) $$
The invariant sections $\omega$ of $g_*{}^p\mathcal{L}'_{D_1}$ satisfying $\xi.\omega = \omega$ correspond to sections of ${}^p\mathcal{L}_D$ on $D$. This action extends to $g_*\Omega^*{}^p\mathcal{L}'_{D_1}$ since $g^*dt/t = \nu du/u$. We have:

$$ \Omega^*{}^p\mathcal{L}_D \simeq (g_*\Omega^*{}^p\mathcal{L}'_{D_1})^\xi $$

The inverse image of a section $f.\tilde{v}$ of Deligne's extension ${}^p\mathcal{L}_D$ is a section of the extension ${}^p\mathcal{L}'_{D_1}$, since $\tilde{v}(u^\nu) = u^b\tilde{v}^1(u)$, hence:

$$ g^*(f.\tilde{v}(t)) = f(u^\nu)\tilde{v}(u^\nu) = f(u^\nu)u^b\tilde{v}^1(u) $$

Let $h(u)\tilde{v}^1(u) = g^*(f.\tilde{v}(t))$, we deduce:

$$ h(u) = f(u^\nu)u^b \text{ and } h(\xi u) = f(u^\nu)\xi^b u^b = h(u)\xi^b. $$

If we develop $h(u) = \sum_i a_i u^i$: $h(\xi u) = \sum_n a_n u^n = \sum_n a_n u^n \xi^b \implies a_n \xi^n = a_n \xi^b$; hence $a_n = 0$ or $\xi^{n-b} = 1, n-b = k\nu$ and finally $n = k\nu + b$.

Reciprocally, sections $h(u)\tilde{v}^1(u)$ where $h(u) = \sum_k a_{k\nu+b} u^{k\nu+b}$ are written as inverse image of sections on $D$.

**Corollary 5.5.** The filtration $F$ on $(\Omega^*{}^p\mathcal{L})_D$ is defined by the invariant sections

$$ F^q(\Omega^*{}^p\mathcal{L})_D \simeq (F^q\Omega^*{}^p\mathcal{L}')^G_{D_1} $$

and extends to a global filtration $F$ on $(\Omega^*{}^p\mathcal{L})_X$. Hence (Theorem 5.1 and Equation 5.36) apply to polarized VHS non necessarily unipotent along $Y$.

The case of $j_*{}^p\mathcal{L}$. The restriction to $D$ of the perverse direct image: $(j_*{}^p\mathcal{L})|_D$ is the invariant perverse subspace of $g_*((j'_*{}^p\mathcal{L}')|_{D_1})$. Hence, the cohomology $\mathbb{H}^*(D^*,{}^p\mathcal{L})$ is isomorphic to the invariant subspace of $\mathbb{H}^*(D_1^*,{}^p\mathcal{L}')$. Since $\mathbb{H}^*(D, j_*{}^p\mathcal{L})$ is defined by truncation, we deduce $j_*{}^p\mathcal{L}$ by the invariance construction as well.

Let $Z \subset Y$ be a sub-divisor (resp. $Z' \subset Y'$ above $Z$), $j_Z : D^n \setminus (Z \cap D^n) \to D^n$ and $j'_{Z'} : D_1^n \setminus (Z' \cap D_1^n) \to D_1^n$, then $\mathbb{H}^*(D^n, j_*j'_Zj_*{}^p\mathcal{L})$ is isomorphic to the invariant subspace of $\mathbb{H}^*(D_1^n, j'_{Z'}j'_{Z'}{}^p\mathcal{L}')$.

The case of $\psi_f^u j_*j_*{}^p\mathcal{L}$. The properties of $\psi_f^u j_*j_*{}^p\mathcal{L}$ may be recovered again from the definition of the filtration $F$ on $(\Omega^*{}^p\mathcal{L})_X$. Hence, the case of $\psi_f^u j_*j_*{}^p\mathcal{L}$ may be deduced from the corollary 5.5.

The reduction to the locally unipotent case apply. The ramified covering $g$ fits in the following diagram

<figure><img src="image_8.png" /></figure>
where $f: D^n \to D$ is defined by $f = t_1^{m_1} \cdots t_p^{m_p}$ and $Z = f^{-1}(0) \subset Y$ are NCD, $f_1 = f \circ g, Y' = f^{-1}(Y), Z' = f^{-1}(Z)$ and $Y \cap f^{-1}(D^*), Y' \cap f_1^{-1}(D^*)$ are relative NCD over $D^*$.

We deduce $\psi_{fj_1*}^u \mathbf{pL}$ as the invariant subspace by the action of $G$ on $g_* \psi_{f_1j_1'}^{u*} \mathbf{pL}'$. Let $N_i := \mathrm{Log} T_i^u, N_i' := \mathrm{Log} T_i', N' := N_1' + \cdots + N_n'$ and $N := N_1 + \cdots + N_n$. The filtration $W(N)$ defined by $N$ on $\psi_{fj_1*}^u \mathbf{pL}$ corresponds to the invariant subspace of the filtration $W(N')$ defined by $N'$ on $\psi_{f_1j_1'}^{u*} \mathbf{pL}'$. The decomposition of $Gr_*^{W(N')} \psi_{fj_1*}^u \mathbf{pL}$ is deduced from the decomposition of $g_* Gr_*^{W(N')} \psi_{f_1j_1'}^{u*} \mathbf{pL}'$ by the action of $G$.

### 5.2.0.2 Weight and Hodge filtrations on $^{p}\psi_{fj_1*}^{p}\mathbf{L}$ and $^{p}\varphi_{fj_1*}^{p}\mathbf{L}$

Let $\mathbf{pL}$ be a quasi-unipotent shifted VHS. Let $\alpha \in [0, 1[$ such that $\exp(-2i\pi\alpha)$ is an eigenvalue of $^{p}\psi_{fj_1*}^{p}\mathcal{L} := \psi_{fj_1*}^{p}\mathcal{L}[-1]$. Set $\beta = \exp(-2i\pi\alpha)$ and $^{p}\psi_{fj_1*}^{\beta} \mathbf{pL}$ the sub-perverse sheaf invariant by $T^s - \beta Id$ where $T^s$ is the semi simple action of $T$. There exists a natural decomposition into a direct sum of perverse sub-sheaves:

$$^{p}\psi_{fj_1*}^{p}\mathcal{L} \simeq \bigoplus_{\alpha \in [0,1[} ^{p}\psi_{fj_1*}^{\beta} \mathbf{pL}, \quad ^{p}\varphi_{fj_1*}^{p}\mathcal{L} \simeq \bigoplus_{\alpha \in [0,1[} ^{p}\varphi_{fj_1*}^{\beta} \mathbf{pL}$$

**Proposition 5.5 (perverse VMHS on $^{p}\psi_{fj_1*}^{p}\mathbf{L}$ and $^{p}\varphi_{fj_1*}^{p}\mathbf{L}$).** With the notations of (Equation 5.36 and Theorem 5.1), let $\mathbf{pL}$ be a shifted VHS quasi-unipotent along $Y$.

There exists a natural structure of perverse VMHS on $^{p}\psi_{fj_1*}^{p}\mathcal{L}$ (resp. $^{p}\varphi_{fj_1*}^{p}\mathcal{L}$) with support on the NCD $Z$ satisfying the following decomposition according to the action of the monodromy. Let $\beta = \exp(-2i\pi\alpha)$, then:

$$(\mathrm{Gr}_a^W {}^p\psi_{fj_1*} {}^p\mathbf{L}, F) \simeq \bigoplus_{\alpha \in [0,1[} (\mathrm{Gr}_a^W {}^p\psi_{fj_1*} {}^p\mathbf{L}, F) \simeq \bigoplus_{\alpha \in [0,1[} \bigoplus_{Z_J \subset Z} ({}^p\mathcal{P}_a^{\beta,J}, F) \\ (\mathrm{Gr}_a^W {}^p\varphi_{fj_1*} {}^p\mathbf{L}, F) \simeq \bigoplus_{\alpha \in [0,1[} (\mathrm{Gr}_a^W {}^p\varphi_{fj_1*} {}^p\mathbf{L}, F) \simeq \bigoplus_{\alpha \in [0,1[} \bigoplus_{Z_J \subset Z} ({}^p\mathcal{Q}_a^{\beta,J}, F) \qquad (5.39)$$

where $^{p}\psi_{fj_1*}^{\beta} \mathbf{pL}$ (resp. $^{p}\varphi_{fj_1*}^{\beta} \mathbf{pL}$) is the perverse sub-sheaf on which the semi-simple monodromy action $T_s$ has eigenvalue $\beta$ and $^{p}\mathcal{P}_a^{\beta,J}$ (resp. $^{p}\mathcal{Q}_a^{\beta,J}$) is an intermediate extension of a polarized VHS of weight $a$ on $Z_J$.

*Proof.* In the case of quasi-unipotent $\mathbf{pL}$, the decomposition into a direct sum over $\alpha$ is proved in ([DeK 73]: Proposition 4.17, Lemma 4.18.5 and Remark 4.18.6). By the same reference, we introduce on $D^*$ the local system $U_\alpha$ of rank 1 with monodromy $e^{2i\pi\alpha}$ and trivial Hodge structure of type $(0,0)$. We have:

$$^{p}\psi_{f}^{u}(j_{1*}^{p}\mathcal{L} \otimes f^{*}U_{\alpha}) \simeq {}^{p}\psi_{f}^{\beta}(j_{1*}^{p}\mathcal{L})$$

where for each Milnor fiber at $z \in Z: {}^{p}\psi_{fj_1*}^{u}(\mathcal{L} \otimes f^*U_\alpha)_z \simeq {}^{p}\psi_{fj_1*}^{\beta}(\mathcal{L})_z$. Then
$$ p\psi_f j_{1*}^p \mathcal{L} \simeq \bigoplus_{\alpha \in [0,1]} p\psi_f^u (j_{1*}^p \mathcal{L} \otimes f^* U_\alpha) \quad (5.40) $$

Hence the proof is reduced to the unipotent case $p\psi_f^u$. However, since $\mathcal{L} \otimes f^*U_\alpha$, is a VHS, quasi-unipotent along $Z$, we apply (§5.2.0.1) to reduce the decomposition of $p\psi_f^u(j_{1*}^p \mathcal{L} \otimes f^*U_\alpha)$ over $Z_J \subset Z$ to the case of a locally unipotent $\mathbf{L}$, where we apply the equations (5.36) and (5.31) to write: $(p\mathcal{P}_a^J, F) = (j_{1*}^p \mathcal{P}_{a-pw+1}^J[\dim X - |J|], F)(1 - |J|)$.

By (§5.1.3.3, Corollary 5.1) the proof apply to give a similar decomposition for $(\mathrm{Gr}_a^W p\varphi_f j_{1*}^p \mathbf{L}, F)$. For $\alpha \neq 0$, the canonical morphism induces an isomorphism $p\psi_f^\beta j_{1*}^p \mathcal{L} \simeq p\varphi_f^\beta j_{1*}^p \mathcal{L}$.

### 5.2.0.3 Perverse VMHS with support

The following structure of perverse VMHS with support on $Z$ is satisfied by $p\psi_f j_{1*}^p \mathbf{L}$ and $p\varphi_f j_{1*}^p \mathbf{L}$.

**Definition 5.4 (Perverse VMHS on $X$ with support on a NCD).** A perverse VMHS is a bifiltered complex $(K', W, F)$ with support on a NCD $Z \subset X$ satisfying the following conditions:

1. $W$ is an increasing perverse filtration on $K'$ defined over $\mathbb{Q}$,

2. $F$ is a decreasing filtration defined over $\mathbb{C}$ such that for all $r \in \mathbb{Z}$,

$$ (\mathrm{Gr}_r^W K', F) \simeq \bigoplus_{Z_J \subset Z} (j_{1*}^p \mathbf{L}_r^J, F) \quad (5.41) $$

is a direct sum of intermediate extension of polarized VHS $p\mathbf{L}_r^J$ of weight $r$ on non empty smooth open subsets $Z_J^*$, with embedding $j: Z_J^* \to Z_J$.

This concept is a generalization of MHC in [De 72] and by extension it is sometimes denoted MHC or PVMHS.

### 5.2.1 *Functoriality of the monodromy filtration*

We illustrate the proof of the decomposition theorem in §6 and its relation to the purity theorem, in the case of $\dim V = 1$.

We deduce the structure of PVMHS on $\psi_f^u j_{1*}^p \mathbf{L}$ by the isomorphism $\psi_f^u j_{1*}^p \mathbf{L} \simeq \Psi^* I^p \mathcal{L}$ and similarly the structure of PVMHS on the vanishing cycles $\varphi_f^u (j_{1*}^p \mathbf{L})$ by the isomorphism $\varphi_f^u (j_{1*}^p \mathbf{L}) \simeq \varphi^* I^p \mathcal{L}$.
### 5.2.1.1 Bigraded polarized HS

We assume the purity theorem 3.1. In the text, this purity theorem is proved by induction on $\dim X$. The induction starts in dimension 0 or 1 with the result on a projective smooth curve [Zu 79].

As a consequence, if we assume the purity theorem in the case $\dim X = n - 1$, we define for $X$ of dimension $n$:

**Definition 5.5.** The limit MHS on $\mathbb{H}^*(Z, \Psi^* I^{p\mathcal{L}})$ is defined by $(\Psi^* I^{p\mathcal{L}}, W, F)$ and transported onto the hypercohomology $\mathbb{H}^*(Z, \psi_f^u j_{f_*}^p \mathbf{L})$ (Equation 4.22) with monodromy endomorphism $N$ induced by the endomorphism $-2i\pi\nu$ on $\Psi^* I^{p\mathcal{L}}$.

We deduce from the definition, the next statement on the functoriality of the monodromy filtration with the derived direct image $f_*$ (this functoriality is used later to extend the induction on the purity theorem to dimension $n$):

**Theorem 5.2 (Monodromy filtration).** *The iterated morphism*

$$N^r : \mathrm{Gr}_{p_w+q+r}^W \mathbb{H}^q(Z, \psi_f^u j_{f_*}^p \mathbf{L}) \simeq \mathrm{Gr}_{p_w+q-r}^W \mathbb{H}^q(Z, \psi_f^u j_{f_*}^p \mathbf{L})(-r)$$

is an isomorphism of Hodge structures induced by $-2i\pi\nu$. Hence, the filtration induced by (Equation 5.5) is the monodromy filtration of $N$.

*Proof.* We recall the statement of (Equation 5.31) in terms of the monodromy filtration $W(\nu)$ on the perverse sheaf $\Psi^* I^{p\mathcal{L}}[-1]$, where the shift by $[-1]$ induces a shift on the weight filtration, hence $W := W(\nu)[p_w - 1]$:

$$(\mathrm{Gr}_a^W \psi_f^u j_{f_*}^p \mathbf{L}, F) := (\mathrm{Gr}_{a-pw+1}^{W(\nu)} \Psi^* I^{p\mathcal{L}}[-1], F) \simeq \oplus_{Z,J \subset Z} (p\mathcal{P}_a^J, F) \quad (5.42)$$

where: $(p\mathcal{P}_a^J, F) = (j_{Z_*} \mathcal{P}_{a-pw+1}^J[\dim X - |J|], F)(1 - |J|)$ (Equation 5.31).

The shift in the degree of the perverse sheaf allows symmetry in the equations and the use of the next result.

