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# Decomposition, local Purity and Perverse variation of MHS

Fouad El Zein, Dung Tráng Lê, Jozef Steenbrink, Xuanming Ye

► To cite this version:

Fouad El Zein, Dung Tráng Lê, Jozef Steenbrink, Xuanming Ye. Decomposition, local Purity and Perverse variation of MHS. In press. hal-04445732

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# Preface

To Pierre Deligne and to the memory of Jean Louis Verdier

The purpose of this volume is to prove the existence of *special structures* on
the cohomology of a family of complex projective varieties, namely *Lefschetz*
*and Hodge decompositions*. We mention:

1. The construction of *perverse extension of degenerating variation of graded polarized mixed Hodge structures (VMHS)* along a normal crossing divisor (NCD).

2. The *decomposition theorem* of the derived image of a perverse extension of degenerating polarized variation Hodge structures (VHS) by a projective morphism on complex algebraic varieties by methods of algebraic topology.

3. The proof of *local purity* on complex varieties adapted from the proof on varieties of characteristic $p > 0$, and an other new proof.

4. The construction of *perverse extension of degenerating variation of polarized Hodge structures (VHS)* on compact complex varieties by application of the decomposition theorem to a desingularization.

5. The definition of perverse variation of mixed Hodge structures (VMHS) on complex algebraic varieties.

Actually, there exist two proofs of the *decomposition theorem*. The first is on algebraic varieties over fields of characteristic $p > 0$ which extends to polarized variation of Hodge structures (VHS) of geometric origin on complex varieties. The second proof is for Hodge modules and equivalently for polarized VHS on complex algebraic varieties.
We give two new proofs by methods of algebraic topology of complex algebraic varieties. The first is based on a generalization of the construction of mixed Hodge complex on the nearby cycles complex.

The second is based on a description of the weight filtration on the log-
arithmic complex $\Omega^*\mathcal{L} := \Omega_X^*(\mathrm{Log}Y) \otimes_{\mathcal{O}_X} \mathcal{L}_X$ with coefficients in Deligne's
extension $\mathcal{L}_X$ of a VHS $\mathbf{L}$ on the complement of a NCD $Y \subset X$.

The local purity theorem, states conditions on the weights of a NCD con-
tracting into an isolated stratum in a Thom-Whitney stratification. As an
application of the decomposition theorem, we adapt the original unpublished
proof by Deligne and Gabber to complex algebraic varieties.

In particular, we prove the *local and global invariant cycle theorems*. We
give also an other direct proof of local purity.

To figure the interest in these results, let us recall the developments after
1950 in this field of algebraic geometry by the schools influenced the work of
*J.P. Serre, A. Grothendieck, P. Deligne and Ph. Griffiths.*

1. **Weight and Hodge filtrations.** The cohomology groups of a com-
plex algebraic variety $X$, constructed with the transcendental topology, are
enriched by the definition of filtrations by sub-spaces reflecting the geometry
and the topology of the variety and its singularities.

In the case of a compact non singular algebraic variety $X$, a Hodge filtra-
tion $F$ is defined on the de Rham cohomology groups $H^i(X, \mathbb{C})$.

Using the conjugate $\bar{F}$ of $F$ with respect to the $\mathbb{Q}$-structure of the co-homology defined by the embedding $H^i(X, \mathbb{Q}) \subset H^i(X, \mathbb{C})$, a Hodge decomposition of the cohomology is defined by the finite direct sum of subspaces $H^{p,q} = F^p \cap \bar{F}^q$ for $p+q=i$.

The Hodge filtration $F$ is already defined on the de Rham complex but the proof of the induced decomposition on the cohomology is based on advanced analysis on varieties. Two homeomorphic varieties which are not isomorphic, give rise to two different decompositions in general, after identification of the cohomology spaces.

In the case of non singular but non compact algebraic varieties, the Hodge
filtration on the de Rham complex does not give a satisfactory structure on
the cohomology. Instead, it is necessary to embed a non singular $X$ into a non
singular compact algebraic variety with a normal crossing divisor (NCD) as
complement (at infinity), which is possible thanks to Hironaka’s desingular-
ization theorem. The presence of the NCD at infinity leads to an additional
filtration $W$ on the cohomology with $\mathbb{Q}$-coefficients called the weight filtra-
tion.

For a first reading, the definition of all terms can be found in the following
book

Claire Voisin: Hodge theory and complex algebraic geometry I, II. *Cambridge
University Press*, 76, 77, 2007.

For an advanced reading, we refer to the foundational work of
Pierre Deligne: Théorie de Hodge II. *Publ. Math. IHES, Paris* 40:5–57, 1972.

Théorie de Hodge III. *Publ. Math. IHES, Paris* 44:6–77, 1975.

Conjecture de Weil II. *Publ. Math. IHES, Paris* 52, 1980.

As well, other basic references are cited in the text.

**2. Mixed Hodge structure (MHS)** In the case of non compact non singular varieties, Deligne introduced a new complex called the logarithmic complex on which both the weight and Hodge filtrations are defined and induce the correct filtrations on the cohomology spaces. We remark that this correct Hodge filtration contains additional information at infinity.

Both weight $W$ and Hodge $F$ filtrations figure in a precise positions in the cohomology space and satisfy strong linear algebraic conditions called mixed Hodge structure (MHS).

Such MHS are also defined on the cohomology of singular algebraic varieties. Thus, the cohomology of any complex algebraic variety has a naturally defined MHS. It is surprising that the theory of MHS is motivated by the concept of purity and weight on $\ell$-adic cohomology of varieties over fields of positive characteristic. In fact, the theory of MHS is motivated by Deligne's results on the Weil conjecture. A "heuristic dictionary" developed by Deligne helps to translate results from one language into the other:

Pierre Deligne: Théorie de Hodge I. *Cong. intern. math.Nice, Tome 1*, 425–430, 1970.

**3. Polarization and Lefschetz decomposition** of the cohomology of complex non singular projective varieties. The cohomology class of an hyperplane section of a non singular projective variety $X$, defines Lefschetz isomorphisms on the cohomology by iteration of the cup-product with its cohomology class. From which we deduce, with Poincaré's duality, a scalar product on the cohomology $H^i(X, \mathbb{C})$ and a Lefschetz decomposition into a direct sum of *polarized subspaces*.

Reciprocally, Deligne's proof of Lefschetz isomorphisms by induction is based on the existence of the polarization (Weil II, 4.1, Lemma 4.1.4).

**4. Variation of mixed Hodge structures (VMHS).** In the case of a family of algebraic varieties defined by an algebraic morphism $f: X \to V$, the cohomology of the fibres form a constructible complex: there exists a stratification $S$ of the complex variety $V$ such that the restriction of the $i$-th cohomology sheaf of the derived image by $f$ of the constant sheaf $\mathbb{Q}_X$ is locally constant on each stratum $S$. The locally constant sheaf on $S$, is determined by a monodromy representation of the fundamental group of $S$ on the $i$-th cohomology of the fibres.

To go beyond topology, in the case of a proper smooth family, the Hodge filtration $F$ of the cohomology space $H^i(X_v, \mathbb{C})$ of non singular complex compact algebraic fibers $X_v$ at $v \in V$, defines a variation of Hodge structures (VHS) on $V$. Griffith gave sense to the fact that the variation of the Hodge filtration is analytic, or in other terms the filtration is defined by analytic sub-bundles of the flat bundle $R^i f_* \mathbb{C} \otimes_{\mathbb{C}} O_V$. The VHS has an additional property
called Griffith's transversality with respect to the connection determined by
the local system of the i-th cohomology $R^i f_* \mathbb{C}$ on $V$.

In the case of a projective family, we obtain a VMHS on each stratum $S$
of a stratification $S$ of $V$.

In this volume we describe results on the degeneration and the extension
of VMHS on the boundary of a stratum $S$ by algebraic topology techniques
in order to give a proof of the decomposition and purity theorems.

In a family, singular fibers occur in general. For example, any pencil of
curves on a surface contains singular special curves. The behavior of the
family near the singular fiber is useful (for example to the classification of
the families).

For this reason, we introduce the concept of *degeneration of the MHS* near
a singularity on the parameter space *V*. At a singularity $v \in V$, the degener-
ation reflects the properties of the family on a punctured neighborhood of *v*.
In particular we can blow up *v* to study the degeneration.

For this reason, the study of the degeneration is reduced in general to the
complement of a normal crossing divisor (NCD) in a non singular variety.

The singularities appear also when it is not possible to extend the local
system or the Hodge filtration of the VHS on a non singular parameter variety
$V$. For example, a singularity of the connection defined by the local system
may exists on a non singular $V$. The problems of the degeneration of the
filtration $F$ are of analytic nature.

**5. Decomposition of derived image of polarized VHS.** The polariza-
tion on a VHS leads to a decomposition into irreducible polarized sub-VHS.

Moreover Lefschetz and Hodge structures on the fibers of a smooth pro-
jective morphism $f: X \to V$, lead to the degeneration of Leray's spectral
sequence of $f$.

This degeneration may be interpreted as a decomposition (in the derived
filtered category $D^+F(V, \mathbb{C})$ of abelian sheaves on $V$), of the complex $K :=
f_*\mathbb{C}_X$ on $V$ defined by the derived image of the constant sheaf $\mathbb{C}$ on $X$, into
its cohomology sheaves. We refer to:

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

**6. Degeneration of VMHS.** In the case of an abstract analytic polarized VHS on a punctured complex disc $D^*$, the main result is the existence of Schmid's limit Hodge filtration $F$ on the space of multiple sections of the local system $\mathbf{L}$ underlying the VHS on $D^*$. We refer to:

Philip Griffiths and Wilfried Schmid: Recent developments in Hodge theory.
*Bombay Colloquium on: Discrete subgroups of Lie Groups*, Oxford University
Press 1973,

for a first reading and to the foundational work of

Wilfried Schmid: Variation of Hodge structure: the singularities of the period mapping. *Invent. Math.* 22:211–319, 1973.
Moreover, the monodromy filtration $W(N)$ defined by the logarithm of
the monodromy and the limit filtration $F$ form a MHS. The VHS is said to
degenerate into a limit MHS.

In the text Weil II by Deligne, as well in the article of Steenbrink, the
theory appears for an algebraic family over curves.

*Graded polarized VMHS.* In the case of an abstract VMHS on $D^*$ with fil-
trations $W$ and $F$, there is no natural limit Hodge filtration $F$. The existence
of a limit Hodge filtration is added by definition of a class of VMHS called
admissible.

In this case Deligne defined a relative monodromy filtration $M := M(N, W)$
and asked for precise properties which must be satisfied by $M$ and the limit
$F$, in particular $(M, F)$ form a MHS on the limit vector space defined by the
local system underlying the VMHS on $D^*$.

The properties required of an admissible VMHS are inspired by the case
of algebraic families and all algebraic families over a punctured disc define
admissible VMHS on $D^*$ called geometric.

We refer to Weil II on varieties over fields of characteristic $p > 0$, where
many problem originated.

In the case of higher dimensional variations, the degeneration of a VHS
 resp VMHS) over a product of punctured discs ($D^{*})^n$ is far more difficult
to study and the results present compatibility relations between the various
degenerations along the components of the NCD at "infinity" in $D^n$.

In this text, degeneration results are recalled but not proved since there
exist many references cited in the text.

**7. Extension of VHS.** In the case of curves, in Weil II by Deligne and
in the corresponding work by Zucker on a complex curve $V$, an answer to the
problem of extension is given along a singularity $v \in V$ of a polarized VHS
on a locally constant sheaf $\mathcal{L}$.

Let $j_*\mathcal{L}$ denotes the sheaf extending $\mathcal{L}$ such that the section at $v$ of $j_*\mathcal{L}$
are defined by the invariant subspace under the action of monodromy near
$v$, with its induced limit MHS at $v$ including the limit Hodge filtration $F$.

The main result is that the cohomology $(H^i(V, j_*\mathcal{L}), F)$ with the induced
filtration $F$ is a pure HS. This is the first example of a Hodge structure not
deduced from HS defined by compact non singular varieties.

In the case where $\dim V > 1$, the main problem is the extension of a VHS
defined on the big stratum $S$ of $V$. For example on a surface $S$, the critical
case is an isolated stratum on the boundary of $S$.

No progress has been made until the work by Goresky and Mac-Pherson
in 1980 on Intersection cohomology satisfying Poincaré duality on topological
spaces eventually singular, appeared.

**8. Perverse sheaves.** The interpretation by Deligne of Intersection co-
homology in the language of derived category and Poincaré-Verdier duality,
leads to the theory of intermediate extensions of a local system and more
generally to the definition of perverse sheaves developed in 1983 on varieties
over fields of strictly positive characteristic. We refer to:

A. Beilinson, J. Bernstein, P. Deligne, O. Gabber: Analyse et Topologie sur les espaces singuliers Vol.I, *Astérisque*, France 100, 1983.

**9. Purity.** The long awaited theory of purity, well understood in the case of curves in Weil II, makes sense in all dimensions as follows:

In the case of an abstract analytic polarized VHS underlying a local system
**L** on a product of punctured complex discs (*D*<sup>*</sup>)<sup>n</sup>, the correct extension is
a complex of sheaves called the intermediate extension of **L** with induced
filtration defined by the limit Hodge filtration *F*.

Globally, if a polarized Hodge filtration ($\mathbf{L}, F$) is given on the complementary of a NCD $Y \subset X$, the intermediate extension is defined in this text as a sub-complex $IC^*\mathcal{L}$ of the logarithmic complex $\Omega^*\mathcal{L} := \Omega_X^*(\mathrm{Log}Y) \otimes_{\mathcal{O}_X} \mathcal{L}_X$ with coefficients in Deligne's extension $\mathcal{L}_X$ with induced filtration $F$ deduced from the degenerate Hodge filtration.

The hypercohomology of the sub-complex ($IC^*\mathcal{L}, F$) on a projective variety $X$ with its induced Hodge filtration carry a pure HS.

After the development of the theory of perverse sheaves the interest shifted
to the property of decomposition of pure perverse sheaves.

The first purity result in the domain is proved over fields of strictly positive
characteristic by Deligne and Gabber in 1981 and has been a motivation of
subsequent work on Hodge theory on complex varieties. We cite among the
consequences, the proof of local and global invariant cycle theorems and local
purity.

In the case of a normal crossing divisor $Y$ in a complex algebraic variety $X$ contracting to an isolated stratum $v \in V$ in a stratified algebraic variety $V$, local purity has consequences on the classical cohomology with support in $Y$ in terms of restrictions on the weights, related to the contractibility of the NCD to a point.

Although the proofs are on varieties over fields of strictly positive charac-
teristic, it is possible to deduce the result in the case where the coefficients
are VHS of geometric origin on complex varieties.

In parallel, the relation to $D$-modules appears in the thesis of Brylinski in
1982, the work of Kashiwara in 1983 and the work of Saito who introduced
Hodge modules in 1988 and proved the property of decomposition of such
modules.

The theory of differential modules is a natural domain to study the degen-
eration and leads to the fact that Hodge modules are direct sums of variations
of intermediate extensions of Hodge structures.

It is surprising that perverse sheaves have been defined and studied first
in the language of derived category.

**10. Perverse extension of VMHS.** *A central problem* is the definition of the extension of a polarized VHS ($\mathbf{L}, F$) on a stratum $S$ of a stratified complex*
variety $V$ ($S$ is non singular but $V$ may be singular along $S$). We give in this
text a definition of the extension of the filtration $F$ by desingularization using
algebraic topology constructions. We apply the result to develop the theory
of perverse VMHS on a singular variety.

A general and useful study of extensions of perverse sheaves along a locally
principal divisor by considering the nearby and vanishing cycles figures in the
work of J. L. Verdier. The application to Hodge theory is written by M. Saito
in the language of $D$-modules.

We develop in the text the results on the extension of graded polarized
VMHS in terms of perverse sheaves and uilding on results of Kashiwara in the
case of NCD, we give an alternative definition in terms of algebraic topology.

**11. Proof of the decomposition and purity theorems.** The main simplification in the exposition of the text consists in the proof of the decomposition and local purity theorems for a projective morphism by reduction to the case of a NCD contracting to an isolated stratum $v$ in $V$ (not merely an isolated singularity) and its application to define the perverse extension of a *VHS* in general.

For this reason, we insist in this text to state the decomposition theorem in
terms of Thom stratification of a morphism $f: X \to V$ of algebraic varieties
and we express the decomposition theorem in terms of the stratification.
The reduction to isolated strata is by induction and restriction to a general
hyperplane section.

In the case of an isolated stratum $v$, the proof is reduced to the case of
existence of a projection of $V$ to the projective line defined by a general
hyperplane section through $v$. The spectral sequence of the nearby cycles
functor with respect to the monodromy weight filtration is relatively simple
to study in this case.

This allows the simple definition of the category of intermediate extension
of variations of Hodge structures and the category of variations of mixed
Hodge structures over singular varieties.

We recall, that even if we start with constant coefficients, the derived
direct image consists of a complex with constructible cohomology sheaves
with additional structure of intermediate extensions of polarized VHS.

Thus, instead of constant coefficients, the general theory is carried with
cohomology coefficients in polarized VHS and more generally VMHS.

**Organization of the book.** The text is divided in three sections.

1. The first section covers three chapters.

*Chapter 1* gives the statements of the local purity theorem, the decomposition theorem in terms of Tom-Whitney stratification and the extension of a polarized VHS.

*Chapter 2* covers the preliminaries on perverse sheaves and on the degeneration of VMHS.
*Chapter 3* centers on perverse sub-sheaves of the logarithmic complex in the case of a normal crossing divisor (NCD) at infinity.

2. The second section covers chapters (4 – 8).

In chapter 4, we introduce in the case of a unipotent VHS: (L, F), the nearby cycles complex $\Psi^*IL$, to define the limit Hodge filtration F and transfer the filtration to the unipotent part of the nearby cycles complex $\psi_f^u j_{f_*}^p L$ with coefficients in the intermediate extension of L.

We prove in chapter 5 the existence of a natural structure of mixed Hodge complex on the nearby and vanishing cycles complex with coefficients in an intermediate extension $j_{f_*}^p L$ of a (shifted) polarized variation of Hodge structure $(L, F)$.

We give a proof of the decomposition and purity theorems in chapter 6 and deduce in chapter 7 the Hodge filtration on perverse intermediate extensions on singular varieties by desingularization, the local and global invariant cycle theorems.

This allows us to adapt in chapter 8 the proof of the local purity theorem by Deligne and Gabber to the case of complex varieties.

3. The third section covers chapters (9 – 12).

In chapter 9, the weight filtration on the logarithmic complex in the case of a NCD at infinity is defined directly along the NCD.