**Bigraded polarized HS:** The weight spectral sequence with respect to the weight filtration

$$E_1^{p,q} = \mathbb{H}^{p+q}(Z, \mathrm{Gr}_{-p}^W \Psi^* I^{p\mathcal{L}}[-1]) \implies \mathrm{Gr}_q^W \mathbb{H}^{p+q}(Z, \Psi^* I^{p\mathcal{L}}[-1]) \quad (5.43)$$

degenerates at rank 2. The proof is based on Hodge theory. Let $n := \dim X$. Since $\dim Z = n - 1$, the $E_1$ terms carry a bigraded polarized HS by the Induction hypothesis. For symmetry reasons we introduce the following family of spaces $L_p^q$ ([Sa 88] Proposition 4.2.2, [St 76], [Shi 93] Lemma 4.2.2):

$$L_p^q := E_1^{-(p+p_w-1), p+p_w-1+q} = \mathbb{H}^q(Z, \mathrm{Gr}_p^W(\nu) \Psi^* I^{p\mathcal{L}}[-1])$$

satisfying the following properties:

1. $L_p^q$ is a HS of weight $p+q+p_w-1$, since we have the decomposition
$$L_p^q \simeq \bigoplus_{J \subset I_Z} \mathbb{H}^q(Z, j_{Z!*} \mathcal{P}_{p-|J|}^J [n - |J|]) (1 - |J|).$$

2. The cup product with the class of an hyperplane section of $X$ induces a morphism $l: L_p^q \to L_p^{q+2}(1)$ such that $l^q: L_p^{-q} \xrightarrow{\sim} L_p^q(q)$ is an isomorphism for $q \ge 0$ by Hard Lefschetz on the components of $Z$.

3. The endomorphism $\nu$ induces the monodromy morphism $\nu: L_p^q \to L_{p-2}^q(-1)$ which commutes with $l: \nu \circ l = l \circ \nu$, such that $\nu^p: L_p^q \xrightarrow{\sim} L_{-p}^q(-p)$ is an isomorphism for $p \ge 0$ by the property of the monodromy filtration $W(\nu)$.

4. We deduce from the self-duality of $\mathbf{L}$ and of the monodromy filtration a scalar product polarizing the HS

$$S: L_{-p}^{-q} \times L_p^q \to \mathbb{Q}(-p\omega + 1)$$

5. Let $0L_p^{-q} = \text{Ker } N^{p+1} \cap \text{Ker } l^{q+1}$, then: $S(x, N^p l^q y)$ induces a polarization of the HS on $0L_p^{-q} \times 0L_p^{-q}$ up to sign.

6. The differential $d_1$ on the terms $E_1$ of the spectral sequence induces a morphism $d: L_p^q \to L_{p-1}^{q+1}$ satisfying $d^2 = 0$, $d \circ l = l \circ d$, $d \circ \nu = \nu \circ d$ and $S(dx, y) = \pm S(x, dy)$.

The above properties form a HL module ([Sa 88] Section 4.2, see also [St 76] §5) such that the following theorem apply:

**Theorem** ([Sa 88] Proposition 4.2.2) Let $L := \bigoplus_{p,q \in \mathbb{Z}} L^{p,q}$ denote a bigraded Hodge Lefschetz polarized differential module, then the cohomology

$$H = \text{Ker } d/\text{Im } d = \bigoplus_{p,q \in \mathbb{Z}} H^{p,q}$$

is also a bigraded Hodge Lefschetz polarized differential module with the induced polarization.

The proof by Saito is a correction to the proof in ([St 76] §5). The statement is also valid in the abelian category of Hodge modules.

The proof of the theorem follows.

### 5.2.1.2 Special case of the Decomposition and Purity theorems

Let $f: X \to \mathbb{P} := \mathbb{P}_\mathbb{C}^1$ be a morphism such that $X_0 := f^{-1}(0)$ is a NCD in $X$. Let $\Sigma \subset \mathbb{P}$ denote the set of critical values of $f$, so $0 \in \Sigma$ and the restriction $f_\mid: X \setminus f^{-1}(\Sigma) \to \mathbb{P}^* = \mathbb{P} \setminus \Sigma$ is smooth. We consider the diagram

$$\begin{array}{ccc}
X_0 \stackrel{i}{\hookrightarrow} & X \stackrel{k}{\hookleftarrow} & X \setminus f^{-1}(\Sigma) \\
\downarrow & \downarrow f & \downarrow \\
0 \stackrel{i'}{\hookrightarrow} & \mathbb{P} \stackrel{k'}{\hookleftarrow} & \mathbb{P}^*
\end{array}
\qquad (5.44)$$

Let $\mathbf{L}$ denote a polarized VHS on the complement of a NCD $Y$ containing $X_0$. We write $Y = X_0 \cup X'$ where $X'$ is the closure of $Y \setminus X_0$ in $X$. We
suppose $X'$ horizontal on $\mathbb{P}^*$ in the sense that $X'_t \subset X_t$ is a NCD in the smooth fiber $X_t$ for $t \notin \Sigma$ and $X'_{|\mathbb{P}^*}$ is topologically locally trivial over $\mathbb{P}^*$ such that $R^i f^* \mathbf{L}_{|\mathbb{P}^*}$ is locally constant over $\mathbb{P}^*$ with fiber $\mathbb{H}^i(X_t, j_{1*} \mathbf{L})$.

Let $K := f_*j_{1*}^{p}\mathbf{L}$. The restriction to $\mathbb{P}^*$ of $^{p}\mathcal{H}^i(K)$ is a shifted polarized VHS and we have $^{p}\mathcal{H}^i(K)|_{\mathbb{P}^*} \simeq \mathbf{L}_{\mathbb{P}^*}^i[1]$.

**Proposition 5.6 (Decomposition).** Let $K := f_*j_{1*}^{p}\mathbf{L}$ be the derived image.
*There exists a canonical splitting*

$$ (^{p}\mathcal{H}^{i}(K), F)_{0} = (L_{0}^{i}, F) \oplus (IC^{*}\mathcal{L}_{\mathbb{P}^{*}}^{i}[1], F)_{0} $$

into the polarized HS $(L_0^i, F)$ and $IC^*\mathcal{L}_{\mathbb{P}^*}^i[1] = k_{1*}^l \mathbf{L}_{\mathbb{P}^*}^i[1]$ with its limit filtration $F$.

We recall that by the uniqueness of the decomposition of $^{p}\mathcal{H}^i(K)$ we have the isomorphism $L_0^i \simeq \mathrm{Im}(\mathbb{H}_{X_0}^i(X, j_{1*}^{p}\mathbf{L}) \to \mathbb{H}^i(X_0, j_{1*}^{p}\mathbf{L}))$.

The proof is local at 0, hence we consider the preimage $X_D := f^{-1}(D) \subset X$ of a disc $D$ with center 0 and coordinate $z$. Since

$$ {}^{p}\psi_{z}^{u} = \psi_{z}^{u}[-1], \quad {}^{p}\psi_{z_{\circ f}}^{u} = \psi_{z_{\circ f}}^{u}[-1] \text{ and } {}^{p}\varphi_{z}^{u} := \varphi_{z}^{u}[-1], \quad {}^{p}\varphi_{z_{\circ f}}^{u} = \varphi_{z_{\circ f}}^{u}[-1] $$

transform perverse sheaves into perverse sheaves, we have:

$$ (^{p}\psi_{z}^{u} {}^{p}\mathcal{H}^{i}(K)) = {}^{p}\mathcal{H}^{i}(f_{|X_{0}^{*}} {}^{p}\psi_{z_{\circ f} j_{1}^{*} {}^{p}\mathcal{L}}^{u}) = H^{i-1}(X_{0}, \psi_{z_{\circ f} j_{1}^{*} {}^{p}\mathcal{L}}) =: H_{\psi} $$

$$ (^{p}\varphi_{z}^{u} {}^{p}\mathcal{H}^{i}(K)) = {}^{p}\mathcal{H}^{i}(f_{|X_{0}^{*}} {}^{p}\varphi_{z_{\circ f} j_{1}^{*} {}^{p}\mathcal{L}}^{u}) = H^{i-1}(X_{0}, \varphi_{z_{\circ f} j_{1}^{*} {}^{p}\mathcal{L}}) =: H_{\varphi}. $$

Moreover, since $\dim X_0 = n-1$, the conditions of Theorem 5.2 are satisfied by induction. Hence, the monodromy filtration on $H_\psi$ is deduced from the monodromy filtration on $\psi_{z\circ f j_1}^{*} \mathbf{pL}$. Let $^{p}w$ be the weight of $^{p}\mathbf{L}$. Then $H_\psi$ (resp. $H_\varphi$) is a nilpotent orbit of weight $^{p}w + i - 1$ (resp. $^{p}w + i$). Hence $H_\psi^m := \mathrm{Gr}_m^{W(N)} H_\psi$ is a polarized HS of weight $^{p}w + i - 1 + m$ and $H_\varphi^{m-1} := \mathrm{Gr}_{m-1}^{W(N)} H_\varphi$ is a PHS of weight $^{p}w + i + m - 1$.

For all $m \in \mathbb{Z}$ the diagrams $H_\psi^m \xrightarrow{C} H_\varphi^{m-1} \xrightarrow{V} H_\psi^{m-2}(-1)$ of polarized HS, induced by the canonical and the variation morphisms (§5.1.3.2), satisfy the conditions of ([Sa 88], Lemma 5.2.15) and (Remark 2.7), hence for all $m$ we have a decomposition

$$ H_{\varphi}^{m-1} = \mathrm{Im} C \oplus \mathrm{Ker} V \qquad (5.45) $$

Since the morphisms $C$ and $V$ are morphisms of MHS, they are strict. Hence we deduce the following decomposition

$$ {}^{p}\varphi_{z}^{u} {}^{p}\mathcal{H}^{i} \simeq \mathrm{Im} (C : {}^{p}\psi_{z}^{u} {}^{p}\mathcal{H}^{i} \to {}^{p}\varphi_{z}^{u} {}^{p}\mathcal{H}^{i}) \oplus \mathrm{Ker} (V : {}^{p}\varphi_{z}^{u} {}^{p}\mathcal{H}^{i} \to {}^{p}\psi_{z}^{u} {}^{p}\mathcal{H}^{i}(-1)) \quad (5.46) $$

where $^{p}\mathcal{H}^i := {}^{p}\mathcal{H}^i(K)$. Let $L^i := {}^{p}\psi_z^u {}^{p}\mathcal{H}^i(K) = {}^{p}\psi_z^u {}^{p}\mathcal{L}_{\mathbb{P}^*}^i = \psi_z^u \mathcal{L}_{\mathbb{P}^*}^i$, then $\mathrm{Im} C \simeq \mathrm{Im}(N : L^i \to L^i)$. Set: $L_0^i := \mathrm{Ker}(V : \mathbb{H}^i(X_0, {}^{p}\varphi_{z\circ f}^u j_1^* \mathbf{pL}) \to L^i(-1))$. We deduce:
$$ {}^{p}\varphi_{z}^{u} {}^{p}\mathcal{H}^{i}(K) = \mathbb{H}^{i}(X_{0}, {}^{p}\varphi_{z_{0}fj_{!}^{*}}^{u} \mathbf{P}) \simeq L_{0}^{i} \oplus (\mathrm{Im} \, N : L^{i} \to L^{i}) \quad (5.47) $$

The term $L^i$ with the monodromy filtration is a nilpotent orbit (Theorem 5.2), hence the term $\mathrm{Im} \, N : L^i \to L^i$ is endowed with the induced MHS as in the definition of the intermediate extension of $k'_{!}(\mathcal{L}^i_{P^*}[1])$ on $\mathbb{P}$.

Remark 5.7. The induced filtration $F$ on $\mathbb{H}^j(\mathbb{P}, k'_{!}(\mathcal{L}^i_{P^*}[1]))$ defines a HS ([Zu 79]), hence $\mathbb{H}^j(\mathbb{P}, K)$ is a direct sum of two HS.

**Corollary 5.6.** Let $K := f_{*}j_{!}^{*}\mathbf{P}\mathcal{L}$. We have the following splitting

$$ K \simeq \bigoplus_{i \in \mathbb{Z}} ({}^{p}\mathcal{H}^{i}(K)[-i] \in D_{c}^{b}(\mathbb{P}, \mathbb{Q})) $$

The iterated cup-product with the class $c_1$ of an hyperplane section of $X$ induces Lefschetz isomorphisms

$$ \eta^i : {}^{p}\mathcal{H}^{-i}(K) \simeq {}^{p}\mathcal{H}^{i}(K)(i) $$

on the components of the decomposition of ${}^{p}\mathcal{H}^{-i}(K)$ including $L_0^i$ by the bigraded polarized setting on ${}^{p}\psi_z^u$ and ${}^{p}\varphi_z^u$ (§5.2.1.1).

Then, the existence of a splitting of the complex $K$ follows from Deligne's general argument.

### 5.2.1.3 Perverse cohomology along a locally principal divisor

Let $j_{!}^{*}\mathbf{P}\mathcal{L}$ be an intermediate extension on the complex algebraic variety $X$. Let $Z$ be a locally principal divisor, $i: Z \to X$, $U = X \setminus Z$ and $j_Z: U \to X$. The complex $i_*i^*j_Z^*j_Z^*j_{!}^{*}\mathbf{P}\mathcal{L}$ occurs in two triangles:

$$ \begin{aligned} j_{Z!}j_Z^*j_{!}^{*}\mathbf{P}\mathcal{L} &\xrightarrow{\mathrm{can}} j_{Z*}j_Z^*j_{!}^{*}\mathbf{P}\mathcal{L} \to i_*i^*j_{Z*}j_Z^*j_{!}^{*}\mathbf{P}\mathcal{L} \xrightarrow{[1]} \\ i^!j_{!}^{*}\mathbf{P}\mathcal{L} &\xrightarrow{I} i^*j_{!}^{*}\mathbf{P}\mathcal{L} \to i^*j_{Z*}j_Z^*j_{!}^{*}\mathbf{P}\mathcal{L} \xrightarrow{[1]} \end{aligned} \quad (5.48) $$

The complexes $j_{Z!}j_Z^*j_{!}^{*}\mathbf{P}\mathcal{L}$ and $j_{Z*}j_Z^*j_{!}^{*}\mathbf{P}\mathcal{L}$ are perverse since $j_Z$ is affine (resp. Stein).

Let $Z$ be a principal divisor and set $K := (j_{!}^{*}\mathbf{P}\mathcal{L})|_U$. The perverse long exact sequences defined by the corresponding triangles, including (Equation 2.16) are reduced to

$$ \begin{aligned} 0 &\to {}^{p}\mathcal{H}^{-1}i^{*}j_{Z*}\mathcal{K} \to \psi_{f}^{u}\mathcal{K}[-1] \xrightarrow{N} \psi_{f}^{u}\mathcal{K}[-1] \to {}^{p}\mathcal{H}^{0}i^{*}j_{Z*}\mathcal{K} \to 0 \\ 0 &\to {}^{p}\mathcal{H}^{-1}i^{*}j_{Z*}\mathcal{K} \to \psi_{f}^{u}\mathcal{K}[-1] \xrightarrow{\mathrm{can}} \varphi_{f}^{u}\mathcal{K}[-1] \to {}^{p}\mathcal{H}^{0}i^{*}j_{Z*}\mathcal{K} \to 0 \quad (5.49) \\ 0 &\to i_{*}{}^{p}\mathcal{H}^{-1}i^{*}j_{Z*}\mathcal{K} \to j_{Z!}\mathcal{K} \xrightarrow{\mathrm{can}'} j_{Z*}\mathcal{K} \to i_{*}{}^{p}\mathcal{H}^{0}i^{*}j_{Z*}\mathcal{K} \to 0 \end{aligned} $$

as $i^*j_{Z*}\mathcal{K}$ has only two non trivial perverse cohomology sheaves ([BBD 83] Corollaire 4.1.10 ii). Recall that $j_{!}^{*}\mathbf{P}\mathcal{L} = j_{Z!}\mathcal{K} := \mathrm{Im}(\mathrm{can} : j_{Z!}\mathcal{K} \to j_{Z*}\mathcal{K})$.
If $Z$ is defined by an equation $f$ and $N := \mathrm{Log} T^u$, we have (([BBD 83], Corollaire 4.1.12) and [Br 82] Proposition 2.3.8, [Br 86]):

$$ 
\begin{align} 
i^* j_{1*} p\mathbf{L}[-1] &\simeq p\mathcal{H}^{-1} i^* j_{Z*} j_{Zj_*}^* p\mathbf{L} \simeq \mathrm{Ker} \mathrm{can} \simeq \mathrm{Ker} N, \notag \\ 
i^! j_{1*} p\mathbf{L}[1] &\simeq p\mathcal{H}^0 i^* j_{Z*} j_{Zj_*}^* p\mathbf{L} \simeq \mathrm{Coker} \mathrm{can} \simeq \mathrm{Coker} N \tag{5.50} \\ 
j_{Z*} j_{Zj_*}^* p\mathbf{L} / j_{1*} p\mathbf{L} &\simeq i_* \mathrm{Coker}(N : p\psi_f^u j_{1*} p\mathbf{L} \to p\psi_f^u j_{1*} p\mathbf{L}(-1)) \notag 
\end{align} 
$$

where $j_{1*} p\mathbf{L}$ is viewed as a sub-complex.

### 5.2.1.4 Hodge theory along a NCD

Let $Y$ be a NCD and $\mathbf{L}$ a local system on $X \setminus Y$ with intermediate extension $j_{1*} p\mathbf{L}$ on the complex algebraic variety $X$. Let $Z$ be a NCD sub-divisor of $Y$, $i: Z \to X$, $U = X \setminus Z$, $j: X \setminus Y \hookrightarrow X$ and $j_Z: U \to X$.

We construct below various logarithmic complexes along a NCD, based on the properties of perverse cohomology sheaves (ch. 3, §3.2.4). We extend the definition of the weight and Hodge filtrations $W$ and $F$ to the logarithmic complexes $i^! j_{1*} p\mathbf{L}$, $i^* j_{1*} p\mathbf{L}$ and $i^* j_{Z*} j_{Zj_*}^* p\mathbf{L}$ using the relations with $p\psi_f^u(j_{1*} p\mathbf{L})$.

Let $Z$ be a principal divisor defined by an equation $f$, the filtration $W(N)$ (resp. $\mathrm{Ker} N$, $\mathrm{Coker} N$) corresponds to the filtration $W(\nu)$ (resp. $\mathrm{Ker} \nu$, $\mathrm{Coker} \nu$) by the isomorphism $\psi_f^u(j_{1*} p\mathbf{L}) \simeq \Psi^*(I^p \mathcal{L})$.

The weight filtration $W$ is deduced from $W(N)$ as follows:

$$ 
\begin{align} 
\mathrm{Gr}_{l+ p_w-1}^W p\psi_f^u j_{1*} p\mathbf{L} &= (\mathrm{Gr}_{l+ p_w}^W \psi_f^u j_{1*} p\mathbf{L})[-1] = \notag \\ 
(\mathrm{Gr}_{l+w'}^W \psi_f^u j_{1*} \mathbf{L})[n-1] &= (\mathrm{Gr}_l^{W(N)} \psi_f^u j_{1*} \mathbf{L})[n-1] \tag{5.51} 
\end{align} 
$$

from which we deduce the weight filtration on $\mathrm{Coker} N | p\psi_f^u(j_{1*} p\mathbf{L})$ (resp. $\mathrm{Ker} N | p\psi_f^u(j_{1*} p\mathbf{L})$) as follows:

$$ W_{l+ p_w-1}(\mathrm{Coker} N) := W(N)_l(\mathrm{Coker} N) = \mathrm{Im}(\mathrm{Ker} N^{l+1} \to \mathrm{Coker} N) \text{ if } l \ge 0 $$

$$ 
\begin{align} 
W_{l+ p_w-1}(\mathrm{Ker} N) &:= W(N)_l(\mathrm{Ker} N) = \mathrm{Ker} N \cap \mathrm{Im} N^{-l} \text{ if } l \le 0, \notag \\ 
W(N)_l(\mathrm{Coker} N) &= 0 \text{ if } l < 0, \quad W(N)_l(\mathrm{Ker} N) = \mathrm{Ker} N \text{ if } l > 0 \tag{5.52} 
\end{align} 
$$

$$ N^l : \mathrm{Gr}_{l+ p_w-1}^W (\mathrm{Coker} N) \simeq \mathrm{Gr}_{-l+ p_w-1}^W (\mathrm{Ker} N)(-l) \text{ if } l \ge 0, $$

Weight filtration on $j_{Z*} j_{Zj_*}^* p\mathbf{L}$.

If $Z$ is a locally principal divisor, we can suppose $Z$ defined locally by an equation $f$. The monodromy weight filtration $W(N)$ on $p\psi_f^u(j_{1*} p\mathbf{L})$ induces a filtration on $\mathrm{Coker} N | p\psi_f^u(j_{1*} p\mathbf{L})$ transported on $i^! j_{1*} p\mathbf{L}[1]$ (resp. on $\mathrm{Ker} N | p\psi_f^u(j_{1*} p\mathbf{L})$ transported on $i^* j_{1*} p\mathbf{L}[-1]$) is independent of the local equation $f$ ([Sa 88], [Sa 90], [ELM 10]). If $Z$ is principal and we consider the filtrations $W$ and $F$ in (Equation 5.49), we have:
$$0 \to \mathcal{H}^{-1}i^*j_{Z*}\mathcal{K} \to \mathcal{H}^u_f\mathcal{K} \xrightarrow{N} \mathcal{H}^u_f\mathcal{K}(-1) \to \mathcal{H}^0i^*j_{Z*}\mathcal{K} \to 0 \quad (5.53)$$

since $N$ drops the weight by 2, hence we write

$$(\text{Coker}(N|_{\mathcal{H}^u_f} j_{Z*} j_{Z*}^* \mathcal{L}, W, F)(-1) \simeq (j_{Z*} j_{Z*}^* j_{Z*}^* \mathcal{L}) / (j_{Z*} j_{Z*}^* \mathcal{L}, W, F).)$$

As a consequence, the weight $W$ on $j_{Z*}j_{Z*}^*j_{Z*}^*\mathcal{L}$ is defined as follows.

Let $q: j_{Z*}j_{Z*}^*j_{Z*}^*\mathcal{L} \to j_{Z*}j_{Z*}^*j_{Z*}^*\mathcal{L}/j_{Z*}^*\mathcal{L} \simeq \text{Coker } N$. We deduce from (Equation 5.52), that there exists unique filtrations $W$ and $F$ on $j_{Z*}j_{Z*}^*j_{Z*}^*\mathcal{L}$, independent of the local equation $f$ of $Z$, satisfying (Equations 5.53, 5.52 and 5.2):

$$W_l(j_{Z*}j_{Z*}^*j_{Z*}^*\mathcal{L}) = 0 \text{ if } l < \mathcal{P}_w, \quad W_l(j_{Z*}j_{Z*}^*j_{Z*}^*\mathcal{L}) = j_{Z*}^*\mathcal{L} \text{ if } l = \mathcal{P}_w; \\ W_{l+\mathcal{P}_w+1}j_{Z*}j_{Z*}^*j_{Z*}^*\mathcal{L} = q^{-1}(W(N)_l \text{Coker } N|_{\mathcal{H}^u_f} j_{Z*}^*\mathcal{L}) \text{ for } l \ge 0; \quad (5.54) \\ \text{Gr}_l^W j_{Z*}j_{Z*}^*(j_{Z*}^*\mathcal{L}) = \text{Gr}_{l-2}^W \text{Coker}(N|_{\mathcal{H}^u_f} j_{Z*}^*\mathcal{L})(-1) \text{ if } l > \mathcal{P}_w.$$

Example: $\text{Gr}_{\mathcal{P}_w+1}^W j_{Z*}j_{Z*}^*(j_{Z*}^*\mathcal{L}) \simeq (\text{Gr}_0^{\mathcal{P}(N)}\psi_f^u j_{Z*}^*\mathcal{L})[n-1](-1)(\text{check for } n=1)$

**Proposition 5.7 (W and F).** Let $Z$ be a NCD sub-divisor of $Y$ in $X$, and $\mathcal{L}$ a polarized VHS of weight $\mathcal{P}_w$ on $X \setminus Y$.

1. There exist unique global filtrations $W$ and $F$ on $i^*j_{Z*}^*\mathcal{L}[-1]$ (resp. $i_{Z*}^*\mathcal{L}[1])$, locally compatible with the filtrations deduced from $W$ and $F$ in Equation 5.52 by the isomorphisms :

$$\begin{align}
(i_{Z*}^*j_{Z*}^*\mathcal{L}[1], W, F) &\simeq (\text{Coker } N|_{\mathcal{H}^u_f} j_{Z*}^*\mathcal{L}, W, F)(-1) \tag{5.55} \\
i_{Z*}^*j_{Z*}^*\mathcal{L}[-1], W, F) &\simeq (\text{Ker } N|_{\mathcal{H}^u_f} j_{Z*}^*\mathcal{L}, W, F) \nonumber
\end{align}$$

2. There exist unique global filtrations $W$ and $F$ on $j_{Z*}j_{Z*}^*j_{Z*}^*\mathcal{L}$ such that $W_{\mathcal{P}_w}j_{Z*}j_{Z*}^*j_{Z*}^*\mathcal{L} = j_{Z*}^*\mathcal{L}$ and locally for $l \ge 0$:

$$i^*\text{Gr}_{l+\mathcal{P}_w+1}^W j_{Z*}j_{Z*}^*(j_{Z*}^*\mathcal{L}) \simeq \text{Gr}_l^{W(N)}\text{Coker } N|_{\mathcal{H}^u_f} j_{Z*}^*\mathcal{L}$$

3. Dually, $i^*j_{Z*}^*\mathcal{L}[-1] \simeq \text{Ker } N|_{\mathcal{H}^u_f} j_{Z*}^*\mathcal{L}$ embeds into $j_{Z*}j_{Z*}^*j_{Z*}^*\mathcal{L}$ and the weight filtration $W$ on $j_{Z*}j_{Z*}^*j_{Z*}^*\mathcal{L}$ is deduced from $W(N)$ on $\text{Ker } N$:
$$\text{Gr}_{\mathcal{P}_w}^W j_{Z*}j_{Z*}^*j_{Z*}^*\mathcal{L} = j_{Z*}^*\mathcal{L} \text{ and locally for } l \le 0$$

$$i^*\text{Gr}_{l+\mathcal{P}_w-1}^W j_{Z*}j_{Z*}^*j_{Z*}^*\mathcal{L} \simeq \text{Gr}_l^{W(N)}\text{Ker } N|_{\mathcal{H}^u_f} j_{Z*}^*\mathcal{L}$$

4. If $X$ is projective, the filtrations $W$ and $F$ induced on the hypercohomology of $X$ by the filtrations on

$$i^*j_{Z*}^*\mathcal{L}[-1], \quad i_{Z*}^*\mathcal{L}[1], \quad j_{Z*}j_{Z*}^*j_{Z*}^*\mathcal{L}, \quad j_{Z*}j_{Z*}^*j_{Z*}^*\mathcal{L}$$

define respectively a structure of mixed Hodge complex.
*Proof of the proposition 5.7.* The definition of the filtrations W and F follows from (Equation 5.53).

The proof relies on the gluing of perverse sheaves as they behave like sheaves (Remark 2.5): morphisms of perverse sheaves on a covering $\mathcal{U}$ of X which coincide on the intersection of two open subsets in $\mathcal{U}$ glue into a global morphism on X ([BBD 83], Corollaires 2.1.21, 2.1.22 and 2.1.23).

Hence, the local definitions of the perverse weight filtration W define a global filtration on X. The Hodge filtrations have been defined globally on $\Omega^*\mathcal{L}$ and on $\Psi^*\mathcal{I}\mathcal{L}$ (Equation 5.28).

An elegant alternative proof is based on the deformation to the normal cone and the specialization functor introduced by Verdier [V 83], which reduces the local construction of the $\psi$ functor in the case of a divisor to the case of a principal divisor and gives a global filtration of W [ELM 10].

*Example 5.4.* 1) The polarization on the primitive spaces is defined by the bilinear product, composition of the duality with the isomorphisms for $l \ge 0$:

$$ \operatorname{Gr}_{l+ p_{w+1}}^{W} i^{\prime} j_{1*}^{p} \mathbf{L}[1] \xrightarrow{\cong} \operatorname{Gr}_{l}^{W(N)} \operatorname{Coker}(N|_{p} \psi_{f}^{u} j_{1*}^{p} \mathbf{L}) \\ \downarrow^{N^{l} \approx} \\ \operatorname{Gr}_{-l}^{W(N)} \operatorname{Ker}(N|_{p} \psi_{f}^{u} j_{1*}^{p} \mathbf{L}) \xrightarrow{=} \operatorname{Gr}_{-l+ p_{w-1}}^{W} i^{\prime} j_{1*}^{p} \mathbf{L}[-1] $$

2) Let $\mathbf{L}$ be a unipotent VHS of weight $w'$ on a punctured disc with center 0 in $\mathbb{C}$, then $(\psi_z \mathbf{L})_0 \simeq L$ is a nilpotent orbit of weight $w'$.

Moreover, $j_*\mathbf{L}/j_*\mathbf{L} \simeq L/NL[-1] \simeq L/NL[-1]$ is of weight $\ge w' - 1$ and $\operatorname{Gr}^W_{w'+1}j_*\mathbf{L} \simeq \operatorname{Prim}\operatorname{Gr}^W_{w'-1}(L)[-1]$.

3) Let $\mathbb{P}^* := \mathbb{P}_C^1 - \Sigma$ with $\Sigma$ finite and $0 \in \Sigma$, $p\mathbf{L}$ be defined on $\mathbb{P}^*$ and $j: \mathbb{P}^* \to \mathbb{P}_C^1$. The weight filtration $W$ on $j_*p\mathbf{L}$ is defined by $W_{p,w} := j_*p\mathbf{L}$. The quotient $j_*p\mathbf{L}/j_*p\mathbf{L}$ is isomorphic to $L/NL := \operatorname{Coker} N: L \to L$ in degree 0, where $L := \psi_z(\mathbf{L})$ is endowed with the limit MHS.

Hence for $i > 0$: $W_{p,w+i}j_*p\mathbf{L}$ corresponds to the filtration induced on $L/NL$ by $W_{w'+i-1}(L/NL)(-1)$ (Equation 5.2).

4) Let $i_0: 0 \to \mathbb{P}_C^1$. The weight filtration $W$ on $i_0^*j_*p\mathbf{L} \simeq (L \xrightarrow{N} L)[1]$ is defined by $W_{i+p_{w-1}} = \operatorname{Gr}_i^W(N)(\operatorname{Ker} N: L \to L)[1]$ (in degree $-1$) if $i \le 0$.