In chapter 10 a combined simple proof of the decomposition and purity theorems is given in terms of Intersection morphism and a relative interpretation of local purity.
# Contents

## Perverse variation of mixed Hodge structure, Local purity and Decomposition

<table><tbody><tr><td>1</td><td>Introduction</td><td>3</td></tr><tr><td>1.1</td><td>Local purity</td><td>3</td></tr><tr><td>1.1.1</td><td>Local purity on complex varieties</td><td>4</td></tr><tr><td>1.2</td><td>Decomposition theorem and Perverse cohomology</td><td>5</td></tr><tr><td>1.2.1</td><td>Decomposition and Intersection morphisms</td><td>7</td></tr><tr><td>2</td><td>Preliminaries</td><td>15</td></tr><tr><td>2.1</td><td>Derived categories</td><td>15</td></tr><tr><td>2.1.1</td><td>Poincaré -Verdier duality</td><td>20</td></tr><tr><td>2.1.2</td><td>Grothendieck - Serre duality of coherent sheaves</td><td>21</td></tr><tr><td>2.1.3</td><td>Constructible sheaves</td><td>22</td></tr><tr><td>2.2</td><td>Perverse sheaves</td><td>26</td></tr><tr><td>2.2.1</td><td>Perverse sheaves defined by the middle perversity</td><td>28</td></tr><tr><td>2.2.2</td><td>ψ<sub>f</sub>j<sub>1*</sub>L and extensions of perverse sheaves</td><td>32</td></tr><tr><td>2.2.3</td><td>Verdier's classification of the extensions</td><td>36</td></tr><tr><td>2.3</td><td>Degeneration of VMHS</td><td>39</td></tr><tr><td>2.3.1</td><td>Nilpotent orbit</td><td>41</td></tr><tr><td>2.3.2</td><td>Infinitesimal mixed Hodge structure (IMHS)</td><td>42</td></tr><tr><td>2.4</td><td>Induced filtrations on the perverse cohomology</td><td>45</td></tr><tr><td>3</td><td>Perverse sub-sheaves of the logarithmic complex Ω<sup>*p</sup>LC</td><td>49</td></tr><tr><td>3.1</td><td>Perverse logarithmic sub-complexes of Ω<sup>*p</sup>LC ≃ j<sub>*</sub>pL</td><td>50</td></tr><tr><td>3.1.1</td><td>Local definitions along a NCD Y ⊂ X</td><td>50</td></tr><tr><td>3.1.2</td><td>The complex Ω<sup>*p</sup>L and the tilde embedding</td><td>52</td></tr><tr><td>3.1.3</td><td>The complex Ω<sup>*p</sup>LC ≃ j<sub>*</sub>L ∈ D<sub>c</sub><sup>b</sup>(X, C)</td><td>54</td></tr><tr><td>3.1.4</td><td>The intermediate extension j<sub>1*</sub>pL ≃ IC<sup>*p</sup>LC ⊂ Ω<sup>*p</sup>LC</td><td>55</td></tr><tr><td>3.2</td><td>The logarithmic complex j<sub>Z*</sub>j<sub>Z</sub><sup>*</sup>j<sub>1*</sub>L ≃ IC<sup>*</sup>LC(log Z)</td><td>56</td></tr></tbody></table>
<table>
  <tbody>
    <tr>
      <td>3.2.1</td>
      <td>Thom isomorphism and Gysin morphism</td>
      <td>59</td>
    </tr>
    <tr>
      <td>3.2.2</td>
      <td>Polarized Variation of HS (VHS)</td>
      <td>60</td>
    </tr>
    <tr>
      <td>3.2.3</td>
      <td>Duality and Polarization</td>
      <td>60</td>
    </tr>
    <tr>
      <td>3.2.4</td>
      <td>Perverse sheaves along a NCD</td>
      <td>65</td>
    </tr>
    <tr>
      <td>3.2.5</td>
      <td>$i_{Z*}i_{Z*}^*j_{Z*}j_{Z*}^*j_{Z*}^*P_L$</td>
      <td>65</td>
    </tr>
    <tr>
      <td>3.2.6</td>
      <td>Combinatorial description</td>
      <td>66</td>
    </tr>
    <tr>
      <td>4</td>
      <td>The Nearby cycles complex $\Psi^*IL$</td>
      <td>69</td>
    </tr>
    <tr>
      <td></td>
      <td>Fouad El Zein, Jozef Steenbrink</td>
      <td></td>
    </tr>
    <tr>
      <td>4.1</td>
      <td>Introduction to chapters 4 – 8 (section 2)</td>
      <td>69</td>
    </tr>
    <tr>
      <td>4.1.1</td>
      <td>Invariant local and global cycles theorems</td>
      <td>72</td>
    </tr>
    <tr>
      <td>4.1.2</td>
      <td>Purity and local purity theorems</td>
      <td>73</td>
    </tr>
    <tr>
      <td>4.2</td>
      <td>The limit Hodge filtration on $\Psi^*IL \simeq \psi_f^u j_{f*} P_L$</td>
      <td>73</td>
    </tr>
    <tr>
      <td>4.2.1</td>
      <td>The limit complex $\Psi^*IL$</td>
      <td>75</td>
    </tr>
    <tr>
      <td>4.2.2</td>
      <td>Proof of proposition ??</td>
      <td>81</td>
    </tr>
    <tr>
      <td>5</td>
      <td>Limit mixed Hodge structure on $\Psi^*IL$</td>
      <td>87</td>
    </tr>
    <tr>
      <td>5.1</td>
      <td>Local proof of theorem ??</td>
      <td>89</td>
    </tr>
    <tr>
      <td>5.1.1</td>
      <td>Infinitesimal mixed Hodge structure on $\Psi_J(L)$</td>
      <td>90</td>
    </tr>
    <tr>
      <td>5.1.2</td>
      <td>Local decomposition of $\mathrm{Gr}_*^{W(N)}\Psi^*IL$</td>
      <td>95</td>
    </tr>
    <tr>
      <td>5.1.3</td>
      <td>Perverse VMHS structure on $\psi_f^u j_{f*} P_L$ and $\varphi_f^u j_{f*} P_L$</td>
      <td>98</td>
    </tr>
    <tr>
      <td>5.2</td>
      <td>Perverse VMHS on $p\psi_f j_{f*} P_L$ and $p\varphi_f j_{f*} P_L$</td>
      <td>102</td>
    </tr>
    <tr>
      <td>5.2.1</td>
      <td>Functoriality of the monodromy filtration</td>
      <td>106</td>
    </tr>
    <tr>
      <td>5.2.2</td>
      <td>Hodge theory on $i^*j_{Z*}j_{Z*}^*j_{Z*}^*P_L$</td>
      <td>115</td>
    </tr>
    <tr>
      <td>6</td>
      <td>Proof of the Decomposition Theorem</td>
      <td>119</td>
    </tr>
    <tr>
      <td>6.1</td>
      <td>Proof of $D(n-1) \Rightarrow D(n)$ for non-singular $X$</td>
      <td>120</td>
    </tr>
    <tr>
      <td>6.1.1</td>
      <td>The crucial case at isolated strata</td>
      <td>121</td>
    </tr>
    <tr>
      <td>6.2</td>
      <td>Nearby and vanishing cycles at an isolated stratum</td>
      <td>121</td>
    </tr>
    <tr>
      <td>6.2.1</td>
      <td>Decomposition of $p\mathcal{H}^i(f_*j_{f*}P_L)$</td>
      <td>127</td>
    </tr>
    <tr>
      <td>6.2.2</td>
      <td>Hodge structure on $\mathbb{H}^*(X, j_{f*}P_L)$ ($\dim V = 0$)</td>
      <td>129</td>
    </tr>
    <tr>
      <td>7</td>
      <td>Perverse VMHS</td>
      <td>133</td>
    </tr>
    <tr>
      <td>7.1</td>
      <td>Perverse variation of mixed Hodge structures</td>
      <td>133</td>
    </tr>
    <tr>
      <td>7.1.1</td>
      <td>Hodge theory along locally principal divisors</td>
      <td>137</td>
    </tr>
    <tr>
      <td>7.1.2</td>
      <td>Nearby and vanishing cycles on a variety $V$</td>
      <td>138</td>
    </tr>
    <tr>
      <td>7.2</td>
      <td>Local and global invariant cycle theorems</td>
      <td>141</td>
    </tr>
    <tr>
      <td>7.2.1</td>
      <td>Global invariant cycle theorem</td>
      <td>144</td>
    </tr>
    <tr>
      <td>8</td>
      <td>Deligne Gabber local purity</td>
      <td>147</td>
    </tr>
    <tr>
      <td>8.0.1</td>
      <td>The crucial case: $i = 0, -1, w = p_w$</td>
      <td>149</td>
    </tr>
  </tbody>
</table>
<table>
  <tbody>
    <tr>
      <td>9</td>
      <td>Weight filtration, decomposition and local purity</td>
      <td>157</td>
    </tr>
    <tr>
      <td></td>
      <td>Fouad El Zein, Dũng Tráng Lê, Xuanming Ye</td>
      <td></td>
    </tr>
    <tr>
      <td>9.1</td>
      <td>Introduction to chapters 9 – 11 (section 3)</td>
      <td>157</td>
    </tr>
    <tr>
      <td>9.2</td>
      <td>The weight filtration W</td>
      <td>161</td>
    </tr>
    <tr>
      <td>9.2.1</td>
      <td>Local weight filtration on Ω*L</td>
      <td>162</td>
    </tr>
    <tr>
      <td>9.2.2</td>
      <td>Decomposition of Gr<sup>W</sup>Ω*L</td>
      <td>165</td>
    </tr>
    <tr>
      <td>9.2.3</td>
      <td>Weight filtration on IC*L</td>
      <td>168</td>
    </tr>
    <tr>
      <td>9.3</td>
      <td>Global definition and properties of the weight W</td>
      <td>168</td>
    </tr>
    <tr>
      <td>9.4</td>
      <td>The logarithmic complex IC*L(Log Z)</td>
      <td>171</td>
    </tr>
    <tr>
      <td>9.5</td>
      <td>Hodge theory on perverse cohomology along a divisor</td>
      <td>173</td>
    </tr>
    <tr>
      <td>9.5.1</td>
      <td>Hodge theory on i*j*Z<jsub>*</sub>j<sub>*</sub>j<sub>*</sub>P<sub>L</sub></td>
      <td>176</td>
    </tr>
    <tr>
      <td>9.6</td>
      <td>The bi-filtered relative complexes IC<sub>f</sub><sup>*</sup>L &sub; Ω<sub>f</sub><sup>*</sup>L</td>
      <td>177</td>
    </tr>
    <tr>
      <td>9.6.1</td>
      <td>The relative Intersection complex IC<sub>f</sub><sup>*</sup>L</td>
      <td>178</td>
    </tr>
    <tr>
      <td>9.7</td>
      <td>Admissible perverse VMHS on X \ Z</td>
      <td>180</td>
    </tr>
    <tr>
      <td>9.7.1</td>
      <td>PVMHS on j<sub>*</sub>P<sub>L</sub></td>
      <td>180</td>
    </tr>
    <tr>
      <td>10</td>
      <td>Relative local purity and Decomposition</td>
      <td>189</td>
    </tr>
    <tr>
      <td>10.0.1</td>
      <td>The case dim V = 0 and dim X = n</td>
      <td>189</td>
    </tr>
    <tr>
      <td>10.0.2</td>
      <td>Intermediate statement D(n, V*) in the case 0 &lt; dim V &le; n</td>
      <td>190</td>
    </tr>
    <tr>
      <td>10.1</td>
      <td>Proof of the relative local purity</td>
      <td>191</td>
    </tr>
    <tr>
      <td>10.1.1</td>
      <td>Preliminaries on hyperplane sections of X</td>
      <td>194</td>
    </tr>
    <tr>
      <td>10.1.2</td>
      <td>The large inequality: w &ge; <sup>p</sup>w for j = 0</td>
      <td>197</td>
    </tr>
    <tr>
      <td>10.2</td>
      <td>The crucial case</td>
      <td>197</td>
    </tr>
    <tr>
      <td>10.3</td>
      <td>The decomposition theorem at v</td>
      <td>201</td>
    </tr>
    <tr>
      <td>10.3.1</td>
      <td>Extension of the perverse cohomology across v</td>
      <td>201</td>
    </tr>
    <tr>
      <td>10.3.2</td>
      <td>Proof of the decomposition at v</td>
      <td>203</td>
    </tr>
    <tr>
      <td>10.3.3</td>
      <td>Singular variety X</td>
      <td>210</td>
    </tr>
    <tr>
      <td>10.3.4</td>
      <td>Decomposition of <sup>p</sup>H<sup>i</sup>(f<sub>*</sub>j<sub>*</sub>P<sub>L</sub>)</td>
      <td>211</td>
    </tr>
    <tr>
      <td>10.4</td>
      <td>Relative local purity and Intersection morphisms</td>
      <td>212</td>
    </tr>
    <tr>
      <td>10.4.1</td>
      <td>Intersection morphism</td>
      <td>213</td>
    </tr>
    <tr>
      <td>11</td>
      <td>Related results</td>
      <td>217</td>
    </tr>
    <tr>
      <td>11.1</td>
      <td>Hodge theory on perverse cohomology</td>
      <td>217</td>
    </tr>
    <tr>
      <td>11.1.1</td>
      <td>Preliminaries</td>
      <td>217</td>
    </tr>
    <tr>
      <td>11.1.2</td>
      <td>Hodge structure on the perverse filtration</td>
      <td>218</td>
    </tr>
    <tr>
      <td>11.1.3</td>
      <td>Mixed Hodge structure (MHS) on the Link</td>
      <td>220</td>
    </tr>
    <tr>
      <td>11.2</td>
      <td>Fibration by normal crossing divisors</td>
      <td>222</td>
    </tr>
    <tr>
      <td>11.2.1</td>
      <td>Fibration by NCD over the strata</td>
      <td>222</td>
    </tr>
    <tr>
      <td>11.2.2</td>
      <td>Existence of fibrations by NCD over the strata</td>
      <td>223</td>
    </tr>
    <tr>
      <td>11.3</td>
      <td>Combinatorial description of the weight</td>
      <td>230</td>
    </tr>
    <tr>
      <td></td>
      <td>References</td>
      <td>238</td>
    </tr>
  </tbody>
</table>
<figure><img src="image_2.png" /></figure>
# Chapter 1
## Introduction

An overview of the domain of research is presented in the preface. This volume is divided in three sections. The first section covers chapters 1 – 3.

In this chapter, the **statement of the local purity and of the decomposition theorems as well the extension of a polarized VHS** on a complex algebraic variety, are given.

In chapter 2, preliminaries on perverse sheaves and variation of mixed Hodge structures may be useful for a first reading.

We develop in chapter three the normal crossing divisor case in details. It is an example of perverse sheaves.

*The Abbreviations and Conventions figure in §2.4.0.2 and the induced filtrations on perverse cohomology in §2.4.*

### 1.1 Local purity

In an unpublished note [DeG 81], Deligne and Gabber announced the following result, known as the purity theorem on a projective variety over a field of positive characteristic:

**Theorem.** Let $\mathbb{F}_q$ be a finite field, $X_0$ a smooth scheme of finite type over $\mathbb{F}_q$, $j: V_0 \hookrightarrow X_0$ an open subset of $X_0$, and $K_0$ a perverse sheaf on $V_0$. If $K_0$ is pure of weight $w$, the complex $j_*K_0$ is pure of same weight.

Let $j: V \to X$ denote the extension of $j: V_0 \to X_0$ over an algebraic closure of $\mathbb{F}_q$ and $K$ the extension of $K_0$ on $V$. We may also say that $j_*K$ is pure of same weight as $K$.

The proof by induction on an hyperplane section leads to the following local purity statement:
**Local purity theorem.** Let $E_0$ be a finite set of zero dimensional points in $X_0$, $V_0 := X_0 \setminus E_0$, $E \subset X$ the extensions over an algebraic closure of $\mathbb{F}_q$ and $K$ the extension on $V$ of a pure complex $K_0$ of weight $pw$ on $V_0$.

Let $B_v$ be a henselization of $X$ at a zero dimensional point $v$ of $E$. The weight $w$ of the cohomology $\mathbb{H}^i(B_v \setminus \{v\}, K)$, satisfies the inequalities:

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

Next, we give the interpretation and application of this theorem on complex varieties. It is a central result in the development of Hodge theory. Moreover, it is related to the proof of the decomposition theorem.

### 1.1.0.1 Deligne's dictionary

According to Deligne [De 71], the henselization of a complex algebraic variety $V$ at a point $v$ corresponds to a ball $B_v$, defined by a local embedding of $V$ in $\mathbb{C}^N$ at $v$, as the intersection with $V$ of a ball of $\mathbb{C}^N$ centered at $v$ of small radius, such that $B_v$ is a cone over a topological space called the Link $L$ of $V$ at $v$ which is a topological invariant independent of the embedding in $\mathbb{C}^N$.

Purity in positive characteristic corresponds to Hodge structure (HS) on cohomology [De 72] and the theory of weight is translated into a weight filtration local purity theorem.

In particular the conditions on the weights above can be translated in the transcendental case into conditions on the weights of a natural filtration $W$ mentioned in the text as the local purity theorem.

#### 1.1.1 Local purity on complex varieties

The Intersection complex on a complex algebraic variety $V$, defined with coefficients in a local system on a non singular locally closed Zariski subset, has been introduced first by a topological construction [GMacP 83].

Later, Deligne extended the definition to algebraic varieties over finite fields (see [Br 82]). The Intersection complex appears as the Intermediate extension in the general theory of perverse sheaves ([BBD 83] Definition 2.1.5 and Proposition 2.1.11).

Let $V^*$ denote a smooth algebraic open dense subset of a complex algebraic variety $V$ of pure dimension, and $j: V^* \hookrightarrow V$ the open embedding. Let $\mathbf{L}$ denote a polarized variation of Hodge structures (VHS) of weight $w'$ on $V^*$.

We attach to $\mathbf{L}$ the perverse sheaf $\mathcal{P}\mathbf{L} = \mathbf{L}[\dim V]$ on $V^*$. Due to the shift in degrees, the perverse sheaf $\mathcal{P}\mathbf{L}$ is concentrated in degree $-\dim V$ and has weight $pw = w' + \dim V$. Let $j_1\mathcal{P}\mathbf{L}$ denote the Intermediate extension of $\mathcal{P}\mathbf{L}$ (see §2.2), $v$ a zero dimensional point of $V$, $i_v: v \to V$ and $k: (V \setminus v) \to X$.
In the text, we prove the local purity theorem in terms of the mixed Hodge structure (MHS) on the cohomology space $H^*(i_v^*k_*k^*j_{1*}^p\mathbf{L})$ (a derived functor RF in the text will be denoted simply by F, in particular $k_*$ stands for $Rk_*$).

We find it more suggestive to say that we put a MHS on the hypercohomology of a punctured small ball $B_v^* := B_v \setminus v$ by fixing an isomorphism $H^*(i_v^*k_*k^*j_{1*}^p\mathbf{L}) \simeq \mathbb{H}^*(B_v \setminus v, j_{1*}^p\mathbf{L})$ for $B_v$ small enough. The local purity theorem is stated in terms of the weight $w$ of a natural MHS as follows:

**Theorem 1.1 (Local purity).** Let $j_{1*}^p\mathbf{L}$ be the intermediate extension of weight $p w$, $v$ a zero dimensional point of $V$, $i_v : v \to V$ and $k : (V \setminus v) \to X$.

The weight $w$ of the space $H^i(i_v^*k_*k^*j_{1*}^p\mathbf{L})$, isomorphic to the Intersection cohomology $\mathbb{H}^i(B_v \setminus \{v\}, j_{1*}^p\mathbf{L})$ of a small punctured ball $B_v$ at $v$, satisfies the relations:

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

The proof in [DeG 81] is adapted to complex variety in chapter 8. A new proof is given in chapters 10 and 10.3. Already, in the simplest case of an isolated singularity $v \in V$, the theorem has important consequences on the weights of the cohomology with support $H_v^i(V, \mathbb{Q})$. A direct proof of these consequences appears in the work of Navarro Aznar [Na 85], where the results are treated in terms of the exceptional NCD in a desingularization and the MHS of the isolated singularity $v$.

## 1.2 Decomposition theorem and Perverse cohomology

The introduction of the category of perverse sheaves and perverse cohomology lead (see §2.2 for a summary) to vast developments in the theory and to the unification of the language and the structure of proofs in positive characteristic as well on complex algebraic varieties.

To state the decomposition theorem in ([BBD 83]) we need to introduce the perverse cohomology sheaves ${}^p H^i(K)$ of a complex $K$ on $V$ (cf §2.2).

Let $X_0$ be a separated scheme of finite type over a finite field $\mathbb{F}_q$ of $q$ elements of characteristic $p$, and $X := X_0 \otimes_{\mathbb{F}_q} \mathbb{F}$ its extension over an algebraic closure $\mathbb{F}$ of $\mathbb{F}_q$.

Let $l \neq p$ and let $j : U \to X$ be the inclusion of a smooth connected open subset of pure dimension $d$ and a $\mathbb{Q}_l$-smooth irreducible sheaf $\mathbb{P}\mathbf{L}$ on $U$.