*Remark 5.8.* The following graded perverse sheaves decompose into direct sum of intermediate extensions of polarized perverse sheaves on smooth components of $Z$.

$$ \operatorname{Gr}_l^W i^* j_1^* p \mathbf{L} [-1] \simeq \operatorname{Prim} \operatorname{Gr}_{l-p_{w+1}}^{W(\nu)} \Psi^* I^p \mathcal{L} [-1] \quad \text{if } l < p_w $$

$$ \operatorname{Gr}_l^W j_{Z^*} j_{Z^*} j_{1*}^* p \mathbf{L} \simeq i_* \operatorname{Prim} \operatorname{Gr}_{l-p_{w+1}}^{W(\nu)} \Psi^* I^p \mathcal{L} [-1] \quad \text{if } l < p_w $$

$$ \operatorname{Gr}_l^W i^\prime j_1^* p \mathbf{L} [+1] \simeq \operatorname{Prim} \operatorname{Gr}_{l-p_{w-1}}^{W(\nu)} \Psi^* I^p \mathcal{L} [-1] (-1) \quad \text{if } l > p_w $$

$$ \operatorname{Gr}_l^W j_{Z^*} j_{Z^*} j_{1*}^* p \mathbf{L} \simeq i_* \operatorname{Prim} \operatorname{Gr}_{l-p_{w-1}}^{W(\nu)} \Psi^* I^p \mathcal{L} [-1] (-1) \quad \text{if } l > p_w $$
where by convention: $W(\nu)_{i-1}(\Psi^*I^p\mathcal{L}[-1]) := (W(\nu)_{i}\Psi^*I^p\mathcal{L})[-1]$,
$\mathrm{Gr}_{l-pw+1}^{W(\nu)}$ Ker $\nu|\Psi^*I^p\mathcal{L}[-1] \simeq \mathrm{Prim}\mathrm{Gr}_{l-pw+1}^{W(\nu)}\Psi^*I^p\mathcal{L}[-1]$ if $l < pw$ and
$\mathrm{Gr}_{l-pw+1}^{W(\nu)}$ Coker $\nu|\Psi^*I^p\mathcal{L}[-1] \simeq \mathrm{Prim}\mathrm{Gr}_{l-pw+1}^{W(\nu)}\Psi^*I^p\mathcal{L}[-1]$ if $l > pw$.

**Corollary 5.7.** Let $Z \subset be \textit{NCD in a projective variety } X \text{ and } I^pL \text{ a shifted polarized VHS of weight } p\mathfrak{w} \text{ on } X \setminus Y$:

1. The induced MHS on $\mathbb{H}^q(X \setminus Z, j_!^* I^p \mathbf{L})$ has weights $w \ge p\mathfrak{w}+q$. Moreover, the natural map $\mathbb{H}^i(X, j_!^* I^p \mathbf{L}) \to W_{p\mathfrak{w}+i} \mathbb{H}^i(X \setminus Z, j_!^* I^p \mathbf{L})$ is surjective.

2. The induced MHS on $\mathbb{H}^q(Z, j_!^* I^p \mathbf{L})$ has weights $w \le p\mathfrak{w} + q$.

3. Dually: $w \le p\mathfrak{w} + q$ on $\mathbb{H}_c^q(X \setminus Z, j_!^* I^p \mathbf{L})$ and $w \ge p\mathfrak{w} + q$ on $\mathbb{H}_W^q(X, j_!^* I^p \mathbf{L})$.

The corollary follows from the purity theorem 4.3. There exists a complex endowed with filtrations $(j_Z^*j_Z^*j_!^*I^p\mathbf{L}, W, F)$, isomorphic to $(IC^*I^p\mathcal{L}(\mathrm{Log}Z), F)$ in the derived filtered category $D^+F(X, \mathbb{C})$, such that: $W_{p\mathfrak{w}}j_Z^*j_Z^*j_!^*I^p\mathbf{L} = j_!^*I^p\mathbf{L}$ and locally, if $l > p\mathfrak{w}$

$$ \mathrm{Gr}_l^W j_Z^* j_Z^*(j_!^* I^p \mathbf{L}) = i_* \mathrm{Gr}_l^W (i_!^* j_!^* I^p \mathbf{L}[1]) = i_* \mathrm{Gr}_{l-pw-1}^{W(\nu)} \mathrm{Coker} \nu|_{\Psi^* I^p \mathcal{L}[-1]} (-1). $$

The complex $(R\Gamma(X, j_Z^*j_Z^*j_!^*I^p\mathbf{L}, W, F)$ is a MHC and the weight spectral sequence

$$ E_1^{p,q} := \mathbb{H}^{p+q}(X, \mathrm{Gr}_{-p}^W j_Z^* j_Z^* j_!^* I^p \mathbf{L}) \Rightarrow \mathrm{Gr}_q^W \mathbb{H}^{p+q}(X \setminus Z, j_!^* I^p \mathbf{L}) $$

degenerates at rank 2. Since $W_{-p}j_Z^*j_Z^*j_!^*I^p\mathbf{L} = 0$ if $-p < p\mathfrak{w}$:

$$ \bullet E_1^{p,q} = \mathbb{H}^{p+q}(X, \mathrm{Gr}_{-p}^W j_Z^* j_Z^* j_!^* I^p \mathbf{L}) = 0 \text{ if } -p < p\mathfrak{w}; $$

$$ \bullet E_1^{-p\mathfrak{w},q} = \mathbb{H}^{q-\mathfrak{w}q}(X, \mathrm{Gr}_{p\mathfrak{w}}^W j_Z^* j_Z^* j_!^* I^p \mathbf{L}); $$

$$ \bullet E_2^{p,q} = \mathrm{Gr}_{-p}^W \mathbb{H}^{p+q}(X \setminus Z, j_!^* I^p \mathbf{L}) = 0 \text{ if } q < p+q + p\mathfrak{w}. $$

The term $E_2^{p,q}$ is a HS of weight $q$.

Since $E_1^{1-p\mathfrak{w},q} = \mathbb{H}^{q+1-\mathfrak{w}q}(X, \mathrm{Gr}_{p\mathfrak{w}-1}^W j_Z^* j_Z^* j_!^* I^p \mathbf{L}) = 0$, the projection

$$ E_1^{-p\mathfrak{w},q} = \mathbb{H}^{q-\mathfrak{w}q}(X, \mathrm{Gr}_{p\mathfrak{w}}^W j_Z^* j_Z^* j_!^* I^p \mathbf{L}) \to E_2^{-p\mathfrak{w},q} = \mathrm{Gr}_{p\mathfrak{w}}^W \mathbb{H}^{q-\mathfrak{w}q}(X \setminus Z, j_!^* I^p \mathbf{L}) $$

is surjective, and $E_2^{-p\mathfrak{w},q} := \mathrm{Gr}_{p\mathfrak{w}}^W \mathbb{H}^{q-p\mathfrak{w}}(X \setminus Z, j_!^* I^p \mathbf{L})$ is a HS of weight $q$ (the weight on the hypercohomology in degree $q-p\mathfrak{w}$ is obtained after a shift on the index of $W$ by the degree $q-p\mathfrak{w}$).

The statements in ii) and iii) follow by duality or standard exact sequences.

**Remark 5.9 (Relation between the weight filtration on $\Psi^*L$ (Proposition 5.1) and the weight filtration on $\Omega^*L$).** i) The isomorphism:

$$ i_* i^! j_!^* I^p \mathbf{L}[1] \simeq \mathrm{Coker} \mathrm{can}/j_*^* I^p \mathbf{L} \simeq \mathrm{Coker} N/\Psi^* I^p \mathbf{L}[-1] \text{ (Equation 5.50)} $$

is equivalent to a de Rham family of canonical isomorphisms for each $J \subset M$:

$$ \mathrm{Coker} (N : \Psi_J L \to \Psi_J L) \simeq Q'_J := \mathrm{Coker} (\mathrm{can} : N_J L \to L). $$

The induced filtration $W'$ on $Q'_J$ by $W(N)$ satisfy:

$$ \mathrm{Gr}_k^{W(N)}(\Psi_J L/NL) \simeq \mathrm{Prim}_k(\mathrm{Gr}_k^{W(N)}\Psi_J L) \simeq \mathrm{Gr}_k^{W'} Q'_J $$
with induced MHS by $M$ defined by $M(N_0)$ on $L_J := L$ shifted by $1 - |J|$ (§5.1.1, Equation 5.5).

ii) The filtration $W(N)$ on $\Psi^*L$ corresponds to a filtration $W'$ on $\Omega^*L$ (see §9.2, Equation 9.10). Then, the local decomposition in terms of $P_k^J L$ (Equations 5.22 and 9.21) corresponds to a local decomposition of $\Omega^*L$ in terms of the spaces $C_k^J L := \text{Coker}(N/P_k^J L)$ (Equation 9.23).

The direct definition of $W'$ in chapter 9, is based on a new filtration $N * W$ on $L$ (Remark 2.21) whose construction from $(L, W)$ by Kashiwara is motivated by the results in (Lemma ??). The filtration on $\Omega^*L$ is studied directly in (ch. 9, Equation 9.6, 9.23, see also (Remark 5.4)).

When $L = \mathcal{L}(x)$ for $x \in Z_J^*$, since $N_0 := \sum_{i \in J} N_i$ vanishes on $C_k^J L$, we deduce, that $C_k^J L = \text{Gr}_0^{W(N_0)} C_k^J L$ is a pure polarized direct summand of weight $k + \delta$ of $\text{Gr}_0^{W(N_0)} \text{Gr}_k^{W'} Q_J'$ with respect to the MHS defined by $W(N_0)$ on the primitive part of $\text{Gr}_k^{W(N_0)} \Psi_J L$ shifted by $k + \delta$.

### 5.2.2 Hodge theory on $i^*j_Z j_Z^* j_{1*} p\mathbf{L}$

Let $W'$ denote the weight filtration on $i^*j_{1*} p\mathbf{L}[-1]$ (resp. $i^!j_{1*} p\mathbf{L}[1]$). We deduce the weight $W$ defined by

$$ (i^*j_{1*} p\mathbf{L}, W) := ((i^*j_{1*} p\mathbf{L}[-1])[1], W'[1]) $$

$$ (i^!j_{1*} p\mathbf{L}, W) := ((i^!j_{1*} p\mathbf{L}[1])[-1], W'[-1]) $$

such that $(i^*j_{1*} p\mathbf{L}, W)$ and $(i^!j_{1*} p\mathbf{L}, W)$ keep the structure of MHC.

Moreover: $W_{p_w-k} i^!j_{1*} p\mathbf{L} = 0$ if $k \ge 1$ and $W_{p_w+k} i^*j_{1*} p\mathbf{L} = i^*j_{1*} p\mathbf{L}$ if $k \ge 0$ and we have filtered morphism:

$$ \text{can} : j_Z j_Z^* j_{1*} p\mathbf{L} \to j_Z j_Z^* j_{1*} p\mathbf{L}, \quad I : i^!j_{1*} p\mathbf{L} \to i^*j_{1*} p\mathbf{L}. \qquad (5.56) $$

where $I : (i^!j_{1*} p\mathbf{L}, W) \to (j_{1*} p\mathbf{L}, W) \to (i^*j_{1*} p\mathbf{L}, W)$ is the composition morphism and $j_{1*} p\mathbf{L}$ is pure of weight $p_w$.

We deduce from the triangles in (Equation 5.48) that $i^*j_{Z*}j_Z^* j_{1*} p\mathbf{L}$ is quasi-isomorphic to the cone over each of the morphisms can and $I$.

**Definition 5.6.** i) The structure of MHC on $K' := i^*j_{Z*}j_Z^* j_{1*} p\mathbf{L}$ is defined as a mixed cone $C_M$ over $I$. $K'$ is not perverse and:

$$ \text{Gr}_r^W C_M(I) = (\text{Gr}_{r-1}^W i^!j_{1*} p\mathbf{L})[1] \oplus \text{Gr}_r^W i^*j_{1*} p\mathbf{L}. $$

**Remark 5.10.** If $Z$ is defined by an equation $f$, we may define $i^*j_{Z*}j_Z^* p\mathbf{L}$ as a mixed cone $C_M$ over $N : p\psi_f^u j_{1*} p\mathbf{L} \to p\psi_f^u j_{1*} p\mathbf{L}$.

**Lemma 5.3.** i) We have a graded long exact sequence:

$$ \text{Gr}_k^W \mathbb{H}^{j+1}(X, i^!j_{1*} p\mathbf{L}) \to \text{Gr}_k^W \mathbb{H}^j(X, i^*j_{1*} p\mathbf{L}) \to \text{Gr}_k^W \mathbb{H}^j(X, i^*j_{Z*}j_Z^* j_{1*} p\mathbf{L}) \xrightarrow{+1} \text{Gr}_k^W \mathbb{H}^{j+1}(X, i^!j_{1*} p\mathbf{L}). $$

ii) We suppose $i^*j_{1*} p\mathbf{L} \subset K'$, then $K'/i^*j_{1*} p\mathbf{L} \cong i^!j_{1*} p\mathbf{L}[1]$.
Let $q: K' \to i'_j_* p_L[1]$. The weight filtration on $K'$ is defined as follows:

$$W_l K' := q^{-1}(W_l i'_j_* p_L[1]) \text{ if } l > p_w \text{ and } \mathrm{Gr}_l^W K' = \mathrm{Gr}_l^W i'_j_* p_L[1]$$

$$W_l K' = W_l i_* j_* p_L = (W_{l-1} i_* j_* p_L[-1])[1] \text{ if } l \le p_w (W_{p_w} = i_* j_* p_L).$$

$$\mathrm{Gr}_l^W K' = \mathrm{Gr}_l^W i_* j_* p_L = (\mathrm{Gr}_l^W i_* j_* p_L[-1])[1].$$

The weight spectral sequence degenerates at rank 2 as follows:

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

$$E_1^{p,q} = \mathbb{H}^{p+q}(X, \mathrm{Gr}_{-p-1}^W i'_j p_L[-1])[1] \text{ if } -p \le p_w.$$

For $p = -p_w - 1$ and $q = p_w$:

$$E_1^{-p_w-1, p_w} = \mathbb{H}^{-1}(X, \mathrm{Gr}_{p_w+1}^W i'_j p_L[1]) \implies \mathrm{Gr}_{p_w}^W \mathbb{H}^{-1}(X, i'_j Z'_* j'_* p_L)$$

$$E_1^{-p_w, p_w} = \mathbb{H}^0(X, \mathrm{Gr}_{p_w}^W i'_j p_L) \implies \mathrm{Gr}_{p_w}^W \mathbb{H}^1(X, i'_j Z'_* j'_* p_L[-1])$$

The terms are dual, since in both cases the cohomology is with value in $\mathrm{Gr}_0^{W(N)}$Coker $N = \mathrm{Gr}_0^{W(N)}$Ker $N$ (see Equation 5.55).

**Remark 5.11.** The spectral sequence of the mixed cone $K' := C_M(\mathrm{can})$ with a structure of MHC defined by

$$W_l K' = W_l i_* j_* p_L = (W_{l-1} i_* j_* p_L[-1])[1] \oplus W_l j_* Z'_* j'_* p_L$$

coincide at rank 2 with the above spectral sequence (see §9.5.1 and the octahedron diagrams (§7.2.0.2 and Equation 10.42)).

### 5.2.2.1 Thom isomorphism and Gysin morphism

With the previous notations ($Z \subset Y \subset X$ and $p_L$ defined on $X \setminus Y$), let $i_W: W \to X$ be a closed embedding of codimension $d$ transverse to $Y$ and the strata of $S$ on $X$, then we have Thom isomorphism

$$i_W^1 j_* p_L[2d] \simeq i_W^* j_* p_L$$

(see [GMacP 88] on stratified varieties and [CaMi 5], Lemma 3.5.4).

Let $W^* := W \setminus (Z \cap W)$, then:

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

In particular *Gysin morphism* is compatible with MHS.