Let $f : X_0 \to V_0$ be a projective morphism. The **purity theorem** states the decomposition of ${}^p H^j(f_*j_{1*}^p\mathbf{L})$ into a finite direct sum of intermediate extension on the projective variety $V$ non necessarily smooth, of $\mathbb{Q}_l$-smooth irreducible sheaves $\mathbb{P}\mathbf{L}_i$ on locally closed Zariski subsets $j_i : U_i \to V$ ([BBD 83], théorème 5.3.8):

$${}^p H^j(f_*j_{1*}^p\mathbf{L}) \simeq \bigoplus_{i \in I} j_{1*}^p \mathbf{L}_i \qquad (1.1)$$
The relative Hard Lefshetz theorem ([BBD 83], théorème 5.4.10) states that the class $\ell \in H^2(X, \mathbb{Q}_l(1))$ of an hyperplane section (or a relatively ample bundle) induces isomorphisms for $i \ge 0$:

$$ \ell^i : \mathcal{H}^i(f_*j_*!^p\mathbf{L}) \xrightarrow{\sim} \mathcal{H}^i(f_*j_*!^p\mathbf{L}) \qquad (1.2) $$

Both equations 1.2 and 1.1 are known as the **Decomposition theorem**.

### 1.2.0.1 Geometric statements

To deduce results concerning geometric statements on schemes of finite type over $\mathbb{C}$ from corresponding statements in characteristic $p > 0$, a general procedure, described in ([BBD 83], §6), applies to perverse sheaves of geometric origin ([BBD 83], §6.2.4).

Let $f: X \to Y$ be a proper morphism and $P$ a simple perverse sheaf over $X(\mathbb{C})$ of geometric origin, there exists a decomposition ([BBD 83], théorème 6.2.5)

$$ f_*P \simeq \bigoplus_i \mathcal{H}^i(f_*P)[-i] \in D_c^b(V(\mathbb{C}), \mathbb{C}) \qquad (1.3) $$

in the derived category [Ve 77]. Moreover, each perverse cohomology $\mathcal{H}^i(f_*P)$ decomposes into a direct sum of simple perverse sheaves of geometric origin.

**Example 1.1.** Let $g: Z \to Y$ be a proper morphism, by definition any simple component of $\mathcal{H}^i(g_*\mathbb{C})$ is of geometric origin.

We recommend strongly to read the full proof in chapter 6 of [BBD 83].

Indeed, the argument consists of general conditions and steps to deduce theorems on algebraic complex varieties from theorems on varieties over fields of characteristic $p > 0$, as an application of Thom-Whitney stratifications.

### 1.2.0.2 Analytic statements

On complex varieties, Intermediate extensions of polarized VHS correspond to pure Intermediate extensions in positive characteristic.

**Purity.** In the case of a polarized VHS $\mathbf{L}$ of weight $w'$ on a Zariski open subset $j: U \to X$ complement of a normal crossing divisor (NCD) $Y$ in a non singular complex projective variety $X$, the Intersection cohomology $\mathbb{H}^i(X, j_*\mathbf{L})$ has an induced pure HS of weight $w' + i$ ([CaKSc 87] and [KaK 87]).

**Decomposition.** Let $f: X \to V$ be a projective morphism of complex algebraic varieties. There exists a decomposition of the perverse cohomology into a finite direct sum of intermediate extensions of polarized VHS $\mathcal{H}^i$ on locally closed Zariski subsets $j_i: U_i \to V$:

$$ \mathcal{H}^i(f_*j_*!^p\mathbf{L}) \xrightarrow{\sim} \bigoplus_{i \in I} j_{i!*}^p\mathbf{L}_i \qquad (1.4) $$
and there exists a decomposition in the category $D_c^b(V, \mathbb{Q})$

$$f_*j_{1*}\mathbf{L} \simeq \bigoplus_{i \in \mathbb{Z}^p} \mathcal{H}^i(f_*j_{1*}\mathbf{L})[-i]. \qquad (1.5)$$

The decomposition is proved for polarized Hodge modules [Sa 88] and [Sa 90] in terms of the theory of differential modules ($\mathcal{D}_X$-modules). We give a proof inspired by [DeG 81].

### 1.2.1 Decomposition and Intersection morphisms

In the text we give a proof of the decomposition theorem in terms of stratifications and Intersection morphisms, by induction on the dimension of $X$, using Hodge theory on the complex of nearby and vanishing cycles in chapter 6. In the last chapters 10 and 11 we give a combined simple proof using the weight filtration on the logarithmic complex $\Omega^*\mathcal{L}$.

*Intersection morphisms.* Let $f: X \to V$ be a projective morphism of algebraic varieties. We express the results in terms of a Thom-Whitney stratification of $f$ (Definition 2.3) adapted to $j_{1*}\mathbf{L}$ in the sense that the restriction of the cohomology groups of $j_{1*}\mathbf{L}$, to the various strata of $X$, are locally constant. The cohomology groups of $f_*(j_{1*}\mathbf{L})$ are locally constant when restricted to the strata of $V$.

Let $D_c^b(V, \mathbb{Q})$ denote the derived category of complexes with constructible cohomology sheaves on $V$, $X_S := f^{-1}(S)$ the inverse image of a stratum $S \in V$, $f_S: X_S \to S$ the restriction of $f$, $i_{X_S}: X_S \to X$ and $i_S: S \to V$ the embeddings.

The intersection morphism in the derived category $D_c^b(V, \mathbb{Q})$

$$I: i_{X_S}^! j_{1*}^p \mathbf{L} \to i_{X_S}^* j_{1*}^p \mathbf{L} \qquad (1.6)$$

is defined by the composition of $i_{X_S}^* i_{X_S}^! j_{1*}^p \mathbf{L} \to j_{1*}^p \mathbf{L}$ with the restriction morphism $j_{1*}^p \mathbf{L} \to i_{X_S}^* i_{X_S}^! j_{1*}^p \mathbf{L}$ ([BBD 83] subsection 1.4.6 and Definition 2.1.2).

**Definition 1.1.** For each stratum $S$, the local system $\mathbf{L}_S^i$ on $S$ is the image of the induced morphism by $I$:

$$\mathbf{L}_S^i := \mathrm{Im} \left\{ R^{-\dim S + i} f_{S*}(i_{X_S}^! j_{1*}^p \mathbf{L}) \xrightarrow{I_S} R^{-\dim S + i} f_{S*}(i_{X_S}^* j_{1*}^p \mathbf{L}) \right\} \qquad (1.7)$$

such that $\mathbf{p}\mathbf{L}_S^i := \mathbf{L}_S^i[\dim S]$ is perverse.

The local systems $\mathbf{L}_S^i$ are necessarily components of any decomposition. The shift by $-\dim S$ in $R^{-\dim S + i} f_{S*}$ is motivated by the conventions on perverse sheaves. The local system $\mathbf{L}_S^i = i_S^* H^{-\ell}(\mathcal{H}^i(K))$ is uniquely defined by the
decomposition. On a stratum $S = U$ open subset of $V$ of dimension $m$, $\mathbf{L}_U^i := R^{-m+i}f_{S_*j_*!\mathbf{P}\mathbf{L}}$.

For example if $X$ is smooth of dimension $n$ and $!\mathbf{P}\mathbf{L} = \mathbb{Q}[n]$. At a point $v \in U$: $(\mathbf{L}_U^i)_v = \mathbb{H}^{n-m+i}(X_v, \mathbb{Q})$, which leads to a Hard Lefschetz formula centered at 0 on the fiber $X_v$ of dimension $n-m$.

Since $X_S \to S$ is a topological fibration $R^i f_{S_*}(Ri_{X_S}^i j_*!\mathbf{P}\mathbf{L})$ and $R^i f_{S_*}(i_{X_S}^* j_*!\mathbf{P}\mathbf{L})$ are local systems on $S$. For a general normal section $N_s$ at $s \in S$ such that $X_{N_s} := f^{-1}(N_s)$ is transversal to the strata of $X$, we have

1. $R^i f_{S_*}(Ri_{X_S}^i j_*!\mathbf{P}\mathbf{L})_s \simeq H_{X_s}^i(X_{N_s}, j_*!\mathbf{P}\mathbf{L})$ is a MHS of weight $w \ge p_w + i$

2. $R^i f_{S_*}(Ri_{X_S}^i j_*!\mathbf{P}\mathbf{L})_s \simeq H_{X_s}^i(X_{N_s}, j_*!\mathbf{P}\mathbf{L})$ is a MHS of weight $w \ge p_w + i$.

Then, $R^i i_{X_S}^i j_*!\mathbf{P}\mathbf{L}$ and $R^i f_{S_*}(i_{X_S}^* j_*!\mathbf{P}\mathbf{L})$ endowed with induced filtrations $W$ and $F$ are variations of MHS (VMHS) (§2.3) and $!\mathbf{P}\mathbf{L}_S^i$ is a shifted VHS of weight $p_w + i$ on each stratum $S$.

### 1.2.1.1 Decomposition theorem on complex varieties

Let $K := f_*j_*!\mathbf{P}\mathbf{L} \in D_c^b(V, \mathbb{Q})$ and $i_S : S \to V$ the embedding of a stratum $S$ in $V$. The cup product with the class $c_1 \in \mathbb{H}^2(X, \mathbb{Z})$ of an hyperplane section of $X$ induces morphisms ([BBD 83], Théorème 5.4.10):

$$ \eta : K \to K[2], \eta : i_{S_*}!\mathbf{P}\mathbf{L}_S^i \to i_{S_*}!\mathbf{P}\mathbf{L}_S^{i+2} \text{ and } \eta : \mathcal{P}\mathcal{H}^i(K) \to \mathcal{P}\mathcal{H}^{i+2}(K). $$

The proof in chapter 6 of the following statement of decomposition is based on the use of nearby and vanishing cycles complexes in [DeG 81], whose construction in chapters 4 and 5 extends Steenbrink's construction.

We do not use the theory of Hodge modules as in [Sa 88] and [Sa 90] but a parallel method by induction on the dimension of $X$. Instead we rely on the abelian category of IMHS ([Ka 86] Prop. 5.2.6). The theorem will be proved by a different method in chapters 10 and 11.

**Theorem 1.2.** Let $f : X \to V$ be a projective morphism of complex algebraic varieties, $!\mathbf{P}\mathbf{L}$ a shifted polarized variation of Hodge structure of weight $p_w$, $S$ a Thom-Whitney stratification of $V$ adapted to $f$, $j_*!\mathbf{P}\mathbf{L}$ and $K = f_*j_*!\mathbf{P}\mathbf{L}$.

i) There exists a decomposition of the perverse cohomology on $V$ for all $i \in \mathbb{Z}$

$$ \mathcal{P}\mathcal{H}^i(K) \xrightarrow{\sim} \bigoplus_{S \in \mathcal{S}} (i_S)_*!\mathbf{P}\mathbf{L}_S^i \quad (1.8) $$

into a direct sum of intermediate extensions of shifted polarized VHS $!\mathbf{P}\mathbf{L}_S^i$ (Equation 1.7) of weight $p_w + i$ on the strata $S$.

ii) Hard Lefschetz: The iterated cup-product with the class of a relatively ample line bundle on $X$

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

is an isomorphism over $V$ for all $i \in \mathbb{N}$. The perverse cohomology $\mathcal{P}\mathcal{H}^{-i}(K)$ is dual to $\mathcal{P}\mathcal{H}^i(K)$ and the duality is compatible with the natural decomposition in i).
iii) The (perverse) spectral sequence with respect to the perverse filtration $p_{\tau}$ on $K$ (Equation 2.2.0.1) degenerates at the term $E_1$. We have a natural decomposition into a direct sum of polarized HS of weight $pw + i + j$

$$Gr_i^{p_{\tau}} \mathbb{H}^{i+j}(X, j_* \mathbf{L}) \simeq \mathbb{H}^j(V, p\mathcal{H}^i(K)) \xrightarrow{\sim} \oplus_{S \in \mathcal{S}} \mathbb{H}^j(V, i_{S_*} p\mathbf{L}_S^i). \quad (1.9)$$

Moreover, there exists an isomorphism in the derived category of constructible sheaves $D_c^b(V, \mathbb{Q})$

$$K \simeq \bigoplus_{i \in \mathbb{Z}} p\mathcal{H}^i(K)[-i]. \quad (1.10)$$

*Remark 1.1.* i) Let $V_0$ be the union of the zero dimensional strata of $V$ and let $v \in V_0$, then by (Equation 1.8)

$H^j(i_v^! p\mathcal{H}^i(K)) \simeq \bigoplus_{S \in \mathcal{S}} H^j(i_v^!(i_S)_* p\mathbf{L}_S^i) = 0$ if $j < 0$ and $H^0(i_v^! p\mathcal{H}^i(K)) \simeq p\mathbf{L}_v^i$ \\
$H^j(i_v^{*} p\mathcal{H}^i(K)) \simeq \bigoplus_{S \in \mathcal{S}} H^j(i_v^{*}(i_S)_* p\mathbf{L}_S^i) = 0$ if $j > 0$ and $H^0(i_v^{*} p\mathcal{H}^i(K)) \simeq p\mathbf{L}_v^i$ \\
hence $p\mathbf{L}_v^i \simeq \text{Im} H^0(i_v^! p\mathcal{H}^i(K)) \to H^0(i_v^! p\mathcal{H}^i(K))$ and by Equation 1.10, we have:

$$p\mathbf{L}_v^i \simeq (\text{Im} H^i(i_v^! K) \to H^i(i_v^! K)).$$

At a general point $s \in S$, we apply this argument to the intersection with a normal section to $S$ at $s$, which explains the interest in the intersection morphisms (Equation 1.7).

ii) We have: $H^0(p\mathcal{H}^i(K)) \simeq \bigoplus_{v \in V_0} p\mathbf{L}_v^i$.

iii) The local systems $p\mathbf{L}_S^i$ are shifted polarized VHS on $S$, $p\mathbf{L}_S^{-i}$ is dual to $p\mathbf{L}_S^i$, and $\eta^i$ is an isomorphism $p\mathbf{L}_S^{-i} \simeq p\mathbf{L}_S^i$ for each $i$ and $S$.

Let $V_l := \bigcup_{S \in \mathcal{S}, \dim S \le l} S$ denote the union of the strata of $V$ of dimension $\le l$ and $k_j : (V \setminus V_j) \to V$. We have:

$$\mathrm{Im} \left( p\mathcal{H}^i(k_{j!} k_j^* K) \to p\mathcal{H}^i(K) \right) = k_{j!_*} k_j^* p\mathcal{H}^i(K) \quad (1.11)$$

iv) The decomposition in (Equation 1.10) is not canonical (see [De 93] for a preferred choice of decomposition).

v) The proof is local on $V$. Hence, along the proof we may suppose $X$ and $V$ projective varieties by reduction to a Zariski affine open subset of $V$ and then take its projective closure. Then, the proof is by induction and reduction to ample hyperplane sections.

### 1.2.1.2 Scheme of the proof

We give here a guide to follow the full proof in chapter 6, based on the necessary technical constructions in chapters 3–5. The new method of proof introduced in the text occurs in the abelian category of infinitesimal mixed Hodge structures (IMHS) defined by M. Kashiwara (ch. 2, §2.3.2) and avoid the axiomatic definition of the category of Hodge $D$-modules in Saito’s proof.
The elements necessary to read the statements such as perverse sheaf, intermediate extension, perverse truncation of a constructible complex of sheaves and perverse cohomology are recalled in chapter 2.

*Induced filtration.* First, to define the Hodge structure on perverse cohomology, we suppose the perverse truncation $p_{\tau_i}$ of $K := f_*j_*!^p \mathcal{L}$ defined by a sub-complex of $K$ (see the remark by Deligne that any morphism of complexes is homotopic to an embedding of complexes in chapter 2 §2.3.2),

Thus, the filtration $f_*F$ is induced on $p_{\tau_i}$ and then projected on $p_{\tau_i}/p_{\tau_{i-1}}$. In this way, we are entitled to apply the theory developed by Deligne in [De 72] on filtered and bi-filtered derived functor. Remark that the sub-complexes defined by the filtration $F$ are not constructible and their perverse filtration is not defined or used in the text. Another option is to use the concept of spectral objects by Verdier ([Ve 77], [De 93]).

*Induction on dim $X$.* If $X$ is singular, we consider a desingularization $\pi : X' \to X$, and then it suffices to prove the theorem for $\pi$ and $\pi \circ f$ ([De 68], Proposition 2.16) (see ch. 6, 6.2.2.2). The case $\dim X = 0$ is trivial while the case $\dim X = 1$ follows essentially from [Zu 79].

We suppose $V$ projective, $X$ smooth and projective. Let $S$ denotes a Thom-Whitney stratification of $f$ (2.1.3.1) (including one Whitney stratification of $V$ and one of $X$). The inverse image of a general section $L$ of $V$ transversal to the strata of $V$ is a non singular subvariety $L'$ of $X$. The perverse cohomology of $f_*K$ has good reduction to $L$.

Hence by induction on $\dim X$, we can deduce the theorem on a neighborhood of $L$ from the case $f': L' \to L$ where $\dim L' = \dim X - 1$.

However, such argument apply over the open subset $V^* := V \setminus V_0$ complement of the isolated strata but does not apply for $L$ through the strata of dimension zero in $V_0$ since then $L$ is not general. In this way the proof is reduced to a local problem concerning a neighborhood at each point $v \in V_0$.

*Proof via nearby and vanishing cycles.* The first proof of purity in [DeG 81], is based on the nearby cycles complex defined by a general section $L := g^{-1}(0)$ through a point $v \in V_0$, that is transversal to the strata outside $v$. In this case the vanishing cycles complex is supported by $v$.

Let $h: U \to D$ be a proper analytic morphism defined on an open analytic subset of $X$ onto a disc $D$ of $\mathbb{C} \supset 0$ such that $U_0 := h^{-1}(0)$ is a NCD in $U$. We use Hodge theory in the context of normal crossing divisors (NCD) to construct a complex $\Psi^*IL \simeq \psi_h j_*!^p \mathcal{L}$ in chapter 4 such that a Hodge filtration $F$ and the weight monodromy filtration $W$ are well defined on $\Psi^*IL$. Moreover $\text{Gr}_k^W \Psi^*IL$ decomposes into a direct sum of intermediate extensions on the smooth intersections of components (see chapter 5).

*Weight spectral sequence.* Returning to $f: X \to V$ and using a Lefschetz pencil intersecting the projective variety $V$, the proof is reduced to the case where there exist $g: V \to \mathbb{P}_C^1$, $v \in L := g^{-1}0$, $L$ is transverse to the strata of $V \setminus v$ and moreover $f^{-1}L$ is a NCD in $X$ consisting of the union of the fiber
$X_v := f^{-1}v$ and the strict transform $L'$ of $L$ (we may need many blowings
up above the fiber $f^{-1}v$ to reach this situation).

In this situation we construct the complex $\Psi^*IL \simeq \psi_{g\circ fj_*}^u p_L$ such that a
Hodge filtration $F$ and the weight monodromy filtration $W$ are well defined
on $\Psi^*IL$ (see chapters 3 and 4).

In this situation, the weight spectral sequence defined by the monodromy
filtration $W$ on the nearby cycle complex $\psi_{g\circ fj_*}^u p_L$ splits into a spectral
sequence degenerating at rank 1 and a spectral sequence of MHS degenerating
at rank 2 (see chapter 5). This is an important simplification of the proof.

Precisely, the terms $E_1$ of the weight spectral sequence involves

1. the hypercohomology of intermediate extensions on the intersections of components of the NCD $X_v$

2. the derived image of intermediate extensions on the "horizontal" smooth component $L'$ above $L$

The terms $E_1$ of the spectral sequence above $v$ form a spectral sequence of Hodge structures $E_1^{p,q}$ of weight $q$ degenerating at rank 2.

The other terms involving the decomposition of $L' \to L$ form a spectral
sequence which degenerates at rank 1 ($E_2^{p,q} = E_1^{p,q}$).

*Proof of the decomposition.* The proof is reduced to the decomposition of the vanishing cycle complex $\varphi_g j_* p_L$ by application of Verdier's classification of extensions of a perverse sheaf along a divisor [V 85-1] (see §2.2.3).