Let $W = H$ of codimension 1, the *Thom isomorphism*:

$$\mathbb{H}^i(H \setminus (Z \cap H), j_* p_L) \xrightarrow{\sim} \mathbb{H}_{H \setminus (Z \cap H)}^{i+2}(X \setminus Z, j_* p_L)(1) \quad (5.57)$$

is given by the residue map:

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

On the one hand we have an isomorphism of the quotient complex with the cohomology with support:
$$ IC^*\mathcal{L}(\log Z \cup H)/IC^*\mathcal{L}(\log Z)) \xrightarrow{\sim} i_H^! IC^*\mathcal{L}(\log Z)[1] $$

defined as a connecting isomorphism. On the other hand, the residue map

$$ R_H : IC^*\mathcal{L}(\log Z \cup H)[1] \to i_{H*} IC^*i_H^*\mathcal{L}(\log Z \cap H) $$

vanishes on $IC^*\mathcal{L}(\log Z)$ and induces an isomorphism on the quotient.
<figure><img src="image_9.png" /></figure>
Chapter 6

Proof of the Decomposition Theorem

The proof of the decomposition theorem (ch.1, Theorem 1.2) and the purity theorem (ch.4, 4.3) is by induction on the dimension of X. We can suppose $f$ surjective, so $\dim V \le \dim X$. We introduce the following assertion:

*D(n): The decomposition theorem applies to projective morphisms*
$f : X \to V$ on irreducible varieties $X$ of dimension $\le n$.

In particular, in the case $\dim V = 0$, the purity theorem holds for projective varieties $X$ of dimension $\le n$ by the same induction.

The assertion $D(0)$ is obvious. Let $X$ be non singular of dimension $n$. The proof is by reduction to a general hyperplane section of $V$ through an isolated stratum $v$ with a NCD as inverse image $f^{-1}(v) := X_v \subset X$. In particular $D(n-1)$ apply by induction on the components of $X_v$ of dimension $n-1$.

Using a Lefschetz pencil of hyperplanes on $V$, the proof is reduced to the case where there exists a morphism $g$ on $V$ to the projective line such that $g(v) = 0$. Let $j_*{}^P L$ be on $X$, the decomposition of $f_*j_*{}^P L$ is reduced by Verdier's criteria, to the decomposition of $\phi_g{}^{PH^i}(f_*j_*{}^P L)$ locally at $v$.

In the case where $g^{-1}g(v)$ is a general hyperplane through $v$, the weight spectral sequence of $\psi_g{}^{PH^i}(f_*j_*{}^P L)$ (Equation 6.12 below) splits into a direct sum of a spectral sequence of MHS concentrated on $v$ degenerating at rank 2 and a spectral sequence degenerating at rank one (Corollary 6.3).

As a consequence of the splitting of the spectral sequence, we only need the category of IMHS (Definition 2.14) to prove the decomposition.

The decomposition theorem follows from the decomposition of ${}^p\phi_g{}^{PH^i}(K)$ (§6.2.0.4) and Verdier's criteria of extension of a perverse sheaf defined on $V \setminus V(g)$ (§2.2.3, Equation 2.18 and §2.2.3.1). Verdier's extension apply in general along a locally principal divisor [V 85-1] but we need only the principal divisor case.

If we consider all possible local projections $g$, the spectral sequence degenerates at rank 2. Hence, the existence of the category of Hodge $D$-modules is needed to prove the degeneration at rank 2 (see [Sa 88, Sa 90]), while for a general projection the proof is reduced to the category of IMHS.
The construction of the category of Hodge perverse sheaves is carried, after the proof of the decomposition theorem in the next chapter, which simplify considerably the proof and the exposition.

In chapters 10 and 11, we give another proof of the decomposition, combined with a direct proof of local purity, based on the weight on the logarithmic complex.

## 6.1 Proof of $D(n-1) \Rightarrow D(n)$ for non-singular X

The proof is divided in two cases according to the dimension of $V$.

### 6.1.0.1 The case $0 < \dim V \le n$

Let $f: X \to V$ be defined on a non singular complex variety $X$ of dimension $n$, $S$ a Thom-Whitney stratification of $f$ adapted to $j_*!^p \mathbf{L}$, $V_0$ the union of zero-dimensional strata of $S$ on $V$, $i_{V_0}: V_0 \to V$, $V^* := V \setminus V_0$.

Assuming $D(n-1, V)$ and $\dim V > 1$, we introduce an intermediate statement $D(n, V^*)$, to deduce the decomposition theorem over $V^*$ by considering hyperplane sections of $V$ transversal to the strata $S \in S$ (avoiding $V_0$).

**Lemma 6.1 (D(n, V*)).** i) Let $K := f_*j_*!^p \mathbf{L}$. The restriction of the $i$-perverse cohomology of $K$ to $V^*$ decomposes into a direct sum of intermediate extensions of shifted polarized VHS $!^p \mathbf{L}_S^i$ of weight $pw + i$ over all strata $S$ of dimension $> 0$.

Let $k: V^* \to V$, with the notations of (ch. 1, Equations 1.7 and 1.8):

$$ {}^p\mathcal{H}^i(k^*K) \xrightarrow{\sim} \bigoplus_{S \in \mathcal{S}, S \subset V^*} k^*i_{S!*}!^p \mathbf{L}_S^i \quad (6.1) $$

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

$$ \eta^i: {}^p\mathcal{H}^{-i}(f_*j_*!^p \mathbf{L})|_{V^*} \xrightarrow{\sim} {}^p\mathcal{H}^i(f_*j_*!^p \mathbf{L})|_{V^*} \quad (6.2) $$

*Proof.* We remark that on a stratum $S$ of dimension $s > 0$, $\mathbf{L}_S^i$ is uniquely defined in the decomposition as $\mathbf{L}_S^i = i_S^* H^{-s}( {}^p\mathcal{H}^i(K))$.

The decomposition (6.1) is checked by induction on a general normal section $N$ through a point $s \in S$, of dimension $< \dim V$ transversal to all strata, since perverse cohomology commutes with shifted restrictions to transversal sections.

Remark that $R^{i-s}f_{S^*}(i_{X_S}^!j_*!^p\mathbf{L})$ (resp. $R^{i-s}f_{S^*}(i_{X_S}^*j_*!^p\mathbf{L})$) is a variation of MHS of weights $w \ge {}^pw + i - s$ (resp. $w \le {}^pw + i - s$) and $\mathbf{L}_S^i$ is a local system
on S, hence the image of $I_S$ is a polarized VHS of weight $w = {}^p w + i - s$ (see also ch. 9, Lemma 9.7).

The isomorphisms in (6.2) follow by induction since the inverse image $X_N$ of $N$, is non singular of $\dim X_N < \dim X$ as $N$ is general.

**Example 6.1.** Over the big stratum $U$ in $V$, the restriction of $f$ to $X_U := f^{-1}(U)$ is smooth and the restrictions of the cohomology sheaves $\mathcal{H}^i(K)$ are locally constant on $U$, hence ${}^p \mathcal{H}^i(K)|_U = \mathcal{H}^{i-\dim V}(K)|_U[\dim V]$ is a shifted polarized VHS on $U$.

By the induction hypothesis $D(n-1)$, the fiber $(R^i f_* j_* \mathbf{L})_v \simeq \mathbb{H}^i(X_v, j_* \mathbf{L})$, at a point $v \in U$, is isomorphic to the intersection cohomology of the restriction of $\mathbf{L}$ to the fiber $X_v$ and Hard Lefschetz applies on $X_v$.

### 6.1.1 The crucial case at isolated strata

The main problem is to extend the decomposition over isolated strata in $V_0$.

**Proposition 6.1.** Assume $D(n, V^*)$ and the fiber $X_v$ at $v \in V_0$ is a NCD.

(i) The decomposition (6.1) of ${}^p \mathcal{H}^i(k^*K)$ over $V^*$ extends to a decomposition over $V$, including the terms $\mathbf{L}_v^i$ (ch. 1, Equation 1.7)

$$ {}^p \mathcal{H}^i(K) \simeq \bigoplus_{v \in V_0} \mathbf{L}_v^i \oplus k_1 k^* {}^p \mathcal{H}^i(K) \quad (6.3) $$

(ii) The Lefschetz morphisms $\eta^i$ are isomorphisms:

$$ \eta^i : {}^p \mathcal{H}^{-i}(K) \xrightarrow{\sim} {}^p \mathcal{H}^i(K) \quad (6.4) $$

The statement ii) implies the existence of a splitting of $K$ in the derived category $D_c^b(V, \mathbb{Q})$ by Deligne's general argument ([De 68] Théorème 1.5, [De 93]).

We recall that the splitting of $K$ means essentially the degeneration of the spectral sequence with respect to the perverse filtration at the $E_1$-term. Since the proof is local at $v$, we suppose $V_0$ reduced to a single stratum $v$. Moreover, we suppose first the fiber $X_v$ a NCD in $X$.

The proof will occupy the rest of the chapter.

## 6.2 Nearby and vanishing cycles at an isolated stratum

The decomposition at isolated stratum $v \in V_0 \subset V$, is based on the properties of perverse nearby and vanishing cycles along a general hyperplane section $H$ through $v$.

We suppose first, there exists a morphism $g : V \to \mathbb{P}$ to the complex projective line $\mathbb{P} := \mathbb{P}_{\mathbb{C}}^1$ such that $g(v) = 0$ and $g^{-1}(0) := H$ is transverse to all strata $S \subset V^* = V \setminus v$ as in the diagram
$$
\begin{equation} \tag{6.5}
\begin{aligned}
& X_v \hookrightarrow Z \xrightarrow{i'} X \hookleftarrow X \setminus Z \\
& \downarrow \quad \downarrow \quad \downarrow f \quad \downarrow \\
& v \hookrightarrow H \xrightarrow{i} V \xrightarrow{k'} V \setminus H \\
& \downarrow \quad \downarrow \quad \downarrow g \quad \downarrow \\
& 0 \to 0 \xrightarrow{i''} \mathbb{P} \to \mathbb{P} \setminus 0
\end{aligned}
\end{equation}
$$

where $Z := (g \circ f)^{-1}(0) = f^{-1}(H) = X_v \cup Z_s$ is a NCD in $X$, $X_v := f^{-1}(v)$,
and $Z_s$ is the strict transform of $H$ defined as the closure of $f^{-1}(H \setminus v)$.
For a general section $H$, $Z_s$ is smooth and $X_v \cap Z_s$ is a NCD in $Z_s$.

Let $I_Z$ (resp. $I_v \subset I_Z$) denote the set of indices of the components $Z_j \subset Z$
 resp. $Z_j \subset X_v$) and let $s \in I_Z$ corresponds to the component $Z_s$, hence
$I_Z = I_v \cup s$. For each $J \subset I_Z$, let $Z_J = \cap_{j \in J} Z_j$, $Z_J^*$ the regular locus of $Z_J$
and set uniformly $j : Z_J^* \hookrightarrow Z_J$.

We suppose the stratification $S$ compatible with $f, H, Z$ and the singularities
$Y$ of $j_* p_L$. Let $i_S : S \to \bar{S}$ denote the embedding of any stratum $S$ in its
closure $\bar{S}$.

6.2.0.1 Decomposition hypothesis

Let (K', W, F) be a perverse VMHS with support on Z (ch. 5, Definition 5.4). By definition

$$
(\mathrm{Gr}_r^W K', F) \simeq \oplus_{J \in I_Z} (j_*{}^P \mathbf{L}_r^J, F) \quad (6.6)
$$

is a direct sum of intermediate extension of VHS $p\mathbf{L}_r^J$ of weight $r$ on $Z_j^*$.

The weight spectral sequence: $E_{W,r}^{p,q} \Rightarrow \mathbb{H}^{p+q}(X, K', F)$ defined by $W$ on $R\Gamma(X, K')$ degenerates at rank 2, since the terms $E_{W,1}^{p,q} := \mathbb{H}^{p+q}(X, \mathrm{Gr}_{-p}^W K', F)$ with their filtrations induced by $F$ are HS of weight $q$ by the assumption $D(n-1)$ applied to intersections of components of $Z$,

We still denote $(f_{|Z})_*$ by $f_*$.
The derived direct image $(f_*K', W, F)$, where
$W$ and $F$ are derived direct image of $W$ and $F$ on $K'$, has support in $H$.
By definition, $(K', W, F)$ satisfy the *decomposition hypothesis* if:

for each component $j_* p \mathbf{L}_r^J$ of $(\mathrm{Gr}_r^W K', F)$ in equation 6.6, the following non-canonical decomposition property is satisfied:

$$
f_* j_*{}^p \mathbf{L}_r^J \simeq \bigoplus_{i \in \mathbb{Z}, S \in \mathcal{S}, S \subset H} i_{S!*} ({}^p \mathbf{L}_r^J)_S^i [-i] \quad (6.7)
$$

where $(p\mathbf{L}_r^J)_S^i$ is the component of the decomposition of $f_*j_*p\mathbf{L}_r^J$ according
to the notations of (ch 1, Equation 1.7).
### 6.2.0.2 The cases $^p\psi_{g\circ fj_*}^p\mathbf{L}$ and $^p\varphi_{g\circ fj_*}^p\mathbf{L}$

The complex $K' := {}^p\psi_{g\circ fj_*}^p\mathbf{L}$ (resp. $K' := {}^p\varphi_{g\circ fj_*}^p\mathbf{L}$) is a perverse VMHS by (ch. 5, §5.1.3.3, Proposition 5.5, and Equation 5.42) and we have

$$ (\mathrm{Gr}_a^W K', F) \simeq \bigoplus_{Z_J \subset X_v} (({}^p\mathcal{P}_a^J, F) \oplus_{Z_J \cap Z_s} ({}^p\mathcal{P}_a^{J\cup s}, F)) \oplus_{Z_s} ({}^p\mathcal{P}_a^s, F) \quad (6.8) $$

where ${}^p\mathcal{P}_a^J$ (resp. ${}^p\mathcal{P}_a^{J\cup s}$ and ${}^p\mathcal{P}_a^s$) are intermediate extensions of VHS on $Z_J$ for $J \subset I_v$ (resp. $Z_J \cup s$, $Z_s$) of weight $a$. Respectively, we have similar decomposition for $K' := {}^p\varphi_{g\circ fj_*}^p\mathbf{L}$ (Theorem 5.1).

Since $Z_s$ is transverse to $X_v$ and to the NCD Y singular set of $^p\mathbf{L}$, the component ${}^p\mathcal{P}_a^s$ vanishes for $a \neq {}^p w - 1$:

$$ ({}^p\mathcal{P}_a^s, F) = ((j_*{}^p\mathbf{L})|_{Z_s}[-1], F) \text{ if } a = {}^p w - 1 \text{ and } {}^p\mathcal{P}_a^s = 0 \text{ if } a \neq {}^p w - 1 \quad (6.9) $$

Moreover, since $\dim H < \dim V$, $K'$ satisfy the decomposition hypothesis with respect to the restriction of $f$ to $Z_J$ for all non empty $J \subset I_Z$.