Singular X. In the case where X is singular, let $j : U \hookrightarrow X$ be a non
singular Zariski open subset, $(L, F)$ a polarized VHS on $U$, $\pi : X' \to X$
a desingularization of X and $j' : U \hookrightarrow X'$. The Hodge filtration on the
intermediate extension $j_* p_L$ on X is defined as a component of $\pi_* j'_* p_L$ by
the decomposition theorem applied to $\pi$.

Relation to the proof based on the theory of Hodge D-modules ([Sa 88] and [Sa 90]). M. Saito covers the weight spectral sequence of any $\psi_g K$; in this case the weight spectral sequence does not split, but degenerates at rank 2 in the category of Hodge modules still to define. Effectively this category is defined axiomatically and then the proof is carried coherently with the definition.

In our exposition, the category corresponding to Hodge $\mathcal{D}$-modules is not needed in the proof of decomposition since we use the degeneration at rank 2 only in the category of MHS. Moreover, since $\varphi$ is concentrated on one point, the decomposition occurs in the abelian category of IMHS ([Ka 86], 4.3, see Definition 2.14). After the proof of the decomposition, the definition of the category of perverse VHS corresponding to Hodge $\mathcal{D}$-modules is developed easily in chapter 7.

The relation between $\mathcal{D}_X$-modules and perverse complexes is given by
Kashiwara's proof that the de Rham complex of a holonomic $\mathcal{D}_X$-module
(see [PS 08] Equation (XIII-24)) is a perverse complex [Ka 74].

This basic result has been completed by Mebkhout [Me 80] and [Me 89]
into a Riemann-Hilbert correspondence between the derived category of
bounded complexes of coherent $\mathcal{D}$-modules with regular holonomic cohomology sheaves and $D_c^b(X, \mathbb{C})$ (see [PS 08], Theorem 13.65).

The intermediate extension $j_*{}^p\mathbf{L}$ in terms of $\mathcal{D}$-modules figure in ([KaK 87], Lemma 3.4.1 and [KaK 86]). The correspondence with the theory in terms of $\mathcal{D}$-modules is not given here. For more information on $\mathcal{D}$-modules (see [Sch 20], [Shi 93] and [PS 08], ch.13 and 14).

### 1.2.1.3 Perverse extension of a polarized VHS

As an application of the decomposition theorem we give the definition of the extension of a polarized VHS ($\mathbf{L}, F$) defined on a stratum $S$ of a stratified complex variety $V$ ($S$ is non singular but $V$ may be singular along $S$).

Let $j: S \to V$ denote the embedding of the locally closed subset $S$ (equivalently $j$ is the embedding into the closure $\bar{S}$ of $S$). Let ${}^p\mathbf{L} := \mathbf{L}[\dim S]$ denote the associated perverse local system (by convention we shift the degree of $\mathbf{L}$ into a complex with cohomology $\mathbf{L}$ in degree $-\dim S$). There exists a unique intermediate extension $j_*{}^p\mathbf{L}$ of ${}^p\mathbf{L}$ by a complex in the category of constructible complexes $D_c^b(V, \mathbb{C})$.

The problem is to construct the correct filtration $F$ on $j_*{}^p\mathbf{L}$, then the abelian category of VHS on singular varieties.

The category of VMHS is defined by the fact that the graded object with respect to the weight $W$ is a polarized VHS.

We give a definition of the extension of the filtration $F$ by desingularization in terms of algebraic topology constructions as follows.

**Theorem 1.3.** Let $(\mathbf{L}, F)$ be a polarized VHS on a non singular open algebraic subset $U$ of a variety $V$. There exists a canonical filtration $F$ on the intermediate extension $j_*{}^p\mathbf{L}$ on $V$ defined in the filtered derived category $D^bF(V, \mathbb{C})$.

Let $j: U \to V$ (resp. $j': U \to V'$) be the inclusion in $V$ (resp. in a desingularization $V'$ of $V$ with complement of $U$ a NCD). We define the extension $(j_*{}^p\mathbf{L}, F)$ on $V'$ by an explicit definition of a sub-complex $IC^*{}^p\mathcal{L}'$ of the derived image $\Omega^*{}^p\mathcal{L}' \simeq j'_*{}^p\mathbf{L}$ with the induced degenerated filtration $F$ in the case of a NCD. Let $\pi: V' \to V$ be the canonical morphism. We push forward $F$ into a filtration $\pi_*F$ of $\pi_*IC^*{}^p\mathcal{L}'$.

Since the perverse filtration ${}^p\tau$ is naturally defined on $\pi_*IC^*{}^p\mathcal{L}'$, the two filtrations ${}^p\tau$ and $\pi_*F$ define a bifiltered complex in the category $D^bF_2(V, \mathbb{C})$ of bifiltered complexes, then a natural induced filtration $F$ is defined on the perverse cohomology ${}^p\mathcal{H}^i(\pi_*j'_*{}^p\mathbf{L})$. We use the decomposition theorem to embed $j_*{}^p\mathbf{L}$ in ${}^p\mathcal{H}^i(\pi_*j'_*{}^p\mathbf{L})$ and to induce the filtration $F$ on $j_*{}^p\mathbf{L}$.
### 1.2.1.4 Local and global invariant cycle theorems

Basic results such as the local and global invariant cycle theorems with constant coefficients ([De 80] §3.6, [NaG 90], resp. [De 72] Corollaire 3.2.18, [De 80] §6.2.11, [Vo 2007] Théorème II.4, 24), are generalized to coefficients in an intermediate extension of polarized VHS (see ch. 7, Theorems 4.1 and 4.2), which allow us to adapt the proof of local purity [DeG 81] to complex varieties in chapter 8. The text give a faithful account of the geometrical constructions in [DeG 81].

However, in chapter 10, we see that the decomposition theorem leads to a simplification of the proof of the local purity theorem.

### 1.2.1.5 Perverse extension of polarized VMHS

The most general definition of polarized VMHS is in terms of $D$-modules by M. Saito [Sa 90]. We mention in (ch. 7, §7.1.1) similar results in terms of PVMHS under the conditions of admissibility and the existence of the relative monodromy filtration. Such conditions should coincide in general with the conditions stated in the theory of mixed Hodge modules.

### 1.2.1.6 Direct construction of the weight filtration and Relative local purity

In chapter 9, we give a direct construction of the weight filtration $W$ on the logarithmic complex $\Omega^*\mathcal{L} \simeq j_*\mathbf{L}$ in terms of the local monodromy along the components of the NCD. The weight filtration $W$ is induced by the monodromy filtration on the nearby cycles complex (see §5.2.1.4).

In chapter 10 a new combined direct geometric proof of the local purity (ch. 1, Theorem 1.1) and the decomposition theorem (ch. 1, Theorem 1.2) is given in terms of the relative version of local purity.

In particular, the proof of local purity differs from the proof in [DeG 81] and an interpretation in terms of Intersection morphisms is given at the end.
<figure><img src="image_3.png" /></figure>
# Chapter 2
## Preliminaries

We start by a summary on derived categories [Ve 77] in order to fix the notations and conventions and the nature of the isomorphisms of complexes in the text (quasi-isomorphisms).

We give a guide to basic results on perverse sheaves introduced in [BBD 83], without proofs. Namely, we recall the definitions of perverse sheaves and perverse cohomology (see §2.2).

In the next chapter, perverse sheaves with singularities along NCD are described. Even in such case one should not under estimate the theoretical study. Indeed, the theory has not been developed until the Intersection cohomology sheaf has been discovered by M. Goresky, R. MacPherson, based on the search for an auto-dual cohomology of singular spaces [GMacP 83]. An exposition on the level of graduate texts is given in [Max 19].

It is remarkable that this abelian category has been defined first in the context of derived category. Its relation to the category of $\mathcal{D}_X$-modules has been established later by J.L. Brylinski and M. Kashiwara [KaS 90], which lead to the theory of Hodge modules by M. Saito ([Sa 88]).

In view of the application to Hodge theory, we recall also results on the local structure of a degenerating polarized variation of Hodge structure (VHS) (resp. Variation of mixed Hodge structure (VMHS)) (see §2.3), which is needed to define the weight and Hodge filtrations on the de Rham complex $\Omega^*\mathcal{L}$ with coefficients in a graded polarized admissible variation of MHS on a local system.

### 2.1 Derived categories

Let $\mathcal{A}$ denote an abelian category and $C(\mathcal{A})$ the abelian category of complexes of $\mathcal{A}$. Two morphisms of complexes $\psi, \varphi: X \to Y$ are homotopic if there exists a family $h_i: X^i \to Y^{i-1}$ such that $d^{i-1} \circ h_i + h^{i+1} \circ d^i = \psi^i - \varphi^i: X^i \to Y^i$. Let $Ht(X, Y)$ denote the subgroup of morphisms $X \to Y$ homotopic to 0.
The category $K(\mathcal{A})$ is defined with the same objects of $C(\mathcal{A})$ but the morphisms of complexes are equivalence classes up to homotopy:

$$ \operatorname{Hom}_{K(\mathcal{A})}(X, Y) := \operatorname{Hom}_{\mathcal{A}}(X, Y)/\operatorname{Ht}(X, Y). $$

Homotopical morphisms $\varphi$ in a class induce on cohomology a unique family of morphisms $\varphi^i : H^i(X) \to H^i(Y)$ for $i \in \mathbb{Z}$.

*Mapping Cone.* The mapping cone of a morphism $\varphi : X \to Y$ is the complex $C(\varphi)$ defined by $d_C^n : C^n := X^{n+1} \oplus Y^n \to C^{n+1} := X^{n+2} \oplus Y^{n+1}$

$$ d_C^n = ((-1)d_X^{n+1} + \varphi^{n+1}, d_Y^n) $$

The translation functor $T$ (denoted also by $[1] : C \to C$) is defined by $(X[1])^n := X^{n+1}$ and $d_{(X[1])}^n := -d_X^{n+1}$.

There exist natural morphisms $i : Y \to C(\varphi)$ and $p : C(\varphi) \to X[1]$ satisfying the following sequence of morphisms

$$ X \xrightarrow{\varphi} Y \xrightarrow{i} C(\varphi) \xrightarrow{p} X[1] $$

called a distinguished triangle. We remark that there exists a morphism in the category $K(\mathcal{A})$: $\phi : X[1] \to C(i)$ such that we have an isomorphism of sequences

$$ \begin{array}{ccc} Y & \xrightarrow{i} & C(\varphi) \xrightarrow{p} X[1] \xrightarrow{-\varphi[1]} Y[1] \\ \big|_{id} & & \big|_{id} & & \big|_{\phi} & \big|_{id} \\ Y & \xrightarrow{i} & C(\varphi) & \xrightarrow{i'} & C(i) & \xrightarrow{p'} Y[1] \end{array} $$

We refer to ([KaS 90] Lemma 1.4.2) for a proof. The morphism $\phi$ is not unique and it is an isomorphism in $K(\mathcal{A})$ (up to homotopy), but not in $C(\mathcal{A})$. As a consequence, $X[1]$ may be constructed as the cone $C(i)$. A distinguished triangle is embedded in a turning sequence of distinguished triangles.

*Remark 2.1.* i) Let $\varphi$ and $\psi$ be two morphisms: $X \to Y$. Given an homotopy $h$ from $\varphi$ to $\psi$, we can deduce an isomorphism $C(h) : C(\varphi) \to C(\psi)$ depending on $h$, hence the construction of the cone in $K(\mathcal{A})$ is unique up to an isomorphism but this isomorphism is not unique.

ii) Since the morphism $\varphi$ is defined up to homotopy, the choice of $C(\varphi)$ is not natural which is a problem in the context of mixed Hodge complexes (MHC) [De 72]. In this case, there exists a problem of compatibility between the rational and complex level of construction.

To obtain the naturality of the mixed Hodge structure, Deligne used the concept of simplicial varieties to avoid compatibility problems.

iii) *Second homotopy.* If $h$ and $h'$ are two homotopies from $\varphi$ to $\psi$, an homotopy from $h$ to $h'$ leads to an isomorphism of diagrams between the corresponding isomorphisms $C(h)$ and $C(h')$.

iv) The category $K(\mathcal{A})$ is not abelian ([Max 19] Example 4.5.3).
The derived category $D(\mathcal{A})$.

The classical cohomology theories of a topological space $X$ may be defined as derived functors of the global section functor $F := \Gamma(X, \cdot)$ on the category of sheaves on $X$ [G 58].

A functor $F : \mathcal{A} \to \mathcal{B}$ of abelian categories extends to a functor $F^\bullet : \mathcal{C}(\mathcal{A}) \to \mathcal{C}(\mathcal{B})$ defined by $F^\bullet(X^\bullet) := Y^\bullet$ where $Y^i := F(X^i)$. A morphism of complexes $\varphi : X^\bullet \to Y^\bullet$ is called a quasi-isomorphism if it induces isomorphisms on cohomology in each degree. We remark that the image of a quasi-isomorphism is not a quasi-isomorphism unless $F$ is exact.

When $\mathcal{A}$ has enough injectives (every object of $\mathcal{A}$ may be embedded in an injective object of $\mathcal{A}$) and $F$ is left exact, for any bounded below complex $X^\bullet$ there exists an injective resolution (a quasi-isomorphism $X^\bullet \to I^\bullet$ and $F^\bullet$ transforms a quasi-isomorphism of complexes of injective objects $I^\bullet \to J^\bullet$ into a quasi-isomorphism.

To construct of the $i$-th satellite functor $R^i F$ (we should write $R^i F^\bullet$), we fix an injective resolution of $X^\bullet$, that is a quasi-isomorphism $X^\bullet \to \mathcal{I}^\bullet$ and define $R^i F(X^\bullet) := H^i(F(\mathcal{I}^\bullet))$ (we should write $R^i F^\bullet(X^\bullet)$).

To construct in general the derived category $D(\mathcal{A})$, we remark that a quasi-isomorphism is not necessarily an isomorphism in $K(\mathcal{A})$.

The purpose of derived categories is to construct a category of complexes where the quasi-isomorphisms are isomorphisms. The right derived functor $RF$ of a left exact functor $F$ is defined as a complex $RF(X) := F(\mathcal{I}^\bullet)$ and not merely its cohomology objects: the satellites $R^i F$. If $F$ is right exact we consider projective resolutions $\mathcal{P}^\bullet \to X$ and define the left derived functor $LF$ by $F(\mathcal{P}^\bullet)$.

Later in the text we often denote also by $F$ the derived functor.

The class of quasi-isomorphisms $X \to Y$ form a multiplicative system $S$ in the sense of Verdier [Ve 77]. By a construction, similar to the construction of $\mathbb{Q}$ out of $\mathbb{Z}$, it is possible to construct a derived category $D(\mathcal{A})$ with the same objects as $K(\mathcal{A})$ but with group of morphisms obtained by inverting quasi-isomorphisms: there exists a functor $Q : K(\mathcal{A}) \to D(\mathcal{A})$ such that the image of a quasi-isomorphism $X \to Y$ is an isomorphism in $\operatorname{Hom}_{D(\mathcal{A})}(X, Y)$.

An important consequence is that any short exact sequence in $K(\mathcal{A})$ is isomorphic to a distinguished triangle in $D(\mathcal{A})$ ([KaS 90] proposition 1.7.5).

### 2.1.0.1 The complex of Homomorphisms $\operatorname{Hom}^\bullet(\cdot, \cdot)$

We mention the following bifunctor on an abelian category $\mathcal{A}$ to the category of abelian groups: $\operatorname{Hom}(\cdot, \cdot) : \mathcal{A}^\circ \times \mathcal{A} \to \operatorname{Ab} : (X, Y) \mapsto \operatorname{Hom}(X, Y)$.

If $X^\bullet$ and $Y^\bullet$ are complexes, we define the complex $\operatorname{Hom}^\bullet(X^\bullet, Y^\bullet)$ by

$$ \operatorname{Hom}^n(X^\bullet, Y^\bullet) = \prod_{p \in \mathbb{Z}} \operatorname{Hom}(X^p, Y^{p+n}) \qquad (2.1) $$
$$d^n = \prod_{p \in \mathbb{Z}} d_p^n, \quad d_p^n : \operatorname{Hom}(X^p, Y^{p+n}) \to \operatorname{Hom}(X^p, Y^{p+n+1}) : d_p^n(\varphi_p) = d_{Y^{p+n}} \circ \varphi_p + (-1)^{n+1} \varphi_{p+1} \circ d_{X^{p+n}}.$$

The group $H^i(\operatorname{Hom}^\bullet(X^\bullet, Y^\bullet)) := H^0(\operatorname{Hom}^\bullet(X^\bullet, Y^\bullet)[i])$ may be identified with the group of classes of morphisms of complexes of degree $i$ modulo the sub-group of morphisms homotopic to zero.

Given a complex $X^\bullet := (X^p, d_p)_{p \in \mathbb{Z}}$, the complex $(X^\bullet)^\bullet \in C(\mathcal{A})^\bullet$ is defined by $(X^\bullet)^r := X^{-r}$ and the complex $\operatorname{Hom}^\bullet$ is defined by the double complex $S^{p,q} := \operatorname{Hom}(X^{-p}, Y^q)$. We deduce a functor defined modulo homotopy of complexes:

$$\operatorname{Hom}^\bullet : K(\mathcal{A})^\bullet \times K(\mathcal{A}) \to K(\mathcal{Ab}).$$

If $Y^i$ is injective for all $i$, then $\operatorname{Hom}^\bullet(\ast, Y^\bullet)$ transforms quasi-isomorphisms into quasi-isomorphisms. Hence, if $\mathcal{A}$ admits enough injective objects, we deduce a bifunctor $\operatorname{Hom}^\bullet : D(\mathcal{A})^\bullet \times D^+(\mathcal{A}) \to D(\mathcal{Ab})$ (this bifunctor may be defined also by projective resolutions on the left). We deduce a right derived functor $\operatorname{RHom}^\bullet(\ast, \ast) : (D(\mathcal{A})^\bullet \times (D^+(\mathcal{A}) \to D(\mathcal{Ab})$ such that:
$$\operatorname{Hom}_{D(\mathcal{A})}(X^\bullet, Y^\bullet) = H^0(\operatorname{RHom}^\bullet(X^\bullet, Y^\bullet)) := \operatorname{Ext}^0_{\mathbb{Z}}(X^\bullet, Y^\bullet) \text{ and} \\ \operatorname{Ext}^i_{\mathbb{Z}}(X^\bullet, Y^\bullet) = H^0(\operatorname{RHom}^\bullet(X^\bullet, Y^\bullet)[i]) \text{ for all } i \in \mathbb{Z}.$$

### 2.1.0.2 Triangulated categories

We list now some properties of the derived category $D(\mathcal{A})$. It is an advantage to abstract these properties into a set of axioms and define directly the concept of triangulated categories.

Let $\mathcal{C}$ be an additive category with a translation functor [1]. A triangle of $\mathcal{C}$ is defined as a sequence of morphisms $X \to Y \to Z \to X[1]$.

**Definition 2.1.** i) A triangulated category consists of an additive category $\mathcal{C}$ with a translation functor and a family of triangles called distinguished triangles satisfying four axioms TR1-4 stated in [Ve 77] such that any triangle isomorphic to a distinguished triangle is in the family.

ii) An additive functor $F : \mathcal{C} \to \mathcal{A}$ to an abelian category is called a cohomological functor if for any distinguished triangle $X \to Y \to Z \to X[1]$, the sequence $F(X) \to F(Y) \to F(Z)$ is exact.

Set $F^k := F \circ T^k$, a long exact sequence follows from the axioms:

$$\dots F^{-1}(Z) \to F(X) \to F(Y) \to F(Z) \to F^1(X) \to F^1(Y) \to F^1(Z) \dots$$

More about triangulated functors and the long exact sequence of cohomology groups defined by a triangle figure in ([KaS 90] §1.7).