Since $f_* {}^p\psi_{g\circ fj_*}^p\mathbf{L} \simeq \psi_g f_* {}^p\varphi_{g\circ fj_*}^p\mathbf{L}$ and $f_* {}^p\varphi_{g\circ fj_*}^p\mathbf{L} \simeq \varphi_g f_* {}^p\varphi_{g\circ fj_*}^p\mathbf{L}$ and since $p\psi_g$ (resp. $p\varphi_g$) are exact functors, we define the filtrations on $p\psi_g K$ (resp. $p\varphi_g K$), where $K = f_* {}^p\varphi_{g\circ fj_*}^p\mathbf{L}$, as follows:

**Definition 6.1.** The filtrations $W$ and $F$ on $p\psi_g K$ (resp. $p\varphi_g K$) are induced by $W$ and $F$ on $f_* K'$:

$$ K' := p\psi_{g\circ fj_*}^p\mathbf{L}, \quad p\psi_g K = f_*K', \quad W_r p\psi_g K := f_* W_r K', \quad F^r p\psi_g K := f_* F^r K' $$

$$ K' := p\varphi_{g\circ fj_*}^p\mathbf{L}, \quad p\varphi_g K = f_*K', \quad W_r p\varphi_g K := f_* W_r K', \quad F^r p\varphi_g K := f_* F^r K' $$

The filtrations $W$ and $F$ on $p\psi_g p\mathcal{H}^i(K)$ (resp. $p\varphi_g p\mathcal{H}^i(K)$) are induced by $W$ and $F$ on $f_* K'$:

$$ p\psi_g p\mathcal{H}^i(K) \simeq p\mathcal{H}^i(f_* K'), \quad \text{resp.} \quad p\varphi_g p\mathcal{H}^i(K) \simeq p\mathcal{H}^i(f_* K') $$

In the case of the vanishing cycles $K' := {}^p\varphi_{g\circ fj_*}^p\mathbf{L}$, all the terms of the decomposition are supported by $X_v$ since $H$ is general through $v$, hence $K'$ is a perverse VMHS supported by $X_v$ and $p\varphi_g K = f_* K'$ is supported by $v$.

**Proposition 6.2.** Let $K := f_* {}^p\varphi_{g\circ fj_*}^p\mathbf{L}$. The isomorphism

$$ {}^p\mathcal{H}^i(p\psi_g K) \simeq {}^p\mathcal{H}^i(f_* {}^p\psi_{g\circ fj_*}^p\mathbf{L}) \quad (\text{resp. } {}^p\mathcal{H}^i(f_* K') \simeq {}^p\varphi_g {}^p\mathcal{H}^i(K)) $$

transports the action of $\nu$ on $K'$ to the action of the monodromy logarithm $N$ on the right hand term. Let $\eta$ denote the cup-product by the class of the hyperplane section of $X$, we have:

$$ \eta^j : \mathrm{Gr}_a^W {}^p\mathcal{H}^{-j}(p\psi_g K) \xrightarrow{\sim} \mathrm{Gr}_a^W {}^p\mathcal{H}^j(p\psi_g K)(j) \quad (\text{resp. in the case of } {}^p\varphi_g K) $$

$$ N^i : \mathrm{Gr}_{p w-1+j+i}^W {}^p\mathcal{H}^j(p\psi_g K) \xrightarrow{\sim} \mathrm{Gr}_{p w-1+j-i}^W {}^p\mathcal{H}^j(p\psi_g K)(-i) $$

$$ \text{resp. } N^i : \mathrm{Gr}_{p w+j+i}^W {}^p\mathcal{H}^j(p\varphi_g K) \xrightarrow{\sim} \mathrm{Gr}_{p w+j-i}^W {}^p\mathcal{H}^j(p\varphi_g K)(-i) $$

We prove first the lemma:
**Lemma 6.2.** i) Let $K' := {}^p\psi_{g_0fj_1*}{}^p\mathbf{L}$ and $w_0 := {}^pw - 1$. If $a \neq w_0$:

$$f_*\mathrm{Gr}_a^W K' = \bigoplus_{i \in \mathbb{Z}} {}^p\mathcal{H}^i(f_* \mathrm{Gr}_{w_0}^W K')[-i] = \bigoplus_{i \in \mathbb{Z}} A_a^i[-i]$$

where $A_a^i$ are PHS of weight $a + i$ supported by $v$.

Let $a = w_0$, $Z_s$ the strict transform of $H$ and $u : (H \setminus v) \to V$:

$$f_*\mathrm{Gr}_{w_0}^W K' = \bigoplus_{i \in \mathbb{Z}} {}^p\mathcal{H}^i(f_*\mathrm{Gr}_{w_0}^W K')[-i] = \\ \bigoplus_{i \in \mathbb{Z}} (A_{w_0}^i[-i] \oplus B_{w_0}^i[-i] \oplus u_{i*} u^* {}^p\mathcal{H}^i((f|_{Z_s})_{j_1*} {}^p\mathbf{L}|_{Z_s}[-1])[-i]) \quad (6.10)$$

where $B_{w_0}^i$ is the polarized HS of weight $w_0 + i$, component of ${}^p\mathcal{H}^i(f_*j_1* {}^p\mathbf{L}|_{Z_s}[-1])$ with support $v$, $A_{w_0}^i$ is also a polarized HS of weight $w_0 + i$ with support $v$ and $u_{i*} u^* {}^p\mathcal{H}^i(f_*j_1* {}^p\mathbf{L}|_{Z_s}[-1]) \simeq u_{i*} u^* {}^p\mathcal{H}^i(f_*j_1* {}^p\mathbf{L})|_H[-1]$ is a direct sum of intermediate extensions of polarized VHS of weight $w_0 + i$ supported by $H$.

i') Respectively, let $K' := \varphi_{g_0fj_1*}{}^p\mathbf{L}$,

$$f_*\mathrm{Gr}_a^W K' = \bigoplus_{i \in \mathbb{Z}} {}^p\mathcal{H}^i(f_*\mathrm{Gr}_a^W K')[-i] = \bigoplus_{i \in \mathbb{Z}} C_a^i[-i]$$

where $C_a^i$ are polarized HS of weight $a + i$ supported by $v$.

**Remark 6.1.** To follow the proof, we explain the indices in the lemma and the degeneration at rank 1 on the restriction to $V \setminus v$. By transversality, the restriction $u_* {}^p\mathcal{H}^j(f_*j_1* {}^p\mathbf{L})[-1]$ to $H \setminus v$ is perverse, direct sum of shifted VHS of pure weight $j + {}^pw - 1 = j + w^0$ over the strata.

By induction, the derived direct image $f_*i_{Z_s}^*j_1* {}^p\mathbf{L}[-1]$ is of weight $w_0$ and decomposes into a direct sum of shifted ${}^p\mathcal{H}^j(f_*i_{Z_s}^*j_1* {}^p\mathbf{L}[-1])$ of pure weight $j+w_0$ whose restriction to $H \setminus v$ coincide with $u_* {}^p\mathcal{H}^j(f_*j_1* {}^p\mathbf{L})[-1]$ by transversality, which is consistent with the degeneration at rank 1.

By the decomposition by induction of $Z_s \to H$, we know more: the weight spectral sequence on $H$ degenerates to $u_* u^* {}^p\mathcal{H}^j(f|_{Z_s}j_1* {}^p\mathbf{L}|_{Z_s}[-1]) \oplus B_{w_0}^j$. The spaces $B_{w_0}^j$ are pure HS of weight $w_0 + j$ with support on $v$.

**Proof.** The Hodge filtration $F$ is induced on ${}^p\mathcal{H}^i(f_*\mathrm{Gr}_a^W K')$ as explained in (ch.1, §2.4). The lemma follows from (Equations 6.8, 6.9) as the terms $A_a^i$ are the image of terms with support in $X_v$, except the derived image $f_* (j_1* {}^p\mathbf{L}|_{Z_s}[-1])$ which has support on $H$. In the case $a = w_0$ we have, by induction, the decomposition on $H$ (see Equations 6.7 and 6.9) :

$${}^p\mathcal{H}^i(f_*j_1* {}^p\mathbf{L}|_{Z_s}[-1]) \simeq u_{i*} u^* {}^p\mathcal{H}^i(f_*j_1* {}^p\mathbf{L}|_{Z_s}[-1]) \oplus ({}^p\mathbf{L}|_{Z_s}[-1])_v^i.$$

Recall that $u_* {}^p\mathcal{H}^i(f_* (j_1* {}^p\mathbf{L}|_{Z_s}[-1])) \simeq u_* {}^p\mathcal{H}^i(f_*j_1* {}^p\mathbf{L})|_H[-1]$ by transversality of $H$ except at $v$. With the notations of (Equations 6.7, 1.7), set:

$$B_{w_0}^i := ({}^p\mathbf{L}|_{Z_s}[-1])_v^i.$$

**Lemma 6.3 (Splitting of the spectral sequence).** i) The perverse weight spectral sequence of $(f_*K', W, F)$

$$E_1^{p,q} := {}^p\mathcal{H}^{p+q}(\mathrm{Gr}_{-p}^W f_* K') \Rightarrow \mathrm{Gr}_q^W {}^p\mathcal{H}^{p+q}(f_* K') \quad (6.11)$$

splits into the direct sum of two spectral sequences:
$$E_1^{p,q} = E_1^{p,q}(v) \oplus E_1^{p,q}(H) \qquad (6.12)$$

where $E(v)$ is a spectral sequence of HS supported by $v$ which degenerates at the $E_2$-term while $E(H)$ is supported by $H$ and degenerates at the $E_1$-term.

i') Respectively, if $K' := {}^p\varphi_{g\circ fj_*}{}^p\mathbf{L}$, the perverse weight spectral sequence $E_1^{p,q}$ is supported by $v$ and degenerates at rank 2.

$$ii') \quad \mathrm{Gr}_q^W {}^p\mathcal{H}^{q-w_0}(f_*K') \simeq E_2^{-w_0,q}(v) \oplus u_{!*}u^* {}^p\mathcal{H}^{q-w_0}(f_{|Z_s*}j_{!*}{}^p\mathbf{L}|_{Z_s}[-1]) ,$$

$$\mathrm{Gr}_q^W {}^p\mathcal{H}^{p+q}(f_*K') \simeq E_2^{p,q}(v) \quad \text{if } p \neq -w_0.$$

$E_2^{p,q}(v)$ is a HS supported by $v$ and $u_{!*}u^* {}^p\mathcal{H}^{q-w_0}(f_*j_{!*}{}^p\mathbf{L}|_{Z_s}[-1])$ is an intermediate extension of a VHS.

ii') Similarly, $K' := {}^p\varphi_{g\circ fj_*}{}^p\mathbf{L}$ is supported by $v$:

$$\mathrm{Gr}_q^W {}^p\mathcal{H}^{p+q}(f_*K') \simeq E_2^{p,q} \quad \text{for all } p, q.$$

*Proof.* i) The weight filtration $W_i$ on $ {}^p\mathcal{H}^j(f_*K') $ is defined by the perverse image of $ {}^p\mathcal{H}^j(f_*W_i) \to {}^p\mathcal{H}^j(f_*K') $, while the Hodge filtration is defined by restriction as explained in (ch. 1, §2.4).

$$\begin{align*}
E_1^{p,q} &= {}^p\mathcal{H}^{p+q}(\mathrm{Gr}_{-p}^W f_*K') \xrightarrow{d_1} E_1^{p+1,q} = {}^p\mathcal{H}^{p+q+1}(\mathrm{Gr}_{-p-1}^W f_*K') \\
&\implies E_1^{p,q} \implies \mathrm{Gr}_q^W {}^p\mathcal{H}^{p+q}(f_*K')
\end{align*}$$

The differentials on the terms $E_1^{p,q}$ are written as follows:

$$\begin{align}
E_1^{-w_0-1,q} \xrightarrow{d_1} E_1^{-w_0,q} \xrightarrow{d_1} E_1^{-w_0+1,q} \nonumber \\
E_1^{-w_0,q} &= E_1^{-w_0,q}(v) \oplus u_{!*}u^* {}^p\mathcal{H}^{q-w_0}(f_*j_{!*}{}^p\mathbf{L}|_{Z_s}[-1]) \tag{6.13} \\
E_1^{-w_0-1,q} &= E_1^{-w_0-1,q}(v), \quad E_1^{-w_0+1,q} = E_1^{-w_0+1,q}(v) \nonumber
\end{align}$$

Set

$$E_1^{p,q}(v) := A_{-p}^{p+q} \quad \text{if } p \neq -w_0, \quad E_1^{-w_0,q}(v) := A_{w_0}^{q-w_0} \oplus B_{w_0}^{q-w_0} \qquad (6.14)$$

then $E_1^{p,q}(v)$ consists of HS in degree 0 supported by $v$, while

$$E_1^{-w_0,q}(H) := u_{!*}u^* {}^p\mathcal{H}^{q-w_0}(f_*j_{!*}{}^p\mathbf{L}|_{Z_s}[-1]), \quad E_1^{p,q}(H) = 0 \quad \text{if } p \neq -w_0. \qquad (6.15)$$

Since $H^0(i_v^*u_{!*}u^* {}^p\mathcal{H}^{q-w_0}(f_*j_{!*}{}^p\mathbf{L}|_{Z_s}[-1])) = 0$, the induced differentials by $d_1$ on $E_1^{p,q}(H)$ vanish.

The spectral sequence of HS $E_1^{p,q}(v)$ degenerates at rank 2 ([De 72]) while the spectral sequence $E_1^{p,q}(H)$ degenerates at rank 1 since its terms vanish if $p \neq -w_0$. The statement ii') is clear.

*Proof of the proposition 6.2.* Concerning the terms $E_1^{p,q}(H)$, the statement is part of the decomposition theorem by induction over $H$ and the degener- ation at rank one of $E_1^{p,q}(H)$:

1) $N$ induces 0 on $ {}^p\mathcal{H}^i(f_*j_{!*}{}^p\mathbf{L}|_{Z_s}[-1])$
2) $\eta^j : {}^p\mathcal{H}^{-j}(f_*j_*{}^p\mathbf{L}|_{Z_s}[-1]) \simeq {}^p\mathcal{H}^j(f_*j_*{}^p\mathbf{L}|_{Z_s}[-1])(j)$ are isomorphisms, hence including the terms $B_{w_0}^i := ({}^p\mathbf{L}|_{Z_s}[-1])_v^i$.

Concerning the statement of the terms $E_2^{p,q}(v)$, the following properties of the terms $E_1^{p,q}(v)$ are satisfied also for $a = w_0$:

$$1) N^a : A_a^i \simeq A_{-a}^i(-a)$$

$$2) \eta^j : A_a^{-j} \simeq A_a^j(j)$$

We refer to ([Sa 88] Proposition 4.2.2) to deduce the corollary as follows. For symmetry reasons, set

$$L_p^q := E_1^{-p,p+q}(v) \qquad (6.16)$$

The following properties of bigraded polarized HS are satisfied (see §5.2.1.1):

1) $L_p^q$ is a HS of weight $p+q$,

2) The cup product with the class of an hyperplane section of $X$ induces a morphism $l: L_p^q \to L_p^{q+2}(1)$ such that $l^q: L_p^{-q} \xrightarrow{\sim} L_p^q(q)$ is an isomorphism for $q \ge 0$ by Hard Lefschetz on the components of $Z$.

3) The endomorphism $\nu: L_p^q \to L_{p-2}^q(-1)$ which commutes with $l: \nu \circ l = l \circ \nu$, such that $\nu^p: L_p^q \xrightarrow{\sim} L_{-p}^q(-p)$ is an isomorphism for $p \ge 0$ by the property of the monodromy filtration $W(\nu)$.

4) We deduce from the autoduality of $\mathbf{L}$ and the auto-duality of the monodromy filtration, a scalar product polarizing the HS

$$S: L_{-p}^{-q} \times L_p^q \to \mathbb{Q}(-w - n + 1)$$

5) The differential $d_1$ on the terms $E_1$ of the spectral sequence induces a morphism $d: L_p^q \to L_{p-1}^{q+1}$ satisfying $d^2 = 0$, $d \circ l = l \circ d$, $d \circ \nu = \nu \circ d$ and $S(dx, y) = \pm S(x, dy)$.

### 6.2.0.3 Reduction to a fibration by $g$

The following reduction is a method introduced by Grothendiek's school ([De 80] §4.2) based on Lefschetz pencils and will be used also later in chapter 8. Let $(H_t, t \in \mathbb{P})$ be a general pencil of hyperplane sections of $V$ with axis $A$ of codimension 2 away from $v$. The only section $H_0$ through $v$ is transverse to the strata of a Whitney stratification $S$ of $V \setminus v$.

We set $H := H_0$. The sections $H_t$, for $t \ne 0$ varying in a small disc $D \subset \mathbb{P}$ with center 0, intersect the strata of a Whitney stratification $S$ of $V$ transversally $(H_t$ is normally embedded in $V \setminus v)$.

The blow-up $\tilde{V}$ of $V$ along $A$ is the variety $\tilde{V}$ of disjoint union of strict transforms $\tilde{H}_t \simeq H_t$ of $H_t$ with natural projections on $V$ and $\mathbb{P}$

$$\tilde{V} = \{(x, t) : x \in X, t \in \mathbb{P}, x \in H_t\}, p_1 : \tilde{V} \to V, p_2 : \tilde{V} \to \mathbb{P}, p_1 : \tilde{H}_t \simeq H_t.$$
The projection $p_2$ defines a fibration on $\tilde{V}$ onto $\mathbb{P}^1$, while the projection $p_1$ induces an isomorphism $\tilde{V} \setminus p_1^{-1}(A) \simeq V \setminus A$. Hence there exists a point $\tilde{v} \in \tilde{V}$ over $v$ such that $p_1$ projects a neighborhood $B_{\tilde{v}}$ of $\tilde{v}$ isomorphically onto a neighborhood $B_v$ of $v$.

We consider the fiber product morphism $\tilde{f}: \tilde{X} \to \tilde{V}$ on $\tilde{X} := X \times_V \tilde{V}$. Since the morphism remains unchanged over a neighborhood of $v$, the decomposition of $f$ at $v$ is equivalent to the decomposition of $f$.

### 6.2.0.4 Decomposition of $p_{\varphi_g} p^{\mathcal{H}^i}(K)$

Let $K := f_*j_*!\mathbb{P}_L$. The decomposition of $p_{\varphi_g} p^{\mathcal{H}^i}(K)$ below (Equation 6.17) follows from (Corollary 6.2) by a method due to M. Saito ([Sa 88], Lemma 5.2.15, see also [Ka 86] Proposition 3.3.1 for example).

**Lemma 6.4.** In the diagram of morphisms of perverse sheaves

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

$p_{\varphi_g} p^{\mathcal{H}^i}(K)$, supported by $v$ in degree 0, is a MHS direct sum of two sub-MHS:

$$p_{\varphi_g} p^{\mathcal{H}^i}(K) \simeq \text{Im can} \oplus \text{Ker var} \quad (6.17)$$

Since $H$ is general, $p_{\varphi_g} p^{\mathcal{H}^i}(K)$ is a MHS supported by $v$ in degree 0 (Proposition 6.2 and Corollary 6.3). The decomposition is in the category of IMHS as it occurs on degree 0:

$$H^0(p_{\psi_g} p^{\mathcal{H}^i}(K)) \xrightarrow{\text{can}} H^0(p_{\varphi_g} p^{\mathcal{H}^i}(K)) \xrightarrow{\text{var}} H^0(p_{\psi_g} p^{\mathcal{H}^i}(K))$$

The proof is carried on $\text{Gr}_*^W p_{\varphi_g} p^{\mathcal{H}^i}(K)$. The morphism can vanishes on the summand of $\text{Gr}_*^W p_{\psi_g} p^{\mathcal{H}^i}(K)$ with support on $H$ with cohomology 0 in degree 0 as intermediate extensions.

For each $m \in \mathbb{Z}$, let $H_{\psi}^m(v)$ denote $\text{Gr}_m^{W(N)} H^0(p_{\psi_g} p^{\mathcal{H}^i}(K))$ with support on $v$. Then $H_{\psi}^m(v)$ is a polarized HS of weight $pw - 1 + i + m$. Similarly, let $H_{\phi}^{m-1} := \text{Gr}_{m-1}^{W(N)} p_{\varphi_g} p^{\mathcal{H}^i}(K)$. Then $H_{\phi}^{m-1}$ is of weight $pw + i + m - 1$.

For all $m \in \mathbb{Z}$ we have diagrams $H_{\psi}^m \xrightarrow{C} H_{\phi}^{m-1} \xrightarrow{V} H_{\psi}^{m-2}(-1)$ induced by the canonical and the variation morphisms. Such diagrams of polarized HS satisfy the conditions of ([Sa 88], Lemma 5.2.15) by (Corollary 6.2). Hence for all $m$ we have a decomposition: $H_{\phi}^{m-1} = \text{Im } C \oplus \text{Ker } V$ from which we deduce the decomposition of $p_{\phi}^u$, since $\text{Im can}$ and $\text{ker var}$ are sub-MHS.
### 6.2.1 Decomposition of $^{p}\mathcal{H}^i(f_*j_*^{\mathcal{P}\mathcal{L}})$

The decomposition is deduced by Verdier's criteria on the extension of perverse sheaves in terms of the diagram defined by the canonical and variation morphisms (Equation 2.22). We stress the splitting of the filtration $F$.

By the reduction to a general hyperplane section, the proof of the decomposition theorem, following the decomposition of the spectral sequence, is limited to the use of the abelian category of IMHS.

**Lemma 6.5.** Let $K := f_*j_*^{\mathcal{P}\mathcal{L}}$, $i: H \to V$, $k: V \setminus v \to V$.

i) There exists a canonical splitting

$$ (^{p}\mathcal{H}^i(K), F) = (L_v^i, F) \oplus (k_{1*}k^*^{p}\mathcal{H}^i(K), F) \qquad (6.18) $$

where: $(L_v^i, F)$ is a pure HS supported by $v$ and $k_{1*}k^*^{p}\mathcal{H}^i(K)$ is an intermediate extension of a polarized VHS on $V \setminus v$ of weight $pw + i$.

Moreover we have induced Hodge Lefschetz isomorphisms
$$ \eta^i: i^*(^{p}\mathcal{H}^{-i}(K), F) \to i^*(^{p}\mathcal{H}^i(K), F). $$

**Proof.** Let $k': V \setminus H \to V$. The proof is based on Verdier's description of the extensions of perverse sheaves by the equivalence of the categories Perv($X$) and $V(H)$ (§2.2.3, Equation 2.18). The decomposition $\varphi \simeq \text{Im } \text{can} \oplus \text{Ker } \text{var}$ (Equation 6.17) corresponds to the extension over $H$

$$ k'_{1*}k'^*^{p}\mathcal{H}^i(K) \oplus \text{Ker } \text{var}. $$

Let $k'' : V \setminus H \to V \setminus v$, hence $k' = k \circ k''$. By construction:

$$ k'_{1*}k'^*^{p}\mathcal{H}^i(K) = k_{1*}(k''_{1*}k'^*^{p}\mathcal{H}^i(K)) $$

(first we extend to $V \setminus v$, then to $V$ according to the stratification defined by $H$ and $v$). Since by hypothesis $H$ is transversal to the stratification away from $v$: $k''_{1*}k'^*^{p}\mathcal{H}^i(K) = k^*^{p}\mathcal{H}^i(K)$, hence:

$$ k'_{1*}k'^*^{p}\mathcal{H}^i(K) = k_{1*}k''_{1*}k'^*^{p}\mathcal{H}^i(K) = k_{1*}k^*^{p}\mathcal{H}^i(K). $$

With the notations of (ch. 1, Equation 1.7) and by the uniqueness of the decomposition $L_v^i := \text{Ker } \text{var}$ is a pure HS and by induction $k^*^{p}\mathcal{H}^i(K)$ is a direct sum of intermediate extensions, which proves (Equation 6.18).

The morphisms $\eta^i: i^*(^{p}\mathcal{H}^{-i}(K), F) \to i^*(^{p}\mathcal{H}^i(K), F)$ are isomorphisms since they are already isomorphisms on $^{p}\psi_g^{p}\mathcal{H}^{-i}(K)$ and on $^{p}\varphi_g^{p}\mathcal{H}^{-i}(K)$, which ends the proof of the lemma.

**Remark 6.2.** In view of the decomposition 6.17 the proof is direct as follows:

**Lemma 6.6.** i) $\text{Ker } \text{var} \simeq \mathcal{H}^0(i_*^{p}\mathcal{H}^i(K))$

$$ ii) \mathcal{H}^k(i_*^{p}\mathcal{H}^i(K)) = \mathcal{H}^{k+1}(i_*^{p}\psi_g^{u}^{p}\mathcal{H}^i(K)) \text{ if } k < -1 $$

$$ iii) 0 \to \mathcal{H}^{-1}(i_*^{p}\mathcal{H}^i(K)) \to \mathcal{H}^0(i_*^{p}\psi_g^{u}^{p}\mathcal{H}^i(K)) \to \text{Im } \text{can} \to 0 $$

where the MHS $\text{Im } \text{can}$ is determined exactly by the terms $E_2^{p,q}(v)$ supported by $v$ (Lemma 6.3, ii) while the split vector space $L_v^i$ is $\text{Ker } \text{var}$.
Since $p\varphi_g^u p\mathcal{H}^i(K)$ is the cone of $sp: i^* p\mathcal{H}^i(K)[-1] \to p\psi_g^u p\mathcal{H}^i(K)$, we have a distinguished triangle:

$$ p\psi_g^u p\mathcal{H}^i(K) \xrightarrow{\text{can}} p\varphi_g^u p\mathcal{H}^i(K)) \xrightarrow{[-1]} i^* p\mathcal{H}^i(K) \to \qquad (6.19) $$

and its long perverse exact sequence

$$ 0 \to p\mathcal{H}^{-1}(i^* p\mathcal{H}^i(K)) \to p\psi_g^u p\mathcal{H}^i(K) \xrightarrow{\text{can}} p\varphi_g^u p\mathcal{H}^i(K) \to p\mathcal{H}^0(i^* p\mathcal{H}^i(K)) \to 0 $$

Since $p\varphi_g^u p\mathcal{H}^i(K) \simeq \text{Im can} \oplus \text{Ker var}$ is a vector space supported by $v$, we deduce the following isomorphism and short exact sequences:

$$ \text{Ker var} \simeq p\mathcal{H}^0(i^* p\mathcal{H}^i(K)) \\ 0 \to p\mathcal{H}^{-1}(i^* p\mathcal{H}^i(K)) \to p\psi_g^u p\mathcal{H}^i(K) \to \text{Im can} \to 0 $$

We remark also, since $var$ is a monomorphism on $\text{Im can}$ and $var \circ can = N$:

$$ \text{Im can} \xrightarrow{\text{var}\sim} \text{Im}(N : p\psi_g^u p\mathcal{H}^i(K) \to p\psi_g^u p\mathcal{H}^i(K)) \\ p\mathcal{H}^{-1}(i^* p\mathcal{H}^i(K)) \simeq \text{Ker}(N : p\psi_g^u p\mathcal{H}^i(K) \to p\psi_g^u p\mathcal{H}^i(K)) $$

Then, by (Equation 5.49) and ([BBD 83] corollary 4.1.12):

$$ p\mathcal{H}^{-1}(i^* p\mathcal{H}^i(K)) \simeq i^*(k'!_* k'^* p\mathcal{H}^i(K))[-1] = \text{Ker}(N|_{p\psi_g^u} p\mathcal{H}^i(K)). $$

Since $p\mathcal{H}^0(i^* p\mathcal{H}^i(K)) \simeq \text{Ker var} = \mathcal{H}^0(i_v^* p\mathcal{H}^i(K))$ is supported by $v$, we deduce $p\tau_{\le -1} i^* p\mathcal{H}^i(K) = \tau_{\le -1} i^* p\mathcal{H}^i(K) = \text{Im can}$ and

$$ i_v^* p\mathcal{H}^i(K) \simeq \tau_{\le -1} i^* p\mathcal{H}^i(K) \oplus \mathcal{H}^0(i_v^* p\mathcal{H}^i(K)). $$

Moreover, we extract from the long cohomology exact sequence, associated to the triangle (Equation 6.19) restricted to $v$, the exact sequence (iii) of the lemma.

**Lemma 6.7 (Splitting of the filtration F).** *The decomposition of $i^* p\mathcal{H}^i(K)$ splits the induced Hodge filtration.*

At this point, we did prove the decomposition into the direct sum of the perverse cohomologies. The splitting of the induced filtration $F$ into the induced filtrations on each term is a consequence of the existence of MHS on $i_v^* p\mathcal{H}^i(K)$.

By (Lemma 6.3) each term $\psi_g^u$ and $\varphi_g^u$ has an induced MHS. The MHS on $i_v^* p\mathcal{H}^i(K) = i_v^* i_H^* p\mathcal{H}^i(K)$ is deduced from the mixed cone of the morphism $\text{can}: \psi_g^u \to \varphi_g^u$. The decomposition in (Equation 6.17) is compatible with MHS. Since $L_v^i = \text{Ker var}$ has a pure HS and its embedding is compatible with the MHS, we deduce the splitting of the MHS and consequently the splitting of the filtration $F$ at $v$.

*Remark 6.3.* The MHS on $i_v^* p\mathcal{H}^i(K) = i_v^* i_H^* p\mathcal{H}^i(K)$ may be defined by desingularization and compatibility of MHS with the perverse filtration.

Such compatibility of Hodge and perverse filtrations is a general property on various cohomology groups on subspaces of $V$ and will be repeatedly useful in the text.
### 6.2.2 Hodge structure on $\mathbb{H}^*(X, j_*^p\mathbf{L})$ (dim $V = 0$)

We refer to ([Zu 79]) in the case where $\dim V = 0$ and $\dim X = 1$.

If $\dim X > 1$, we use a Lefschetz fibration by a pencil of hyperplane section to reduce the proof to the case where $f$ admits a factorization by a morphism to the projective complex line and the constant map:

$$X \xrightarrow{p_2} \mathbb{P} \xrightarrow{c_p} \text{spec } \mathbb{C}, \quad f = c_p \circ p_2$$

Let $(X_t, t \in \mathbb{P})$ be a pencil of hyperplane sections of $X$ with axis $A$ of codimension 2. The disjoint union of the hyperplane sections $X_t$ forms an algebraic variety known as the blow-up $\tilde{X}$ of $X$ along $A$ with projections $p_1$ and $p_2$

$$\tilde{X} = \{(x, t) : x \in X_t, t \in \mathbb{P}\}, p_1 : \tilde{X} \to X, p_2 : \tilde{X} \to \mathbb{P}, p_1 : \tilde{X}_t \simeq X_t$$

([De 80], 4.3.1). Let $j' : U \to \tilde{X}$ denote the embedding of an open subset of definition of the polarized VHS $\mathbf{L}$. If the pencil is general with axis $A$ transverse to a stratification of $(X, j_*^p\mathbf{L})$, then $j'_*^p\mathbf{L} \simeq p_1^*j_*^p\mathbf{L}$ and $p_1^*j'_*^p\mathbf{L} \simeq p_1^*p_1^*j_*^p\mathbf{L}$ since $p_1$ is a fibration by simply connected projective lines $\mathbb{P}$: The decomposition follows a from §6.1.0.1 since $\dim \tilde{X} = \dim X = n$, hence we have in the derived category:

$$p_{1*}j'_*^p\mathbf{L} \simeq j_*^p\mathbf{L} \oplus {}^p\mathcal{H}^2(p_{1*}j'_*^p\mathbf{L})[-2] \simeq j_*^p\mathbf{L} \oplus i_A^*j_*^p\mathbf{L}[-2] \quad (6.20)$$

We have: $\text{Gr}_2 {}^p\mathbb{H}^k(\tilde{X}, j'_*^p\mathbf{L}, F) \simeq \mathbb{H}^{k-2}(A, j_*^p\mathbf{L}, F)$ and ${}^p\tau_0 \text{Gr}_2 {}^p\mathbb{H}^k(\tilde{X}, j'_*^p\mathbf{L}, F) \simeq \text{Gr}_2 {}^p\mathbb{H}^k(X, j_*^p\mathbf{L}, F)$.