### 2.1.0.3 Derived category of sheaves $D^+(X, A)$ on a locally compact space $X$

Let $A$ a commutative Noetherian ring (for some results we need to add that $A$ has finite homological dimension but we may suppose $A = \mathbb{Z}$ or $\mathbb{Q}$). We
mention classical functors on the category of sheaves $Sh(X, A)$ ($Sh(X)$ for short) and refer to a basic reference such as [G 58] or to [KaS 90] for a full study.

1) Tensor product $\ast \otimes \ast$ in $\mathcal{A} := Sh(X, A)$:

Let $\otimes_Z : \mathcal{A} \times \mathcal{A} \to \mathcal{A} : (B_1, B) \mapsto B_1 \otimes_A B$ denote the tensor product in $\mathcal{A}$.
The complex tensor product of two complexes $B_1^\bullet$ and $B^\bullet$ in $K^-\mathcal{A}$ is

$$B_1^\bullet \otimes_A B^\bullet \in K^-\mathcal{A} \text{ defined by } (B_1^\bullet \otimes_A B^\bullet)^n = \sum_{p+q=n} A^p \otimes_A B^q$$

with differential $d := d_{B_1} + (-1)^n d_B$. This bifunctor is well defined up to homotopy and it is right exact.

If $F$ is a flat sheaf, the functor $F \otimes \ast$ is exact on $\mathcal{A}$. Since there exists flat resolutions in $D^-\mathcal{A}$, we deduce the left derived functor

$$\otimes_Z^L : D^-\mathcal{A} \times D^-\mathcal{A} \to D^-\mathcal{A}.$$

2) $f_*$ and $f^*$. Let $f : X \to V$, the direct image of a sheaf $F$ on $X$ is a sheaf denoted $f_*F$ on $V$. The functor $f_*$ is left exact with a derived functor $Rf_*$ often denoted also by $f_*$.

The inverse image of a sheaf $G$ on $V$ is a sheaf denoted $f^*G$ on $X$. The functor $f^*$ is exact and its derived functor $Rf^*$ is often denoted by $f^*$ (we use the notation $f^*$ instead of $f^{-1}$ on the level of the category $Sh(V, A)$).

We have an isomorphism of groups $\operatorname{Hom}(G, f_*F) \simeq \operatorname{Hom}(f^*G, F)$. It is an example of the notion of adjoint functors: we say that $f^*$ is left adjoint to $f_*$ and $f_*$ is right adjoint to $f^*$. On the level of morphisms of sheaves, we have: $\operatorname{Hom}(G, f_*F) \simeq f_*\operatorname{Hom}(f^*G, F)$.

In the case of a a subset $Z$ of $X$, let $j : Z \hookrightarrow X$. We write

$$F|_Z := j^*F \text{ and } \Gamma(Z, F) := \Gamma(Z, j^*F).$$

3) *Direct image with proper support* $f_1$. The support of a section $s$ of a sheaf $F$ on $U$ is the complementary $S$ in $U$ of the open subsets $V \subset U$ such that $s|_V = 0$, hence $S := \text{supp}_U s$ is closed in $U$:

$$\text{supp}_U s := \{x \in U : s_x \neq 0\}$$

The support of a sheaf $F$ on $X$, is the subset $S \subset X$ complementary of open sets $U \subset X$ such that $F|_U = 0$, hence $S := \text{supp} F$ is closed in $X$.

Let $f : X \to V$ be a continuous map. We say that $f$ is proper if the image of a closed set of $X$ is closed in $V$, the fibers of $f$ are compact and two distinct points in a fiber have two disjoint neighborhoods in $X$. If $X$ and $V$ are locally compact, $f$ is proper if the inverse of any compact set in $V$ is compact in $X$. The sub-sheaf $f_!F \subset f_*F$ is defined by setting for open sets $U \subset V$:

$$\Gamma(U, f_!F) := \{s \in F(f^{-1}U) : f|_{\text{supp } s} : \text{supp } s \to U \text{ is proper}\}$$

This sheaf is called the direct image with proper supports.

In the case where $V$ is reduced to a unique point, we write

$$\Gamma_c(X, F) := \{s \in \Gamma(X, F) : \text{supp } s \text{ is compact and Hausdorff}\}.$$

If $X$ and $V$ are locally compact then: $(f_!F)_x := \Gamma_c(f^{-1}(x), F)$.

4) *Extension by 0*. Let $Z$ be locally closed in $X$, $j : Z \to X$ and $F$ a sheaf on $Z$, then the sheaf $j_!F$ on $X$ is the unique sheaf on $X$ satisfying:
$$ (j_!F)|_Z = F, (j_!F)|_{X\setminus Z} = 0 \qquad (2.2) $$

If $j$ is a closed immersion $j_!F = j_*F$.

**Remark 2.2.** Let $F^\bullet$ and $G^\bullet$ be two bounded complexes of sheaves on $X$ such that $H^i(F^\bullet) = 0$ for $i > n$ and $H^i(G^\bullet) = 0$ for $i \le n$, then the following natural map is an isomorphism ([GMacP 83] Proposition 1.15)

$$ \operatorname{Hom}_{D^b(X)}(F^\bullet, G^\bullet) \xrightarrow{\sim} \operatorname{Hom}_{\operatorname{Sh}(X)}(H^n(F^\bullet), H^n(G^\bullet)) $$

### 2.1.1 Poincaré - Verdier duality

Sheafified version of the duality. Let $F$ and $G$ be two sheaves on a topological space $X$. We define the presheaf $\operatorname{Hom}_A(F, G)$ (which is already a sheaf) by the correspondance : $U \mapsto \operatorname{Hom}_A(F|_U, G|_U)$.

The bifunctor $\operatorname{Hom}_{\operatorname{Sh}(X,A)}(*, *)$ is defined by

$$ (F, G) \to \operatorname{Hom}_A(F, G) \in \operatorname{Sh}(X, A). $$

Let $C^+(X, A)$ denote the category of sheaves bounded below complexes, $F^\bullet$ and $G^\bullet$ be in $C^+(X, A)$, we deduce from (Equation 2.1) the definition of the complex of sheaves $\operatorname{Hom}^\bullet(F^\bullet, G^\bullet) \in C^+(X, A)$.

The derived bifunctor of $\operatorname{Hom}_{\operatorname{Sh}(X,A)}(*, *)$ is defined by taking injective resolutions of $G$ on the right side or projective resolutions on the left side. We deduce

$$ \begin{gathered} (D^+(X, A))^\circ \times D^+(X, A) \ni (F, G) \to R\operatorname{Hom}^\bullet(F, G) \in D^+(X, A) \\ R\operatorname{Hom}^\bullet(F, G) \simeq R\Gamma(X, R\operatorname{Hom}^\bullet(F, G)) \qquad (2.3) \\ \operatorname{Hom}_{D^+(X,A)}(F,G[n]) = \mathbb{H}^n(X, R\operatorname{Hom}^\bullet(F, G)) \end{gathered} $$

For $F$ the constant sheaf defined by $A$, $\operatorname{Hom}_{D^+(X,A)}(A_X, G[n]) = \mathbb{H}^n(X, G)$.

**Theorem 2.1 (Duality [Ve 65]).** Let $f: X \to V$ be a continuous map of locally compact spaces. Assume $f_1$ has finite cohomological dimension. There exists a functor $f_1^!: D^+(V, A) \to D^+(X, A)$ right adjoint to the functor $Rf_1!: D^+(X, A) \to D^+(V, A)$ satisfying for $F \in \operatorname{Ob}(D^+(X, A))$ and $G \in \operatorname{Ob}(D^+(V, A))$:

$$ R\operatorname{Hom}^\bullet(Rf_1!F, G) \simeq Rf_*\operatorname{Hom}^\bullet(F, f^!G) \text{ in } D^+(V, A) \qquad (2.4) $$

$$ \operatorname{Hom}_V^\bullet(Rf_1!F, G) \simeq \operatorname{Hom}_X^\bullet(F, f^!G) \qquad (2.5) $$

$$ \operatorname{Hom}_{D^+(V,A)}(Rf_1!F, G) \simeq \operatorname{Hom}_{D^+(X,A)}(F, f^!G) \qquad (2.6) $$

To summarize the statement $f^!$ is called right adjoint to $Rf_1$.

In particular, we recover Equations 2.5 and 2.4 by taking $R\Gamma(V, *)$ (resp. $R^0\Gamma(V, *)$) in Equation 2.4. We refer to [Ve 65] and [KaS 90] for the construction of $f^!$ and the proof of the duality isomorphism.
The equation 2.6 helps to define $f^!$ directly on the level of complexes. For any open subset $U \subset X, j: U \subset X$, set $F = j_!\mathbb{Z}_U$; we have:

$$R\Gamma(U, f^!G) := R\operatorname{Hom}_X^\bullet(j_!\mathbb{Z}_U, f^!G) \simeq R\operatorname{Hom}_V^\bullet(Rf_!j_!\mathbb{Z}_U, G).$$

**Definition 2.2 (dualizing complex).** i) Let $f: X \to \{e\}$ be the map to a point. The complex $f^!A \in D^+(X, A)$ is denoted by $\mathbb{D}_{X,A}$ or simply $\mathbb{D}_X$ and called the dualizing complex on $X$.

ii) The (Verdier) dual of a complex of sheaves $F \in \mathcal{O}bD^+(X, A)$ is:

$$\mathcal{D}_X F := R\operatorname{Hom}^\bullet(F, \mathbb{D}_X) = R\operatorname{Hom}^\bullet(F, f^!A)$$

$\mathcal{D}_X : D^+(X, A) \to D^+(X, A) : F \mapsto \mathcal{D}_X F$ is called the dualizing functor.

*Properties of the dualizing functor.* Let $f: X \to V$, it follows from duality and the adjoint properties: for all $F \in \mathcal{O}b^+(X, A)$, $G \in \mathcal{O}b^+(V, A)$

$$\mathcal{D}_X f^*G \simeq f^!(\mathcal{D}_V G), \mathcal{D}_V(f_*F) \simeq f_!(\mathcal{D}_X F) \qquad (2.7)$$

*Example 2.1.* i) In the case of an oriented manifold $X$ of (real-)dimension $n$, $A = \mathbb{Q}$ and $V$ is reduced to a point

$$\mathbb{D}_X := f^!\mathbb{Q} = \mathbb{Q}_X[n], \mathcal{D}_X F := R\operatorname{Hom}^\bullet(F, \mathbb{Q}_X[n])$$

and for any open subset $U \subset X$, by duality:

$$\operatorname{Hom}_{\mathbb{Q}}^\bullet(R\Gamma_c(U, F), \mathbb{Q})) \simeq R\Gamma(U, \mathcal{D}_X F), \mathbb{H}^{-q}(U, \mathcal{D}_X F) \simeq \operatorname{Hom}_{\mathbb{Q}}(\mathbb{H}_c^q(U, F), \mathbb{Q}).$$

ii) Let $A = \mathbb{Q}$ and $F := \mathcal{L}$ a local system, then $R\operatorname{Hom}^\bullet(\mathcal{L}, \mathbb{Q}_X)$ is reduced to a unique sheaf $\check{\mathcal{L}} := \operatorname{Hom}(\mathcal{L}, \mathbb{Q}_X)$. Taking the $(-i)$-th group of cohomology, we have Poincaré duality:

$$\operatorname{Hom}_{\mathbb{Q}}(H_c^i(X, \mathcal{L}), \mathbb{Q}) \simeq H^{n-i}(X, \check{\mathcal{L}})$$

**Proposition 2.1.** ([Ve 76b] §1.2) Let $\phi$ be a paracompact family of support and $U$ an open subset of a complex algebraic variety, there exist isomorphisms:

$$H_p^\phi(U, \mathbb{Z}) \xrightarrow{\sim} \mathbb{H}_{p,\phi}^{BM}(U, \mathbb{Z}) \xrightarrow{\sim} \mathbb{H}_\phi^{-p}(U, \mathbb{D}_{X,\mathbb{Z}})$$

of the homology groups (resp. Borel Moore homology) with support in $\phi$ and the cohomology of $\mathbb{D}_{X,\mathbb{Z}}$ with support in $\phi$.

### 2.1.2 Grothendieck - Serre duality of coherent sheaves

The development of duality in the frame of derived category started earlier with Grothendieck’s extension of Serre duality [Gro 57]. Let $V$ be a projective complex variety and $D^+(V, \mathcal{O})$ the category of sheaves of $\mathcal{O}$-modules.

**Theorem 2.2 (Grothendieck).** ([Gro 57] Proposition 5) Let $Y \subset V$ be an algebraic subset of dimension $n-p$ in a non singular complex algebraic variety $V$ of dimension $n$ and $F$ a coherent module on $Y$, then:
i) The sheaves $\mathcal{E}xt_{\mathcal{O}_V}^{p+i}(F, \Omega_V^n)$ vanish for $i < 0$, and are independent of the choice of $V$ and the embedding of $Y$ for $i \ge 0$.

$$ \text{ii)} \quad \mathcal{E}xt_{\mathcal{O}_V}^p(F, \Omega_V^n) \simeq \operatorname{Hom}_{\mathcal{O}_Y}(F, \mathcal{E}xt_{\mathcal{O}_V}^p(\mathcal{O}_Y, \Omega_V^n)). $$

iii) If $Y$ is non singular of dimension $n - p$ in $V$ of dimension $n$:

$$ \mathcal{E}xt_{\mathcal{O}_V}^{p+i}(\mathcal{O}_Y, \Omega_V^n) = 0 \text{ if } i \neq 0 \text{ and } \mathcal{E}xt_{\mathcal{O}_V}^p(\mathcal{O}_Y, \Omega_V^n) \simeq \Omega_Y^{n-p}. $$

Moreover: $\mathbb{H}^n(V, \Omega_V^n)$, of rank 1, is generated by the cohomology class of $V$.

iv) Duality for coherent sheaves: Let $F$ be coherent on $V$, then:

$$ \mathrm{Ext}_{\mathcal{O}_V}^{n-p}(F, \Omega_V^n) \simeq (\mathrm{Hom}_{\mathbb{C}}(\mathbb{H}^p(V, \Omega_V^n), \mathbb{C})) $$

With the notation of the theorem above, Grothendieck introduced the concept of dual with value in a "dualizing complex $\mathbf{K}_Y := \mathcal{R}\mathcal{H}om_X^\bullet(\mathcal{O}_Y, \Omega_V^n[n])$ which is a basic improvement of the statements i) - iii) involving extension groups $\mathcal{E}xt^*(\mathcal{O}_Y, \mathbf{K}_X)$ ([Gro 58], [Gro 66]), [Ha 66] chapters III, VI).

For this purpose, it was necessary to construct the derived category ([Ve 77]), in order to state duality theorems in general.

In the case of a proper morphism $f: X \to Y$ of locally Noetherian preschemes and a complex of modules $F^\bullet$ on $X$ with quasi-coherent cohomology, the duality theorem in the category $D_{qc}(X)$ is stated as follows [Ha 66] chapter VII, §3):

$$ \mathcal{D}_X F^\bullet := \mathcal{R}\mathcal{H}om_X^\bullet(F^\bullet, \mathbf{K}_X) \simeq (\mathcal{R}\mathcal{H}om_Y^\bullet(Rf_*F^\bullet, \mathbf{K}_Y)) := \mathcal{D}_Y Rf_*F^\bullet $$

### 2.1.3 Constructible sheaves

#### 2.1.3.1 Thom-Whitney stratification $\mathcal{S}$ of $f$

Let $V$ be a complex algebraic variety, and $\mathcal{S} = (S_\alpha)$ an algebraic Whitney stratification by a family of locally closed algebraic non singular sub-varieties called strata $S_\alpha$ of $V$ ([LeT 83], [GMacP 88], [Ve 76]). The subspaces $V_l = \cup_{\dim S_\alpha \le l} S_\alpha$ form an increasing family of closed algebraic subsets of $V$ of dimension $\le l$, with index $l \le n$, where $n$ is the dimension of $V$,

$$ V_{-1} = \emptyset \subset V_0 \subset V_1 \subset \cdots \subset V_{n-1} \subset V_n = V \qquad (2.8) $$

$(V_i \setminus V_{i-1}$ is a union of strata of $\mathcal{S}$ of dimension $i$ but $V$ may be singular along $S_\alpha$). In addition, whenever $S_\alpha \subseteq \bar{S}_\beta$ Whitney conditions are satisfied in order to guarantee that the variety $V$ is topologically equisingular along each stratum.

Let $f: X \to V$ be an algebraic map. If $Z \subset V$ is a sub-space of $V$, let $X_Z := f^{-1}(Z)$, in particular for each strata $S$ of $\mathcal{S}$ let $X_S = f^{-1}(S)$.

**Definition 2.3.** A Thom-Whitney stratification of the algebraic map $f: X \to V$ consists of two Whitney stratifications one for $V$ and one for $X$ such that the inverse $X_S$ of each stratum $S$ of $V$ is a union of connected strata of
**Lemma 2.1 (Properties of a constructible complex).** Let $F \in D_c^b(V, \mathbb{Q})$ and $B_\epsilon(x)$ a small ball with center $x$

$$ \begin{align} \mathbb{H}^q(B_\epsilon(x) \setminus x, F) &\simeq \mathbb{H}^q(\partial B_\epsilon(x), F) \nonumber \\ \mathcal{H}^q(F)_x &\simeq H^q(i_x^*F) \simeq \mathbb{H}^q(\dot{B}_\epsilon(x), F) \simeq \mathbb{H}^q(B_\epsilon(x), F) \tag{2.9} \\ \mathbb{H}^q(i_x^!F) &= \mathbb{H}_x^q(X, F) = H_c^q(\dot{B}_\epsilon(x), F) \simeq \mathrm{Hom}_{\mathbb{Q}}(H^{-q}(i_x^*\mathcal{D}_X F) \nonumber \end{align} $$

The boundary $\partial B_\epsilon(x)$ of a general ball intersect transversally the strata. The first isomorphism follows from the local topological invariance due to constructibility and a deformation of $B_\epsilon(x) \setminus x$ onto $\partial B_\epsilon(x)$ ([Max 19] §7.2).

*Dévissage.* Let $j_i : V_i \setminus V_{i-1} \to V$, then the sheaf $j_i!F|_{V_i \setminus V_{i-1}}$ (extension by zero) is constructible and there exists on $F$ a filtration by constructible sub-sheaves with successive quotients of the type $j_i!F|_{V_i \setminus V_{i-1}}$.

We refer to [Ve 76b] for more properties of constructible sheaves defined on analytic varieties.

**Proposition 2.2.** i) The extension $\mathcal{E}xt_{A_X}^i(F, G) := R^i\mathrm{Hom}_{A_X}^\bullet(F, G)$ of two constructible sheaves is constructible.

ii) The dualizing complex $D_X$ is constructible.

iii) Let $f : X \to V$ be a submersive morphism of analytic varieties, then: for $G \in \mathrm{Ob}D^+(V, A)$, $f^!G \simeq f^*G \otimes^L f^!A_V$.

iv) Let $f : X \to V$ be an algebraic map, and $K$ a constructible complex of sheaves on $X$, then $f_*K$ is constructible on $V$ ($R^qf_*K$ are constructible sheaves). In particular $Rf_*\mathbb{Q}_X$ is constructible on $V$.

In general proofs on algebraic varieties are easier. The proof of i) is by dévissage. Since locally $X$ can be embedded in a non singular variety $Z$, we deduce ii) from i) and the lemma

**Lemma 2.2.** If $X \subset Z$ is closed in $Z$, then $\mathbb{D}_X \simeq R\mathrm{Hom}^\bullet(\mathbb{Z}_X, \mathbb{D}_Z)$.

iii) If $f = \mathrm{pr}_V : Z \times V \to V$, $\mathrm{pr}_Z$ is the projection on $Z$ and $g : Z \to \{e\}$ is the projection to a point $f^!A_V \simeq \mathrm{pr}_Z^*g^!A_V$ ([Ve 65] §5).

We deduce iv) from the existence of Thom-Whitney stratifications.

### 2.1.3.3 Intermediate extensions

To motivate the definition of perverse sheaves later, we state here a definition of the intermediate extension of a local system given on a Zariski open subset. The topological equisingularity along a stratum is the basic property used in Intersection cohomology [GMacP 83]. By the *local normal triviality* property of the stratification in (Equation 2.8) at $x \in V_i \setminus V_{i-1}$, there exists a neighbourhood $U_x \subset V$ and a compact $2n - 2i - 1$ dimensional topologically stratified space

$$ L = L_{2n-2i-1} \supset \dots \supset L_3 \supset L_1 \supset L_{-1} = \emptyset \tag{2.10} $$
with open cone $\check{c}L := L \times [0, 1)/L \times \{0\}$ and a homeomorphism

$$ \varphi: U_x \xrightarrow{\sim} \mathbb{R}^{2i} \times \check{c}L \qquad (2.11) $$

since $V_i \setminus V_{i-1}$ is a topological manifold of dimension 2i.

*Deligne's construction of the intermediate extension.* Let $i \ge 1$ and

$$ U_n = V \setminus V_{n-1}, \quad U_0 = V, \quad U_i = V \setminus V_{i-1}, \quad j_i: U_i \hookrightarrow U_{i-1}, $$

**Definition 2.6 (intermediate extension).** Let $\mathbf{L}$ be a local system on $U_n$ and set $\mathbf{p}\mathbf{L} := \mathbf{L}[n]$. We define the intermediate extension $j_{i_*}\mathbf{p}\mathbf{L}$ on $V$ inductively (here $j_{i_*} := Rj_{i_*}$)

$$ j_{i_*} \mathbf{p}\mathbf{L} := \tau_{\le -1} j_{i_*} \circ \cdots \circ \tau_{\le -i} j_{i_*} \cdots \circ \tau_{\le -n} j_{n_*} \mathbf{p}\mathbf{L} \qquad (2.12) $$

Let $S \subset V_i \setminus V_{i-1}$ be a non singular stratum and $x \in S \subset U_i \setminus U_{i+1}$. There exists a local embedding $U_x \subset \mathbb{C}^N$ and a complex linear subspace $H_x \subset \mathbb{C}^N$ normal to $S$ at $x$, intersecting the strata of $V_j$ near $x$ transversally. We can suppose $U_x$ as in equation 2.11 and $\check{c}L = U_x \cap H_x$. We have a diagram

$$ B_x \cap S \xrightarrow{\sigma_i} B_x \cap U_i \xleftarrow{j_{i+1}} (B_x \cap U_{i+1}) \setminus S \quad \text{where } j_{i+1} : U_{i+1} \cap U_x \to U_i \cap U_x $$

**Lemma 2.3.** i) $(\mathcal{H}^k(j_{i_*} \mathbf{p}\mathbf{L}))|_{V_i} = 0$ if $k \ge -i, \forall i \ge 0$.

Precisely, if $x \in S \subset V_i \setminus V_{i-1}$:

$$ (\mathcal{H}^k(j_{i_*} \mathbf{p}\mathbf{L}))_x = \mathbb{H}^k(L_{2n-2i-1}, j_{i+1_*} \mathbf{p}\mathbf{L}) \text{ if } k \le -i-1, 0 \text{ if } k \ge -i. $$

ii) Duality: $\mathcal{D}j_{i_*} \mathbf{p}\mathbf{L} \simeq j_{i_*} \mathcal{D} \mathbf{p}\mathbf{L}$. If $\mathbf{p}\mathbf{L}$ is auto-dual, the intermediate extension is auto-dual, which is the case of a polarized local system. In particular: $\mathcal{H}^k i'_i(j_{i_*} \mathbf{p}\mathbf{L}) = 0$ if $k \le -i, \forall i \ge 0$.

i) We have $U_{i+1} \cap U_x = U_x \setminus S$. Let $j'_{i+2} : U_x \setminus (V_{n-1} \cap U_x) \to U_{i+1} \cap U_x$ and $j'_{i+1} : U_x \setminus (V_{n-1} \cap U_x) \to U_x$. We have by induction:

$$ (j_{i_*} \mathbf{p}\mathbf{L})|_{U_x} = j'_{i+1_*} \mathbf{p}\mathbf{L} = \tau_{\le -i-1} j_{i+1_*} (j'_{i+2_*} (\mathbf{p}\mathbf{L}|(U_x \setminus (V_{n-1} \cap U_x)))) = \tau_{\le -i-1} j_{i+1_*} \circ \cdots \circ \tau_{\le -n} j_{n_*} (\mathbf{p}\mathbf{L}|(U_x \setminus (V_{n-1} \cap U_x))). $$

By definition $(j_{i+1_*}j'_{i+2_*}(\mathbf{p}\mathbf{L}|U_x))_x \simeq R\Gamma(U_x \setminus S \cap H_x, j'_{i+2_*}(\mathbf{p}\mathbf{L}|U_x))$

which, by constructibility and retraction of $U_x \setminus S$ onto $\check{c}L_{2n-2i-1} \setminus x$ and finally onto $L_{2n-2i-1}$, reduces to the hypercohomology of the link:

$$ R\Gamma(\check{c}L_{2n-2i-1} \setminus x, j_{i+1_*}(\mathbf{p}\mathbf{L}|U_x)) \simeq R\Gamma(L_{2n-2i-1}, j_{i_*} \mathbf{p}\mathbf{L}). $$

The lemma i) follows after truncation.

ii) The properties here give a characterization of the intermediate extension and apply to the dual complex ([GMacP 83] §6, [Max 19] 6.2.7 and 6.4.1).

**Example 2.2.** Let $\mathbf{L}$ be defined on $(\mathbb{C}^*)^n \xrightarrow{j} \mathbb{C}^n$ by a vector space $L$ and $n$ nilpotent endomorphisms, then the fiber $i_x^*j_{i_*} \mathbf{L}$ at $x = 0$ is defined by the complex

$$ L \to \dots \to N_{k_1} L \to \bigoplus_{k_1 < \dots < k_i} N_{k_1} \to \dots \to N_{k_{i-1}} L \to \dots \to N_{k_{i-1}} \to N_{k_i} L $$
with differentials defined by $N_r$ for $r \notin \{k_1, \dots, k_i\}$ (see Definition 3.4). The dual complex is defined by various embedding $N_r^* N_{k_1}^* \cdots N_{k_i}^* L^* \hookrightarrow N_{k_1}^* \cdots N_{k_i}^* L^*$. Recall that by duality $\operatorname{Hom}_{\mathbb{Q}}(H^{-i}(i_x^* j_*!^p \mathbf{L}), \mathbb{Q}) = H^i(i_x^! j_*!^p \check{\mathbf{L}})$.

The above constructions including the direct image $j_{i*}$ and the truncation $\tau_{\le -i}$ along the strata and the properties stated in the lemma give a motivation to the next definition of perverse sheaves.

## 2.2 Perverse sheaves

From now on, we write $F$ instead of $RF$ to denote the derived functors. We present a guideline to the abelian category of perverse sheaves for the middle perversity, including truncation functors and perverse cohomology. We are interested in the definition in terms of stratifications, but we start with an abstract definition of $t$-structures on abelian categories ([BBD 83], chapter 1.3, théorème 1.4.10) (see also [KaS 90] chapter X).

**Definition 2.7 (t-structure).** Two full subcategories $D^{\le 0}$ and $D^{\ge 0}$ of a triangulated category $D$, define a $t$-structure on $D$ if the following conditions are satisfied. Set $D^{\le n} = D^{\le 0}[-n]$ and $D^{\ge n} = D^{\ge 0}[-n]$:

* $D^{\le -1} \subset D^{\le 0}$ and $D^{\ge 1} \subset D^{\ge 0}$

* $\operatorname{Hom}_D(X, Y) = 0$ for $X \in D^{\le 0}$ and $Y \in D^{\ge 1}$

* For any $X \in \operatorname{Ob} D$ there exists a distinguished triangle $X_0 \to X \to X_1 \xrightarrow{+1} $ with $X_0 \in D^{\le 0}$ and $X_1 \in D^{\ge 1}$

We have long exact sequences for all $Y \in \operatorname{Ob} D$:

$\operatorname{Hom}_D(Y, X_1[-1]) \to \operatorname{Hom}_D(Y, X_0) \to \operatorname{Hom}_D(Y, X_1)$ and
$\operatorname{Hom}_D(X_0[1], Y) \to \operatorname{Hom}_D(X_1, Y) \to \operatorname{Hom}_D(X, Y) \to \operatorname{Hom}_D(X_0, Y)$.

**Example 2.3.** The pair of the two strict full subcategories $^{p}D^{\ge 0}(V)$ and $^{p}D^{\le 0}(V)$ of the category of constructible sheaves $D_c^b(V, \mathbb{Q})$ defined in 2.2.1.1 satisfy the above definition as proved in ([BBD 83], Proposition 2.1.3 and Corollary 2.1.4 applied to the middle perversity).

### 2.2.0.1 Truncation functor $\tau_{\le n}$ and $\tau_{\ge n}$

We deduce the existence of truncation functors ([KaS 90] Proposition 10.1.4):
$\tau_{\le n}: D \to D^{\le n}$ right adjoint to the inclusion $D^{\le n} \to D$ and $\tau_{\ge n}: D \to D^{\ge n}$ left adjoint to the inclusion $D^{\ge n} \to D$ satisfying:

$\forall X \in \operatorname{Ob} D^{\le n}, Y \in \operatorname{Ob} D: \operatorname{Hom}_{D^{\le n}}(X, \tau_{\le n} Y) = \operatorname{Hom}_D(X, Y)$ and

$\forall Y \in D^{\ge n}, X \in \operatorname{Ob} D: \operatorname{Hom}_{D^{\ge n}}(\tau_{\ge n} X, Y) = \operatorname{Hom}_D(X, Y)$

For any $X \in \operatorname{Ob} D$ there exists an exact triangle
$$ \tau_{\le 0} X \to X \to \tau_{> 0} X \xrightarrow{+1} $$

where $\tau_{\le 0} X \in D^{\le 0}$ and $\tau_{> 0} X \in D^{\ge 1}$ depend functorially on $X$. Set

$$ \tau_{\le n} X = (\tau_{\le 0}(X[n]))[-n], \tau_{> n} X = (\tau_{> 0}(X[n]))[-n]. $$

The two full sub-categories with the three properties above, and the truncation functors $\tau_{\le n}$, $\tau_{\ge n}$ form the t-structure. We have for all $n, m \in \mathbb{Z}$:

$$ \tau_{\le n}(X[m]) = (\tau_{\le n+m}(X))[m], \tau_{\ge n}(X[m]) = (\tau_{\ge n+m}(X[n]))[m]. $$

### 2.2.0.2 The (perverse) t-cohomology

The full subcategory $P$ of $D$ defined by $\text{Ob } P := \text{Ob } D^{\le 0} \cap \text{Ob } D^{\ge 0}$ is called the heart of $D$ or the category of perverse objects of $D$.

**Proposition 2.3.** i) The heart $P := D^{\le 0} \cap D^{\ge 0}$ is an abelian category.

ii) If $X \to Y \to Z \xrightarrow{+1}$ is a distinguished triangle with $X$ and $Z$ in $P$, then $Y$ is in $P$.

iii) If $0 \to X \to Y \to Z \to 0$ is an exact sequence in $P$, then there exists a unique morphism $\partial : Z \to X[1]$ such that $X \to Y \to Z \xrightarrow{\partial} X[1]$ is a distinguished triangle in $D$.

The definition of perverse cohomology $\mathcal{H}^*$ (or $^t\mathcal{H}^*$) is by truncation

$$ \mathcal{H}^0(X) := \tau_{\ge 0} \tau_{\le 0}(X) \in P, \mathcal{H}^n(X) := \mathcal{H}^0(X[n]) = \tau_{\ge 0}(\tau_{\le 0} X[n]) $$

The functor $\mathcal{H}^0 : D \to P$ defined on $D$ to the abelian category of perverse sheaves on $V$ is cohomological ([BBD 83] Th'eorème 1.3.6):

A distinguished triangle $A \to B \to C \xrightarrow{[1]}$ in $D$ gives rise to the long exact sequence of perverse cohomology sheaves, where $\mathcal{H}^i := \mathcal{H}^0 \circ [i]$:

$$ \dots \to \mathcal{H}^i(A) \to \mathcal{H}^i(B) \to \mathcal{H}^i(C) \to \mathcal{H}^{i+1}(A) \to \dots $$

In particular this is a sequence of complexes with morphisms in the abelian category $P$. The kernel and cokernel are computed as objects in the abelian category $P$ ([BBD 83], Proposition 1.2.2).

### 2.2.0.3 Left and right t-exact functors

Let $D_i$ be two triangulated categories for $i = 1$ and $i = 2$, with t-structures $(D_i^{\le 0}, D_i^{\ge 0})$ and perverse heart $P_i := D_i^{\le 0} \cap D_i^{\ge 0}$, $\varphi : P_i \to D_i$.

**Definition 2.8.** i) A functor $F : D_1 \to D_2$ is left (resp. right) t-exact if $F(D_1^{\ge 0}) \subset D_2^{\ge 0}$ (resp. $F(D_1^{\le 0}) \subset D_2^{\le 0}$) is exact if it is left and right t-exact.

ii) Set $\varphi_1 : P_1 \to D_1$ and define $^pF := {}^p\mathcal{H}^0 \circ F \circ \varphi_1 : P_1 \to P_2$.

**Lemma 2.4.** If $F$ is left (resp. right) t-exact, then $^pF : P_1 \to P_2$ is left (resp. right) exact on the perverse heart abelian sub-categories.
### 2.2.1 Perverse sheaves defined by the middle perversity

The concept of perversity introduced in [GMacP 83] is topological. However, on a complex algebraic variety $V$, the special case called middle perversity is adapted to Hodge theory.

A complex of sheaves of $\mathbb{Q}$-vector spaces $K^\bullet \in D^b(V, \mathbb{Q})$ is constructible with respect to a given stratification $S$ of $V$, if the restriction of its co-homology sheaves $i_S^*H^iK$ to the strata $S$ of a stratification $S$ are locally constant sheaves of finite dimension. Let $D_c^b(V, \mathbb{Q})$ denote the derived category of bounded complexes with constructible cohomology on $V$. We give the definition of middle perversity in terms of a stratification.

**Definition 2.9 (Perverse sheaves with respect to a stratification $S$).**
Let $S$ be a complex stratification of a complex variety $V$. A complex $K^\bullet$, constructible with respect to $S$, is called perverse (with respect to the middle perversity) if the following conditions are satisfied:

$$ \forall S \in S, H^i i_S^* K^\bullet = 0 \text{ for } i > -\dim S, \text{ and } H^i i_S^! K^\bullet = 0 \text{ for } i < -\dim S $$

where $i_S^*$ is the restriction to $S$ and $Ri_S^!$ (denoted $i_S^!$) is the derived functor of the functor of sections with support in $S$. Verdier duality transforms $i_S^!$ into $i_S^*$ in $D_c^b(V, \mathbb{Q})$.

Due to Thom-Mather first isotopy theorem (§2.1.3.1) and the fact that the complex is constructible, the groups $H^k(i_S^*K^\bullet)$ and $H^k(i_S^!K^\bullet)$ are locally constant. Let $N_{S,x}$ be a normal section of $S$ at $x$. The fibers at $x \in S$ are:

$$ H^k(i_S^*K^\bullet)_x = H^k(i_x^*(i_{N_{S,x}}^*K^\bullet)) \text{ and } H^k(i_S^!K^\bullet)_x = H^k(i_x^!(i_{N_{S,x}}^*K^\bullet)). $$

Let $x \in S$, since for any locally constant sheaf $\mathbf{L}$ on $S: H_x^i(S, \mathbf{L}) = 0$ for $i < 2\dim S$, we deduce: $H_x^i(X, K^\bullet) := H^i(i_x^!K^\bullet) = 0$ for $i < \dim S$.

This definition is useful in the text where most of the proofs are by induction on the dimension of the strata. Perverse sheaves on a singular variety are independent of the stratification.

#### 2.2.1.1 The t-structure defined by the middle perversity

Let $\mathcal{A} := C^+(V, \mathbb{Q})$ denote the abelian category of complexes of sheaves bounded at $-\infty$). Let $\mathcal{K}\mathcal{A}$ denote the category with the same objects but with group of morphisms defined by homotopy classes of morphisms of complexes. The derived category $D^+(V, \mathbb{Q})$ is defined with the same objects but with the group of morphisms constructed by inverting the multiplicative system of quasi-isomorphisms (§2.1).

The derived category of bounded complexes with constructible cohomology sheaves $D_c^b(V, \mathbb{Q}) \subset D^+(V, \mathbb{Q})$ is the full sub-category defined by complexes with bounded constructible cohomology sheaves (vanishing cohomology in degrees $i > \infty$ and $i < \infty$).
The definition of perverse sheaves is based on the introduction of the t-structure with two full sub-categories ($^pD_{\bar{V}}^{\le 0}$, $^pD_{\bar{V}}^{\ge 0}$) of $D_c^b(V, \mathbb{Q})$ defined by the following conditions:

$$ \mathrm{Ob}(^pD_{\bar{V}}^{\le 0}) := \{K^{\bullet} \in D_c^b(V, \mathbb{Q}) : \forall j, \dim_{\mathbb{C}} \mathrm{supp} H^j(K^{\bullet}) \le -j\} $$

where $H^j$ is the local cohomology sheaf of degree $j$. Respectively, let $D$ denotes the Verdier dual, then

$$ \mathrm{Ob}(^pD_{\bar{V}}^{\ge 0}) := \{K^{\bullet} \in D_c^b(V, \mathbb{Q}) : \forall j, \dim_{\mathbb{C}} \mathrm{supp} H^j(DK^{\bullet}) \le -j\}. $$

*Remark 2.3.* i) The duality functor $D$ interchanges $^pD_{\bar{V}}^{\ge 0}$ and $^pD_{\bar{V}}^{\le 0}$, since for all $x \in V \cdot D_V \circ i_x^* \simeq Ri_x^! \circ D_V$ or $i_x^*(DVF) \simeq \mathrm{Hom}^\bullet(Ri_x^!F, \mathbb{Q})$ and $Ri_x^!(DVF) \simeq \mathrm{Hom}^\bullet(i_x^!F, \mathbb{Q})$.

ii) The proof of the equivalence with the definition 2.9 is based on Thom-Mather first isotopy theorem §2.1.3.1 and the fact that the complex is constructible. In particular, the support of $H^j(K^{\bullet})$ is a closed analytic subset (resp. algebraic), hence this definition is equivalent to the definition 2.1.2 in [BBD 83].

**Definition 2.10.** i) The category $P_V := {}^pD_{\bar{V}}^{\le 0} \cap {}^pD_{\bar{V}}^{\ge 0}$ of perverse sheaves is the full sub-category of $D_c^b(V, \mathbb{Q})$ whose objects are in $\mathrm{Ob} {}^pD_{\bar{V}}^{\le 0} \cap \mathrm{Ob} {}^pD_{\bar{V}}^{\ge 0}$.

ii) the perverse cohomology of a complex $K \in \mathrm{Ob}D_V$ is denoted by

$$ {}^p\mathcal{H}^0(K) := {}^{\tau}_{\tau \ge 0} {}^{\tau}_{\tau \le 0}(K) \in P_V, \quad {}^p\mathcal{H}^n(K) := {}^p\mathcal{H}^0(K[n]) = {}^{\tau}_{\tau \ge 0}(\tau_{\le 0} K[n]) $$

*Remark 2.4.* i) The perverse cohomology is a cohomological functor ${}^p\mathcal{H}^0: D_c^b(V, \mathbb{Q}) \to P_V$ with value in the abelian category of perverse sheaves on $V$, where $p$ refers to the specific perversity (the auto-dual perversity in this case).

ii) The family of functors ${}^p\mathcal{H}^i$ is conservative: a morphism $f: A \to B \in D_c^b(V, \mathbb{Q})$ is an isomorphism if and only if ${}^p\mathcal{H}^i(f): {}^p\mathcal{H}^i(A) \simeq {}^p\mathcal{H}^i(B)$ for all $i \in \mathbb{Z}$ ([BBD 83, Proposition 1.3.7]).