First we deduce a HS on $(\mathbb{H}^k(\tilde{X}, j'_*^p\mathbf{L}), F)$ since the decomposition theorem apply to $p_2$ and [Zu 79] applies to all intermediate extensions of polarized VHS of the perverse cohomologies of $(p_{2,*}j'_*^p\mathbf{L}, F)$ on $\mathbb{P}$. Hence

$$1. \ ({}^p\tau_2 \text{Gr}_2 {}^p\mathbb{H}^k(\tilde{X}, j'_*^p\mathbf{L}), F) = (\text{Gr}_2 {}^p\mathbb{H}^k(\tilde{X}, j'_*^p\mathbf{L}), F) \text{ is a HS.}$$

$$2. \ {}^p\tau_1 \text{Gr}_2 {}^p\mathbb{H}^k(\tilde{X}, j'_*^p\mathbf{L}), F) = (\text{Gr}_2 {}^p\mathbb{H}^k(X, j_*^p\mathbf{L}), F)$$

Moreover, and $\text{Gr}_2 {}^p\mathbb{H}^k(\tilde{X}, j'_*^p\mathbf{L}), F) \simeq (\text{Gr}_2 {}^p\mathbb{H}^{k-2}(A, j_*^p\mathbf{L}), F)$ is a HS by induction since $\dim A < \dim X$. Hence $(\text{Gr}_2 {}^p\mathbb{H}^k(X, j_*^p\mathbf{L}), F)$ is a HS as the kernel of a morphism of HS. Recall, that the decomposition is natural for constant coefficients (see [Vo 2007], Theorem 7.31).

**Corollary 6.1.** An hyperplane section defines a polarization on $\mathbb{H}^k(X, j_*^p\mathbf{L})$. Let $\eta$ denote the cup-product with the class of a hyperplane section of $X$, then $\eta^i$ defines an isomorphism: $\mathbb{H}^{-i}(X, j_*^p\mathbf{L}) \simeq \mathbb{H}^i(X, j_*^p\mathbf{L})$.

**Remark 6.4.** The HS on $\mathbb{H}^k(X, j_*^p\mathbf{L})$ coincides with the HS defined in [KaK 87], [CaKSc 87] and [De 71]. The proof follows from [KaK 86] where the HS are shown to coincide due to the property of auto-duality.

This ends the induction process on $\dim X$ for non singular $X$.
### 6.2.2.1 Purity on a singular variety X

We complete the proof of the purity theorem in the case of a singular variety $X$ (ch. 4, §4.1.2, Theorem 4.3) by reduction to a complete non-singular algebraic variety $X'$. We consider the finite covering of the variety $X$ by the disjoint union of its irreducible components to reduce the proof to the case where $X$ is irreducible and of dimension $n$ and $\mathbf{L}$ a polarized VHS of weight $w'$ on a non singular Zariski-open subset $U \subset X$. The weight of $\mathbf{pL} := \mathbf{L}[\dim X]$ is $pw := w' + n$.

Let $Y = X \cup U$, $\pi : X' \to X$ a non singular modification of $X$ such that $Y' := \pi^{-1}(Y)$ is a NCD. Let $j' : U \hookrightarrow X'$, then $j'_* \mathbf{pL}$ is endowed with a Hodge filtration $F$ induced by the isomorphism with $\text{IC}^{*\mathbf{p}\mathcal{L}}$ on $X'$.

There exists on $\mathbf{p}\mathcal{H}^0(\pi_*j'_*\mathbf{pL})$ a Hodge filtration $F$ induced by the push-forward filtration $\pi_*F$. The Hodge filtration $F$ on $j_*\mathbf{pL}$ is induced by the decomposition of $\mathbf{p}\mathcal{H}^0(\pi_*j'_*\mathbf{pL})$ compatible with $F$. This Hodge filtration in the derived filtered category $D^b F(X, \mathbb{C})$ is independent of the choice of the designularization $X'$ up to isomorphism.

**Lemma 6.8.** The perverse filtration $\mathbf{p}\tau$ on $\mathbb{H}^k(X', j'_*\mathbf{pL})$ with respect to the desingularization map $\pi : X' \to X$ is a filtration by sub-HS.

The definition of the filtration $F$ is obtained by the decomposition theorem applied to the morphism $\pi : X' \to X$.

As in the previous case, $\mathbf{p}\tau_0\mathbb{H}^k(X', j'_*\mathbf{pL})$ is a Hodge sub-structure since the quotient $\mathbb{H}^k(X', j'_*\mathbf{pL}) / \mathbf{p}\tau_0\mathbb{H}^k(X', j'_*\mathbf{pL})$ is a HS by induction on dimension, as the components of the decomposition over strata of dimension strictly lower than $\dim X$ appears in the quotient.

By a similar argument $\mathbf{p}\tau_{-1}\mathbb{H}^k(X', j'_*\mathbf{pL})$ is a sub-HS, hence $Gr_0^{\mathbf{p}\tau}\mathbb{H}^k(X', j'_*\mathbf{pL})$ is a HS which contains $\mathbb{H}^k(X, j_*\mathbf{pL})$ as a natural sub-HS (see also a general method to prove that $\mathbf{p}\tau$ is a filtration by sub-HS (Equation 10.7)).

**Corollary 6.2 (Purity Theorem on a singular variety X).** Let $\mathbf{L}$ be defined on a Zariski open subset of an algebraic complex and compact variety. Then, the Hodge filtration $F$ induces a HS pf weight $pw + k$ on $\mathbb{H}^k(X, j_*\mathbf{pL})$.

For any complex algebraic variety, there exists a birational morphism $g : X' \to X$ where $X'$ is projective and non-singular over $\mathbb{C}$ [De 68], 5.3.4). The decomposition theorem extends to $g$ since it is local on $X$. By the lemma, the HS on $\mathbb{H}^k(X, j_*\mathbf{pL})$ is a direct summand of $Gr_0^{\mathbf{p}\tau}\mathbb{H}^k(X', j'_*\mathbf{pL})$.

The independence of the choice of the designularization $X'$ up to isomorphism is proved by 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. $X' \to X'_2$). Then the HS on $\mathbb{H}^i(X, j_*\mathbf{pL})$ deduced from $X'_1$ or $X'_2$ coincides with the HS deduced from $X'$.
### 6.2.2.2 Decomposition theorem in the case of a singular variety X

The proof is by reduction to the non singular case. Let $f: X \to V$ and $j_*{}^pL$ on the singular variety $X$. The induced Hodge filtration $F$ on $j_*{}^pL$ on $X$ is deduced from a non singular modification $X'$ of $X$ (see Corollary 6.2). We have a diagram

$$ X' \xrightarrow{\pi} X \xrightarrow{f} V, \quad h := f \circ \pi \qquad (6.21) $$

There exist Thom-Whitney stratifications $S$ of $f$, $\pi$ and $h := f \circ \pi$, compatible with $Y := X \setminus U$ and the NCD $Y' := \pi^{-1}(Y)$, and Whitney stratifications of $X$, $X'$ and $V$ underlying the Thom-Whitney stratifications $S$.

Let $V_0$ (resp. $X_0$) denote the set of zero dimensional strata in $V$ (resp. $X$) and $V_m$ (resp. $X_m$) the maximal strata. We suppose the following conditions:

1) The inverse image $X'_{V_0}$ (resp. $X'_{X_0}$) is a NCD

2) The inverse image $Y'_{V_m}$ (resp. $Y'_{X_m}$) is empty or a relative NCD.

where relative means that the fiber $Y'_v$ at each point $v \in V_m$ (resp. $v \in X_m$) is a NCD in the non singular fiber $X'_v$.

These conditions may be easily satisfied over isolated strata in $V_0$. It is even possible to ask that the inverse image of each stratum $S$ of $V$ is a NCD in $X'$ and a fibration by NCD (chapter 12 §11.2, [EL 14]).

The decomposition theorem for $f$ is deduced from the case of $\pi$ and $h = f \circ \pi$ after ([De 68], Proposition 2.16) as follows.

Recall that $j' : U \to X'$ denote the embedding and $j'_*{}^pL$ is endowed with a Hodge filtration $F$. Let $K' := \pi_*j'_*{}^pL$. By the decomposition theorem applied to $\pi$:

$$ K' \simeq \bigoplus_i {}^pH^i(K')[-i] \in D_c^b(X) $$

Let $K := f_*K' = (f \circ \pi)_{*}j'_*{}^pL$. By the decomposition theorem applied to $f \circ \pi$:

$$ K \simeq \bigoplus_i {}^pH^i(K)[-i] \in D_c^b(V) $$

From a) we deduce $f_*K' \simeq \bigoplus_i f_* {}^pH^i(K')[-i]$. Then by ([De 68], Corollary 1.12) for all $k$ we have

$$ f_* {}^pH^k(K') \simeq \bigoplus_i {}^pH^i(f_* {}^pH^k(K'))[-i] \in D_c^b(V) \qquad (6.22) $$

**Corollary 6.3 (Decomposition).** Let $f: X \to V$ and $j_*{}^pL$ on $X$ (Equation 6.21). There exists a decomposition in the derived category

$$ f_*j_*{}^pL \simeq \bigoplus_i {}^pH^i(f_*j_*{}^pL)[-i] \qquad (6.23) $$

We apply (Equation 6.22) for $k = 0$, since $j_*{}^pL$ is a summand of ${}^pH^0(K')$ where $K' := \pi_*j'_*{}^pL$.
Chapter 7

Perverse VMHS

Let $V$ be a complex projective variety possibly singular. We define the abelian category $\mathcal{IH}$ of intermediate extension of polarized VHS with support on irreducible sub-varieties of $V$ (Lemma 7.1) and the category of perverse variation of mixed Hodge structures PVMHS (Definition 7.2).

The weight spectral sequence of PVMHS degenerates in general at rank 2 (Corollary 7.2) with $E_1$ terms in $\mathcal{IH}$ (§7.1).

For each closed subvariety $S \subset V$, there exists a MHS on $\mathbb{H}^i(S, j_*{}^p\mathcal{L})$ of weights $w \le {}^pw+i$, respectively a MHS on $\mathbb{H}_S^i(V, j_*{}^p\mathcal{L})$ of weights $w \ge {}^pw+i$.

At a closed point $v$, with $i_v : v \to V$ and $k : V \setminus v \to V$, there exists a MHS on $\mathbb{H}^i(i_v^*k_*k^*j_*{}^p\mathcal{L})$ (§7.1.1).

Let $g : U' \to \mathbb{C}$ be a projective morphism defined on an open subset $U' \subset V$. We define a PVMHS on ${}^p\psi_gj_*{}^p\mathcal{L}$ and ${}^p\varphi_gj_*{}^p\mathcal{L}$ by desingularization. Then, we can state the local and global Invariant cycle theorems (§7.2).

At the end of the chapter (§9.7) we recover results of M. Saito ([Sa 90]) on mixed Hodge modules in terms of PVMHS (see Corollary 7.2).

**7.1 Perverse variation of mixed Hodge structures**

We define the abelian category *IH* of intermediate extensions of polarized VHS. Then, we deduce the definition of perverse variations of MHS as extensions of objects of *IH*.

**7.1.0.1 Hodge filtration by desingularization**

Let $U$ be a non singular open dense subset of an irreducible projective variety $V$ and ${}^p\mathcal{L}$ a polarized VHS on $U$ of weight ${}^p w$. Let $\pi : \tilde{V} \to V$ be a desingularization of $V$ with exceptional divisor a NCD $Y$ inverse image of $V \setminus U$. Let $j : U \to V$ and $j' : U \to \tilde{V}$ denote the open embeddings into $V$ and $V'$.
A Hodge filtration $F$ is defined on $j'_* p_L$, from which we deduce by push-forward a filtration $\pi_* F$ on $\pi_* j'_* p_L$.

We recall that the Hodge filtration $F$ on $p\mathcal{H}^0(\pi_*j'_* p_L)$ is induced by the filtration $\pi_* F$ on $V$. Since $j'_* p_L$ is a summand of $(p\mathcal{H}^0(\pi_*j'_* p_L), F)$ by the decomposition theorem (ch.2, §2.4), we deduce again the Hodge filtration by restriction of $F$.

The construction is performed in the bifiltered derived category $D_{p\tau,F}^b(V, \mathbb{C})$. The filtration $F$ induced on $j'_* p_L$ does not depend in $D_F^b(V, \mathbb{C})$ on the choice of the desingularization of $V$. A functor $T$ is derived using bigraded bifiltered $T$-acyclic resolutions of $(\pi_*j'_* p_L, p_\tau, F)$.

### 7.1.0.2 The category $\mathcal{IH}$ of Hodge intermediate extensions

Let $j : U \to V$ denote the embedding of a smooth open subset of $V$. A morphism of polarized VHS (PVHS) $g : (p_L, F) \to (p_{L'}, F)$ on $U$ extends uniquely to the intermediate extensions $j_* g : (j_* p_L, F) \to (j_* p_{L'}, F)$ on $V$.

**Lemma 7.1.** *The category $\mathcal{IH}_V$ of filtered intermediate extensions of polarized VHS on smooth open subsets of $V$ of the form $(j_* p_L, F)$ with support $V$ and morphisms of intermediate extensions is abelian.*

*Proof.* We prove for example the existence of the kernel and image of a morphism. Let $g : (p_L, F) \to (p_{L'}, F)$ be a morphism of polarized VHS on the non singular open subset $U \subset V$ and $j : U \to V$. Let $j' : U \to \tilde{V}$ denote the embedding of $U$ in the desingularization of $V$ and $Y = \cup_{i \in I} Y_i$ the NCD complement of $U \subset \tilde{V}$.

Let $u : p_K \to p_L$ denote the kernel of $g$ (resp. $v : p_L \to p_I$ the image).

**Sub-lemma.** *The morphism $g$ (resp. $u$, $v$) extends into a morphism of intermediate extensions of polarized VHS with induced filtration $F$ by push-forward from $j'_* g : (j'_* p_L, F) \to (j'_* p_{L'}, F)$ (resp. $j'_* u : j'_* p_K \to j'_* p_L$ and $j'_* v : j'_* p_L \to j'_* p_I$) on $\tilde{V}$.*

Moreover $j'_* u$ and $j'_* v$ are kernel and image of $j'_* g$.

The proof is local. Let $J \subset I$, locally at a point of $Y_J$, we have a sequence of nilpotent orbits : $0 \to K \to L \to I \to 0$ endowed with monodromy weight filtration. With the notations of (ch.3, Definition 3.4), by induction on the length of $J$ and the vanishing cycle lemma (see ch. 5, Lemma 5.1), the sequences:

$$0 \to N_J K \to N_J L \to N_J I \to 0, \quad 0 \to \text{Ker } N_J K \to \text{Ker } N_J L \to \text{Ker } N_J I \to 0 \tag{7.1}$$

are exact. The graded exact sequence $\text{Gr}_*^{W(N_J)}$ defined by the weight filtration, split into primitive exact sequences, from which we deduce the exactness in (Equation 7.1). We deduce a distinguished triangle of the intermediate extensions

$$j'_* p_K \xrightarrow{j'_* u} j'_* p_L \xrightarrow{j'_* v} j'_* p_I \xrightarrow{[1]}$$