Moreover $K \in \mathrm{Ob}(^pD_{\bar{V}}^{\le 0})$ (resp. $\mathrm{Ob}(^pD_{\bar{V}}^{\ge 0})$) if and only if ${}^p\mathcal{H}^i(K) = 0$ for $i > 0$ (resp. $i < 0$).

iii) On a singular variety $V$ the usual cohomology sheaf $H^i(K)$ of a complex $K \in \mathrm{Ob}D_c^b(V, \mathbb{Q})$ is a constructible sheaf different than the perverse cohomology sheaf ${}^p\mathcal{H}^i(K)$ which is a complex of sheaves. The objects in $P_V$ are called perverse sheaves, despite the fact that they are complexes of sheaves.

*Example 2.4.* Let $\mathbf{L}$ be a local system on the big stratum $U \subset V$ of equal dimension $n$ and $j: U \to V$, then ${}^p\mathbf{L} := \mathbf{L}[n]$ is perverse on $U$. If $V \setminus U$ is a local principal divisor, $j_1{}^p\mathbf{L}$ and $j_*{}^p\mathbf{L}$ are perverse on $X$. The intermediate extension $j_*{}^p\mathbf{L}$ is perverse.
### 2.2.1.2 Gluing t-structures along a stratification

In a typical proof by induction over a strata, we have a set of inclusions $U \hookleftarrow X \hookleftarrow F$ of an open subset $U$ and its complement a closed subset $F$. Whenever $F$ is a principal divisor, $j$ is affine.

Usual functors are defined between the triangulated categories $D_c^b(U)$, $D_c^b(X)$ and $D_c^b(F)$: the direct image $j_*$, $i_*$, the inverse image $j^*$, $i^*$, the extension $j_!$ by 0, and $i^1$ the support in $F$ ([BBD 83], 1.4.1).

**Theorem 2.3 ([BBD 83] Theorem 1.4.10).** Let $U \hookleftarrow X \hookleftarrow F$ be defined by an open subset $U$ and its closed complement $F$ in an algebraic variety $X$. If a t-structure $(^pD_U^{\le 0}, {}^pD_U^{\ge 0})$ (resp. $(^pD_F^{\le 0}, {}^pD_F^{\ge 0})$) is defined on $U$ (resp. on $F$), there exists a unique t-structure on $X$ defined by

$$ {}^pD_X^{\le 0} := \{K \in D_c^b(X) : j^*K \in {}^pD_U^{\le 0} \text{ and } i^*K \in {}^pD_F^{\le 0}\} $$

$$ {}^pD_X^{\ge 0} := \{K \in D_c^b(X) : j^*K \in {}^pD_U^{\ge 0} \text{ and } i^1K \in {}^pD_F^{\ge 0}\} $$

**Middle perversity t-structure.** Let $V$ be a complex algebraic variety and let $S$ be a Whitney-stratification on $V$ and $V_i$ the union of strata of complex dimension $\le i$. We have a family of locally closed sub-varieties: $V := V_n \supset \dots \supset V_m \supset V_{m-1} \supset \dots \supset V_0 \supset V_{-1} = \emptyset$.

The t-structure on $D_c^b(V_m, \mathbb{Q})$ is defined by increasing induction on $m$.

If $m=0$ we consider the standard t-structure, where a perverse sheaf is just a locally constant sheaf in degree $-\dim V_0$ on the non singular variety $V_0$. For $m > 0$, we consider the diagram

$$ U = (V_m \setminus V_{m-1}) \hookrightarrow V_m \hookleftarrow V_{m-1} = F $$

By induction, we suppose the perverse t-structure defined on $D_c^b(F, \mathbb{Q})$. On $D_c^b(U, \mathbb{Q})$ we put the standard t-structure shifted by $m$:

$$ {}^pD_U^{\le 0} := D_U^{\le -m}, \quad {}^pD_U^{\ge 0} := D_U^{\ge -m}. $$

The perverse t-structure on $D_c^b(V_m, \mathbb{Q})$ is defined by gluing the two t-structures on $D_c^b(F, \mathbb{Q})$ and $D_c^b(U, \mathbb{Q})$.

**Remark 2.5.** We mention the following properties:

1. Let $\mathcal{F} \in \text{Ob} {}^p\mathcal{D}^{\le 0}$ and $\mathcal{G} \in \text{Ob} {}^p\mathcal{D}^{\ge 0}$ on $V$, then $H^i(\text{RHom}(\mathcal{F}, \mathcal{G})) = 0$ for $i < 0$. In particular $U \mapsto \text{Hom}_{D(U)}(\mathcal{F}|_U, \mathcal{G}|_U)$ is a sheaf ([KaS 90] Proposition 10.2.7).

2. If $f: X \to V$ is smooth of relative dimension $d$, $f^* \circ [d]$ is exact, moreover if the fibers are non empty, then $f^* \circ [d] : P_V \to P_X$ is fully faithful ([BBD 83] Proposition 4.2.5).

3. **Gluing perverse sheaves.** Given an open covering $V = \cup_{i \in I} U_i$ a family of perverse sheaves $\mathcal{F}_i$ on $U_i$ and isomorphisms $f_{ij}: \mathcal{F}_{j|U_{ij}} \simeq \mathcal{F}_{i|U_{ij}}$ on the double intersections of $U_i$ satisfying the cocycle condition $(f_{ij|U_{ijk}}) \circ$
$(f_{jk}|_{U_{ijk}}) = (f_{ik}|_{U_{ijk}})$ on the triple intersections of $U_i$, there exists a
perverse sheaf $\mathcal{F}$ on $V$ and $f_i : \mathcal{F}|_{U_i} \simeq \mathcal{F}_i$ such that $f_{ij} \circ f_{j|U_{ij}} = f_i|_{U_{ij}}$.
Moreover the family $(\mathcal{F}, f_i)$ is unique up to isomorphisms ([KaS 90]
Proposition 10.2.9).

If $I$ is finite, the proof is by induction on $|I|$. In case $I = \{1, 2\}$, we define $\mathcal{F}$ by the distinguished triangle $i_{U_{12}}\mathcal{F}_1|_{U_{12}} \xrightarrow{(a,b)} i_{U_{11}}\mathcal{F}_1 \oplus i_{U_{21}}\mathcal{F}_2 \to \mathcal{F} \xrightarrow{1} \mathcal{F}$ where $a$ is natural, $b$ is deduced from $f_{21}$. If $I = \mathbb{N}$, we can suppose $U_n \subset U_{n+1}$ and take the inductive limit of the family $F_n$ constructed by induction.

4. **Lefschetz-Artin's vanishing theorem:** The behavior of perverse sheaves and affine morphisms is basic in the text, we use freely the results ([BBD 83] §4.1). In particular, if $X$ is an affine variety (resp. Stein) ([KaS 90], Theorem 10.3.8, [Max 19], Corollary 8.6.6):

$$
\mathbb{H}^j(X, \mathcal{F}) = 0 \text{ for } j > 0 \text{ if } \mathcal{F} \in \mathrm{Ob}\mathbb{P}\mathcal{D}^{\le 0}
$$

$$
\mathbb{H}_c^j(X, \mathcal{F}) = 0 \text{ for } j < 0 \text{ if } \mathcal{F} \in \mathrm{Ob}\mathbb{P}\mathcal{D}^{\ge 0}
$$

Moreover, let $f: X \to Y$ be an affine morphism, then $f_*$ is right $t$-exact and $f_!$ is left $t$-exact ([BBD 83] §4.1).

5. Let $j: U \to X$ be a Zariski open subset and $\mathcal{F} \in P_X$, we have distinguished triangles in $D_c^b(X)$:

$$
j_*j^*\mathcal{F} \to \mathcal{F} \to i_*i^*\mathcal{F} \xrightarrow{\pm 1} \quad \text{and} \quad i_*i^*\mathcal{F} \to \mathcal{F} \to j_*j^*\mathcal{F} \xrightarrow{\pm 1}
$$

In the case where $U \subset X$ is the complement of a principal divisor $D$,
$j: U \to X$ is an affine embedding, hence $j_*\mathcal{F}, j_*\mathcal{F}$ are perverse. Let
$i: D \to X$, we deduce the following exact sequences in the category of
perverse sheaves $P_X$ ([BBD 83] Corollaries 4.1.10, 4.1.12)

$$
0 \to i_* {}^p\mathcal{H}^{-1}(i^*\mathcal{F}) \to j_* j^* F \to \mathcal{F} \to i_* {}^p\mathcal{H}^0(i^*\mathcal{F}) \to 0
$$

$$
0 \to i_* {}^p \mathcal{H}^0 (i^! \mathcal{F}) \to \mathcal{F} \to j_* j^* \mathcal{F} \to i_* {}^p \mathcal{H}^1 (i^! \mathcal{F}) \to 0
$$

$$
i^* j_* \mathcal{F} [-1] = {}^p \mathcal{H}^{-1} (i^* j_* \mathcal{F}) = {}^p \mathcal{H}^0 (i^! j_* \mathcal{F})
$$

$$
i^! j_* \mathcal{F}[1] = {}^p\mathcal{H}^0(i^* j_* \mathcal{F}) = {}^p\mathcal{H}^1(i^! j_* \mathcal{F})
$$

2.2.1.3 Intermediate extensions

Let $U \stackrel{j}{\hookrightarrow} X$ be an algebraic open subset with complement $F \stackrel{i}{\hookrightarrow} X$. A perverse sheaf $\mathcal{F} \in P_X$ extends a perverse sheaf $\mathcal{G} \in P_U$ if $j^*\mathcal{F} \simeq \mathcal{G}$. There may be many extensions. For example both ${}^p\mathcal{H}^0(j!\mathcal{G})$ and ${}^p\mathcal{H}^0(j_*\mathcal{G})$ extend $\mathcal{G}$.

**Definition 2.11.** There exists a unique perverse sheaf $\mathcal{F}$ on $X$ extending a
perverse sheaf $\mathcal{G}$ on $U$ satisfying the following equivalent properties:

1. $i^*\mathcal{F} \in {}^p\mathcal{D}_F^{\le -1}$ and $i^!\mathcal{F} \in {}^p\mathcal{D}_F^{\ge 1}$.

2. $\mathcal{F}$ is the image in $P_X$ of the morphism ${}^p\mathcal{H}^0(j!\mathcal{G}) \to {}^p\mathcal{H}^0(j_*\mathcal{G})$.
$\mathcal{F}$ is denoted by $j_*\mathcal{G}$ and called the Intermediate extension of $\mathcal{G}$.

Remark 2.6. i) The following exact sequences associated to $\mathcal{F}$ in $P_X$ and $F \subset X$

$$0 \to i_* \mathcal{H}^0(i^!\mathcal{F}) \to \mathcal{F}, \quad \mathcal{F} \to i_* \mathcal{H}^0(i^*\mathcal{F}) \to 0,$$

define the biggest sub-object (resp. the biggest quotient of $\mathcal{F}$) with support in $F$. Hence:

$$j_*\mathcal{G} = \mathcal{P}\mathcal{H}^0(j_*\mathcal{G})/i_*\mathcal{P}\mathcal{H}^0(i^!j_*\mathcal{G}) = \text{Ker} (\mathcal{P}\mathcal{H}^0(j_*\mathcal{G}) \to i_*\mathcal{P}\mathcal{H}^0(i^*j_*\mathcal{G}))$$

where $i_* \mathcal{P}\mathcal{H}^0(i^!j_*\mathcal{G})$ is the biggest sub-object of $\mathcal{P}\mathcal{H}^0(j_*\mathcal{G})$ in $P_F$ and $i_* \mathcal{P}\mathcal{H}^0(i^*j_*\mathcal{G})$ is the biggest quotient of $\mathcal{P}\mathcal{H}^0(j_*\mathcal{G})$ in $P_F$.

1) $j_*\mathcal{G}$ has no non trivial sub-object with support in $F$.

2) $j^!\mathcal{G}$ has no non trivial quotient-object with support in $F$.

3) $j_*\mathcal{G}$ has no non trivial sub-object or quotient-object with support in $F$.

ii) Let $F$ be a locally principal divisor in $X$ and $\mathcal{G} \in P_U$. The derived extension $j_*\mathcal{G}$ of $\mathcal{G}$ by 0 and the derived direct image $i_*\mathcal{G}$ are perverse on $X$ ([BBD 83] Corollaire 4.1.10) and $j_*\mathcal{G} := \text{Im}(j_*\mathcal{G} \to j_*\mathcal{G})$ ([BBD 83] Définition 1.4.22).

iii) Let $U \stackrel{j}{\hookrightarrow} Z \stackrel{i}{\hookrightarrow} X$ denote a closed subvariety $Z$ and $U$ a smooth open Zariski subset of $Z$. Let $\mathcal{G} \in P_U$ and $k = i \circ j : U \to X$, then we write $k_* \mathcal{P}\mathcal{G}$ for $i_*j_*\mathcal{P}\mathcal{G}$ on $X$.

If $\mathcal{G}$ be a simple object in $P_U$, then $k_* \mathcal{P}\mathcal{G}$ is a simple object of $P_X$.

iv) Let $\mathcal{G} := \mathbf{L}$ be a local system, we have: $D_X(j_*\mathbf{L}) \simeq j_*\tilde{\mathbf{L}}$ where $\tilde{\mathbf{L}}$ is the linear dual of $\mathbf{L}$.

**Theorem 2.4.** i) On a complex algebraic variety $X$, the category of perverse sheaves $P_X \subset D_c^b(X, \mathbb{Q})$ is abelian (although $D_c^b(V, \mathbb{Q})$ is not abelian), stable under Verdier duality, Noetherian and Artinian ([BBD 83] Théorème 1.3.6).

ii) The simple perverse sheaves are intermediate extensions of irreducible local systems defined on non singular open subsets of irreducible closed algebraic sub-varieties of $X$ and any perverse complex sheaf has finite length.

### 2.2.2 $\psi_f j_*\mathbf{L}$ and extensions of perverse sheaves

Let $X$ be a complex analytic variety, $f : X \to D$ a non constant analytic morphism to a disc in $\mathbb{C}$ such that the analytic restriction $f_\mid : X_{D^*} \to D^*$ is a topological fibration over the punctured disc $D^*$ with small radius. This is the case if $f$ is proper or $f$ is the restriction of an algebraic morphism of complex algebraic varieties.

Let $K$ be an object of the derived category $D_c^+(X^*, \mathbb{Q})$, i.e $K$ is a complex of sheaves of $\mathbb{Q}$-vector spaces, bounded below, with locally constant cohomology sheaves on the strata of a Thom - Whitney stratification of $X$.
$$i^*K \xrightarrow{sp} \psi_f K \xrightarrow{can} \varphi_f K \to i^*K[1] \qquad (2.15)$$

Since the natural morphism $sp: i^*K \to \psi_f K$ is $T$-equivariant. The action of $T$ extends to the cone and defines an action on $\varphi_f K$ denoted also by $T$.

Then, the action $Id - T: \psi_f K \to \psi_f K$ is well defined on the cone $\varphi_f K$ and induces a (topological) variation denoted $var: \varphi_f K \to \psi_f K$ such that $var \circ can = Id - T$.

**Proposition 2.4 (Invariance under monodromy).** There exists a natural distinguished triangle:

$$i^*Rj_*K \xrightarrow{sp} \psi_f K \xrightarrow{Id-T} \psi_f K \xrightarrow{[1]} i^*Rj_*K[1] \qquad (2.16)$$

The monodromy $T$ acts on $Rp_*p^*K$. The derived functor of the invariant sub-complex under the action of $T$ is by definition the derived functor $RHom_{\mathbb{Z}[T]}(\mathbb{Z}, Rp_*p^*K)$ where $\mathbb{Z}[T]$ acts trivially on $\mathbb{Z}$ (it is denoted also by $R\Gamma(\mathbb{Z}, Rp_*p^*K)$). Such derived functor is computed after a result of Hochschild-Serre, which yields the following triangle in $D_c^+(X^*, \mathbb{Q})$ (see [Br 86] Théorème 1.2)

$$K \to Rp_*p^*K \xrightarrow{Id-T} Rp_*p^*K \xrightarrow{1} \qquad (2.17)$$

We apply $i^*Rj_*$ to the triangle 2.17 and deduce the proposition from $\psi_f K = i^*Rj_*Rp_*p^*K$ (Equation 2.14):

**Corollary 2.1.** There is an exact sequence of perverse sheaves on $X_0$

The shift by $[-1]$ is explained by the fact that $\psi_f^u K[-1]$ is perverse. The corollary follows from the long exact sequence associated to the triangle in equation 2.16.

**Proposition 2.5 (Duality).** There is a non canonical isomorphism in the derived category on $X_0: D_{X_0}(\psi_f K[-1]) \simeq \psi_f(D_X(K))[-1]$.

*Proof.* See ([Br 86] Lemme 1.4). The proof by after Beilinson and Bernstein is based on a concept similar to the sections of finite determination introduced by Deligne. For a general complex $F$ of objects of an abelian category $\mathcal{A}$ on which an automorphism $T$ acts, and for each non zero element $P \in \mathbb{Q}[T, T^{-1}]$.

Let $F(P)$ denotes the simple complex associated to the double complex $P: F \to F$.

If $P$ divides $Q$, there exists a morphism of double complexes $F(P) \to F(Q)$ defined by the morphism of double complexes: $(F \xrightarrow{P} F) \xrightarrow{(Id, Q/P)} (F \xrightarrow{Q} F)$. We suppose that the inductive limit exists in $\mathcal{A}$ and define:

$F_{\text{fin det}}: \lim F(P)$ directed by divisibility of polynomials.

There is a natural morphism $F_{\text{fin det}} \to F$. We have
**Lemma 2.5.** Let $F$ be a bounded complex with constructible cohomology in the category of sheaves of $\mathbb{Q}$-vector spaces on $X_0$. Then: $F_{\text{fin det}} \xrightarrow{\sim} F$ is an isomorphism

*Proof.* The functor: $F \to F_{\text{fin det}}$ being exact, the proof may be reduced by "dévissage" to the case where $F$ is a constructible sheaf in degree 0. Then, we check the lemma on the fibres of $F$ at any point $x$: $(F_{\text{fin det}})_x \simeq (F_x)_{\text{fin det}}$ as $\mathbb{Q}[T, T^{-1}]$-modules of finite length (by constructibility over $\mathbb{Q}$).

Let $B := \mathbb{Q}[T, T^{-1}]$, and $M$ a $B$-module of finite length with attached complex $M_{\text{fin det}}$, then

$$H^0(M_{\text{fin det}}) \simeq \varinjlim_P \text{Hom}_B(B/(P), M) = M,$$

$$H^1(M_{\text{fin det}}) \simeq \varinjlim_P \text{Ext}_B^1(B/(P), M) = 0$$

since every element of $M$ is annihilated by some polynomial in $B$.

**Lemma 2.6.** i) $\psi_f K := i^*Rj_*p_*(p^*K) \simeq i^*Rj_*p_*(p^*K)_{\text{fin det}}$

$$\text{ii) } (p_*p^*K)_{\text{fin det}} \simeq (p_!p^*j^*K)_{\text{fin det}}[1]$$

In the diagram (Equation 2.13), $p_!\mathbb{Q}_{\bar{X}^*}$ is a locally constant sheaf of free $\mathbb{Q}[T, T^{-1}]$-modules of rank 1. In this case the previous lemma does not apply since the module $M$ is a free $\mathbb{Q}[T, T^{-1}]$-module and not of finite length.

The complex $M_P$ is quasi-isomorphic to $(M/P.M)[-1]$ (non zero cohomology in degree 1). We have:

$$(p_!\mathbb{Q}_{\bar{X}^*})_{\text{fin det}}[1] \simeq \varinjlim_P \mathbb{Q}[T, T^{-1}]/(P).$$

ii) While $p_*\mathbb{Q}_{\bar{X}^*} \simeq \mathbb{Q}[[T, T^{-1}]] \otimes_{\mathbb{Q}[T, T^{-1}]} p_!\mathbb{Q}_{\bar{X}^*}$ is a locally constant sheaf of $\mathbb{Q}[T, T^{-1}]$-modules locally isomorphic to $\mathbb{Q}[[T, T^{-1}]]$

The complex $\mathbb{Q}[[T, T^{-1}]](P)$ is non-canonically isomorphic to $\mathbb{Q}[T, T^{-1}]/(P)$ (the elements annihilated by $P$ form a non zero sub-module just the opposite of the previous case of a free $\mathbb{Q}[T, T^{-1}]$-module and since any non zero polynomial $P$ is invertible, the cohomology in degree 1 vanish). Moreover, we can define a commutative diagram of quasi-isomorphisms

$$\begin{tikzcd}
\mathbb{Q}[T, T^{-1}]/(P) \arrow[r, "Q/P"] & \mathbb{Q}[T, T^{-1}]/(Q) \\
\mathbb{Q}[[T, T^{-1}]](P) \arrow[u, "q.iso."'] \arrow[r, "\sim"] & \mathbb{Q}[[T, T^{-1}]](Q)
\end{tikzcd}$$

$$\text{hence } (p_*\mathbb{Q}_{\bar{X}^*})_{\text{fin det}} \simeq \varinjlim_P \mathbb{Q}[T, T^{-1}]/(P) \simeq (p_!\mathbb{Q}_{\bar{X}^*})_{\text{fin det}}[1].$$

We deduce lemma ii) since $p_*p^*K \simeq K \otimes (p_*p^*\mathbb{Q}_{X^*})$.

*Proof of the proposition.* Representing $K$ by a complex of injective sheaves over $\mathbb{Q}[T, T^{-1}]$ we deduce from the lemma

$$\psi_f K \simeq i^*Rj_*(p_*p^*K)_{\text{fin det}} \simeq i^*Rj_*(p_!p^*K)_{\text{fin det}}[1] = i^*Rj_*(p_!p^*K)[1]$$

$$\text{hence: } (\psi_f(DK))[\text{-1}] = (i^*Rj_*p_*p^*(DK))[\text{-1}] = i^*Rj_*p_!p^*(DK)_{\text{fin det}} =$$
$$D(i^!Rj_*p_*p^*K)_{\text{fin det}} = D(i^*Rj_*p_*p^*K[-1])_{\text{fin det}} = D(\psi_f K[-1])$$
since $i^*j_*p_*p^*K[-1] \simeq i^!j_*p_*p^*K$.

The perversity follows from Artin's theorem and duality ([Br 86] Théorème 1.2, see also [Max 19] Theorem 10.3.13) and the references there:

**Proposition 2.6 (Perversity).** i) Let $K$ be a perverse sheaf on $X$, then $\psi_f K[-1]$ is perverse on $X_0$.

ii) $\varphi_f K[-1]$ is perverse on $X_0$.

iii) $\psi_f K[-1], \varphi_f K[-1]: P_X \to P_{X_0}$ are exact functors of perverse sheaves.

i) The proof is local on $X_0$. Let $U_x$ be a small ball with center $x \in X_0$. Since the Milnor fiber $X_{f,x}$ is a principal hypersurface in $U_x$, $K[-1]$ restricts to a perverse sheaf on the Stein subvariety $X_{f,x}$. We deduce by Artin's theorem (Remark 2.5) that the ordinary cohomology $\mathbb{H}^i(X_{f,x}, \psi_f K[-1]) \simeq (\psi_f K[-1])_x$ vanish for $i > 0$.

Since the cohomology $\mathcal{H}^k(P)$ sheaves of any perverse sheaf $P$ on a variety $Y$ vanish for $k > 0$, we deduce that the perverse cohomology $\mathbb{P}\mathcal{H}^i(\psi_f K[-1])$ vanish for $i > 0$. Since this inequality apply to the dual sheaf $D_X K$, we deduce by duality that $\mathbb{P}\mathcal{H}^i(\psi_f K[-1])$ vanish for $i < 0$; hence $\psi_f K[-1]$ is perverse.

ii) $\varphi_f K[-1]$ is perverse. We deduce from the perverse exact sequence defined by the triangle in (Equation 2.15), the exact sequence

$$0 \to \mathbb{P}\mathcal{H}^{-2}(\varphi_f K) \to \mathbb{P}\mathcal{H}^{-1}(i^*K) \xrightarrow{\text{can}-1} \mathbb{P}\mathcal{H}^{-1}(\psi_f K) \to \mathbb{P}\mathcal{H}^{-1}(\varphi_f K) \to \mathbb{P}\mathcal{H}^0(i^*K) \to 0$$

where $\mathbb{P}\mathcal{H}^{-1}(\psi_f K) = \psi_f K[-1]$ and $\mathbb{P}\mathcal{H}^i(\psi_f K) = 0$ if $i \neq -1$.

We prove $\text{can}_-1$ is injective. By the corollary 2.1, it is enough to prove that $\mathbb{P}\mathcal{H}^{-1}(i^*K) \xrightarrow{\text{can}-1} \mathbb{P}\mathcal{H}^{-1}(i^*Rj_*j^*K)$ is injective. Since $\mathbb{P}\mathcal{H}^i(i^*K) = 0$ for $i \le -2$ ([BBD 83], Corollary 4.1.10), the functor $K \to \mathbb{P}\mathcal{H}^{-1}(i^*K)$ from perverse sheaves on $X$ to perverse sheaves on $X_0$ is left exact.

Let $N$ denotes the perverse kernel of $K \to Rj_*j^*K$. Since the sequence of perverse sheaves $0 \to N \to K \to Rj_*j^*K$ is exact we deduce:

$$0 \to \mathbb{P}\mathcal{H}^{-1}(i^*N) \to \mathbb{P}\mathcal{H}^{-1}(i^*K) \to \mathbb{P}\mathcal{H}^{-1}(i^*Rj_*j^*K) \text{ is exact.}$$

Since $N$ is supported by $X_0$, $i_*i^*N = N$ is perverse and $\mathbb{P}\mathcal{H}^{-1}(i^*N) = 0$, hence $h_{-1}$ is injective.

iii) The assertion is admitted.

### 2.2.3 Verdier's classification of the extensions

Extensions of perverse sheaves along a locally principal divisor $D \subset X$ occur in the work of Deligne, Mac Pherson and Verdier in various ways. The description provided by Verdier [V 85-1] is useful to construct perverse sheaves;
in particular the case of singularities along a NCD. It is conveniently applied to prove the decomposition properties in Hodge theory.

We recall the result only in the case of an extension along an effective principal divisor defined by an equation $f$. Let $\psi_f^u K[-1]$ (resp. $\varphi_f^u K[-1]$) denotes the sub-sheaf on which the action of $T$ is unipotent.

The result of Verdier covers the case of locally principal divisor $Y \subset X$ (and even a closed subvariety $Y$ in general), useful in the construction of the weight filtration in the open case $X \setminus Y$ out of the monodromy filtration along the cone $C_Y X$ of $Y$ in $X$ ([V 85-1], [Sa 90], [ELM 10]).

The extension is expressed in terms of the nearby cycle functor $\psi_f^u[-1]$ and the vanishing cycle $\varphi_f^u[-1]$ (Equation 2.15).

Let $\mathrm{Perv}(X)$ denote the abelian category of perverse sheaves on an algebraic variety $X$ and let $Z \subset X$ be a principal divisor defined by an equation $f = 0$, $U := X \setminus Z$, $i: Z \to U$ and $j: U \to X$.

We associate to $Z$ the category $\mathrm{V}(Z)$ consisting of:

1. A perverse sheaf $G \in \mathrm{Perv}(U)$ on $U$

2. A diagram in the category $\mathrm{Perv}(Z)$

$$ \psi_f^u(G)[-1] \xrightarrow{a} \Phi \xrightarrow{b} \psi_f^u(G)[-1] : b \circ a = \mathrm{Id} - T \quad (2.18) $$

where $T$ is the monodromy of $\psi_f^u(G)[-1]$.

3. Morphisms are couple of morphisms of diagrams and over $U$.

The triple $(a, \Phi, b)$ is the variable in $\mathrm{V}(Z)$ which determines the extension.

**Lemma 2.7.** The functor $\mathrm{V}: \mathrm{Perv}(X) \to \mathrm{V}(Z)$

$$ \mathrm{V}: F \mapsto (j^*F, \{\psi_f^u F[-1] \xrightarrow{\mathrm{can}} \varphi_f^u F[-1] \xrightarrow{\mathrm{var}} \psi_f^u F[-1]\}) $$

is an equivalence of categories.

*Proof.* We adapt the proof ([V 85-1], §2 Proposition, §4 Corollary 1) to our case. By definition $\psi_f^u F = \psi_f^u j^*F$. Since $F$ is perverse, the complexes $\psi_f^u F[-1]$ and $\varphi_f^u F[-1]$ are perverse sheaves on $Z$. We recall the triangles where $\mathrm{var}$ is defined by $\mathrm{Id} - T$:

$$ \to i'F \to \varphi_f^u F[-1] \xrightarrow{\mathrm{var}} \psi_f^u F[-1] \xrightarrow{[1]} \to i'F \to F \to j_*j^*F \xrightarrow{[1]} \quad (2.19) $$

where the first triangle is dual to (Equation 2.15).

*V is faithful:* If $m: F \to G$ satisfies $\mathrm{V}(m) = 0$, then by definition $j_*j^*m = 0$ and $\psi_f^u(m) = 0$, $\varphi_f^u(m) = 0$. Hence by the above first triangle $i^!m = 0$ and by the second $m = 0$.

*V is fully faithful:* Let $F, G$ be perverse on $X$ and let $m': V(F) \to V(G)$ be a morphism in $\mathrm{V}(Z)$, hence $m' = (u, \alpha, \beta, \alpha)$ where $u: j^*F \to j^*G$, $\alpha = \psi_f^u(u)$ and $\beta = \varphi_f^u(u)$. We construct $m: F \to G$ with image $\mathrm{V}(m) = m'$ as follows.
We deduce from $\beta : \varphi_f^u F[-1] \to \varphi_f^u G[-1]$ and $\alpha : \psi_f^u F[-1] \to \psi_f^u G[-1]$ a morphism $\gamma : i^! F \to i^! G$ compatible with the first triangle and then a morphism $m : F \to G$ compatible with $\gamma$ and $j_*j^*u$ in the second triangle. $V$ is essentially surjective: Given a diagram $\psi_f^u(G)[-1] \xrightarrow{a} \Phi \xrightarrow{b} \psi_f^u(G)[-1]$ and $G \in \mathrm{Per}(U)$, we construct $F \in \mathrm{Per}(X)$ such that $\varphi_f^u F[-1] \simeq P$.

First we construct a triangle $\to F_1 \to \Phi \xrightarrow{b} \psi_f^u F[-1] \xrightarrow{[1]}$ with support in $Y$ (where the complex $F_1$ is expected to be $i^! F$) with associated perverse exact sequence $0 \to \mathcal{P}\mathcal{H}^0(F_1) \to \Phi \xrightarrow{b} \psi_f^u G[-1] \to \mathcal{P}\mathcal{H}^1(F_1) \to 0$. Hence, we have $\mathrm{Ker} b \simeq \mathcal{P}\mathcal{H}^0(F_1)$, $\mathcal{P}\mathcal{H}^1(F_1) \simeq \mathrm{Coker} b$.

Then we deduce $F_2$ satisfying the second triangle $F_1 \to F_2 \to j_*G \xrightarrow{[1]}$ such that we have a long exact sequence of perverse sheaves:

$$0 \to \mathcal{P}\mathcal{H}^0(F_1) \to \mathcal{P}\mathcal{H}^0(F_2) \to \mathcal{P}\mathcal{H}^0(j_*G) \to \mathcal{P}\mathcal{H}^1(F_1) \to \mathcal{P}\mathcal{H}^1(F_2) \to 0$$

where $\mathcal{P}\mathcal{H}^{-1}(j_*G) = 0 = \mathcal{P}\mathcal{H}^1(j_*G)$ and $\mathcal{P}\mathcal{H}^0(j_*G) = j_*G$ since $j$ is affine. We deduce the exact sequence

$$0 \to \mathrm{Ker} b \to \mathcal{P}\mathcal{H}^0(F_2) \xrightarrow{v} j_*G \to \mathrm{Coker} b \to \mathcal{P}\mathcal{H}^1(F_2) \to 0$$

In particular, the restriction of $v$ to $U$ is an isomorphism and $\mathcal{P}\mathcal{H}^0(F_2)$ is a perverse extension of $j_*G$ since. Moreover, $F_1 \simeq i^! \mathcal{P}\mathcal{H}^0(F_2)$ since $\mathrm{Ker} b \simeq i^! \mathcal{P}\mathcal{H}^0(F_2)$, $\mathrm{Coker} b \simeq i^! \mathcal{P}\mathcal{H}^1(F_2)$ and $\Phi \simeq \varphi_f^u F[-1]$.

**Remark 2.7.** In the text, we will use a variation morphism $\mathrm{var}$ instead of the topological variation $\mathrm{var}$ defined by $N' := \pm(T_j^u - I)$ acting on $\psi_f^u$ and $\mathrm{var} : \varphi_f^u \to \psi_f^u$ satisfying: $\mathrm{var} \circ \mathrm{can} = N'$.

In chapters 4 and 5 we introduce the action $\nu$ on $\Psi^*\mathcal{I}\mathcal{L} \simeq \psi_f^u \mathbf{L}$ (§4.21) with Hodge filtration. By (Equation 3.1), $-2i\pi\nu$ induces $N = \mathrm{Log}T$ on $\psi_f^u \mathbf{L}$.

The action of $N$ on $\psi_f^u K$ induces a morphism of mixed Hodge structure up to a shift. Since $\mathrm{sp}$ is equivariant for $N$, the action of $N$ extends to $\varphi_f^u K$ (instead of $T$ in topology) and we introduce a geometrical $\mathrm{var} : (\varphi_f^u, W, F) \to (\psi_f^u, W, F)(-1)$ such that: $\mathrm{var} \circ \mathrm{can} = N$, instead of the $\mathrm{var}$ defined by $N'$.

However, there exists a polynomial $A$ such that $N = A(N')$ (see Equation 3.2) and $N'$ is nilpotent. Since $A$ is invertible as a series and since $N$ is nilpotent, we find a polynomial $B$ such that $N' = B(N)$.

We deduce that $\mathrm{ker} N = \mathrm{ker} N'$, and $N'$ is also compatible with MHS. Moreover, the monodromy filtrations $W(N), W(N')$ and $W(\nu)$ defined by the nilpotent endomorphisms $N, N'$ or $\nu$ (§2.3.0.1) coincide and are all defined over $\mathbb{Q}$.

For each $k > 1$, $N^k : \mathrm{Gr}_{-k}^{W(N)} L^u \xrightarrow{\sim} \mathrm{Gr}_k^{W(N)} L^u$ is equal to $N'^k$ or $\nu^k$ up to constants, which explains that there is no danger to cite references with one of these conventions.
### 2.2.3.1 Applications

Since the applications in view are in Hodge theory, we use henceforth the *var* defined by $N$ instead of $N'$.

Due to the equivalence of the categories Perv($X$) and $V(Z)$, the following correspondence will be useful in various proofs in the text (*var* denotes two morphisms, one on $\psi_f$ and one on $\varphi_f$).

1) $j_!G$: The case where $a : \psi_f(G)[-1] \xrightarrow{\sim} \Phi$ is an isomorphism and $b = \mathrm{var} \circ a^{-1}$: $\psi_f(G)[-1] \xrightarrow{a} \Phi \xrightarrow{b} \psi_f(G)[-1]$ corresponds to the extension $j_!G$ of $G$ by zero across $Z$ (ker $a = \ker N = i_Z^*G[-1] = 0$) such that we have the diagram where *can* is an isomorphism

$$ (\psi_f G[-1] \xrightarrow{a=can} \varphi_f G[-1] \xrightarrow{b=var} \psi_f G[-1]) \qquad (2.20) $$

2) $j_*!G$: The case where $a$ is an epimorphism and $b$ a monomorphism such that $\Phi = \mathrm{Im}(\psi_f(G)[-1] \xrightarrow{N} \psi_f(G)[-1])$. Then, the diagram

$$ (\psi_f G[-1] \xrightarrow{a=can} \varphi_f G[-1] \xrightarrow{b=var} \psi_f G[-1]) \qquad (2.21) $$

corresponds to the intermediate extension $j_*!G$.

3) The case where $G|_U = 0$ corresponds to the perverse sheaf $\Phi$ concentrated on $Z$.

4) A decomposition of the diagram in (Equation 2.18) corresponds to a decomposition of the extension. In particular, we the following Verdier's decomposition criteria:

$$ \Phi \simeq \mathrm{Im} \, can \oplus \mathrm{Ker} \, var \iff j_*!G|_U \oplus \mathrm{Ker} \, var \qquad (2.22) $$

is used in the text with $\mathrm{Ker} \, var$ concentrated on $Z$.

5) The extension $j_*!G|_U$ corresponds to the case where $b$ is an isomorphism and $a = b^{-1} \circ \mathrm{var}$ ($b \circ a = N$) (Equation 9.43) and the diagram where $var$ is an isomorphism

$$ (\psi_f G[-1] \xrightarrow{a=can} \varphi_f G[-1] \xrightarrow{b=var} \psi_f G[-1]) \qquad (2.23) $$

6) In the case with an increasing filtration $W^0$ on $G$ such that the relative monodromy $M := M(N, W^0)$ exists on $\psi_f G[-1]$, the filtration of weight

$$ (N * W^0)_k := NW_{k+1}^0 + (M_k \cap W_{k+1}^0) \psi_f^u G[-1] $$

lifts to $\varphi_f^u G[-1]$ and corresponds to a filtration $W$ on $j_*!G|_U$ such that $(W_k)$ extends $W_k^0$ (§9.7.1.1).
## 2.3 Degeneration of VMHS

The theory of nilpotent orbit (resp. admissible variation of MHS) is needed in the next chapters to define the limit Hodge filtration and the weight filtration. The theory, inspired by Deligne, appears in [Sc 73, GrSc 73, Gr 74, CaKSc 86, CaKSc 87, CEGL 14] (resp. [De 80], I.8.15, [E 83], [StZ 85]) but we adopt the notations and terminology of [Ka 86] summarized below.

The admissibility of a VMHS on $X^* := X \setminus Y$ refers to asymptotic properties of $(\mathbf{L}, W, F)$ along the NCD. Such asymptotic properties are expressed on Deligne's extension $\mathbf{L}_X$ of $\mathbf{L} \otimes_Q \mathcal{O}_{X^*}$ defined along $Y$ in terms of the 'multivalued' horizontal sections of $\nabla$ on $X^*$. The extension $\mathcal{L}_X$ is a locally free analytic sheaf of modules, hence algebraic if $X$ is projective [De 70].

By construction, the connection $\nabla$ on $\mathcal{L}_{X^*}$ extends to $\mathcal{L}_X$ with logarithmic singularities along $Y$. The residues of the logarithmic singularities are endomorphisms on the restrictions $\mathcal{L}_{Y_i}$ along the components $Y_i$ of $Y$. The connection is uniquely characterized by the choice of the logarithm of the eigenvalues of the monodromy (see §3.1).

The fibre $L := \mathcal{L}(x) := \mathcal{L}_{X,x} \otimes_{\mathcal{O}_{X,x}} \mathbb{C}$ of the vector bundle $\mathcal{L}_X$ is viewed as the space of the 'multivalued' horizontal sections of $\mathbf{L}$ at $x$ (sections of a universal covering of the complementary of $Y$ in a ball $B_x$ at $x$).

The extension over $Y$ of the Hodge filtration of a polarizable variation of HS in the case of a punctured disc constructed by Schmid [Sc 73, GrSc 73], ([Gr 74] Chapter IV) is a fundamental asymptotic property, that is required by assumption in the case of a graded polarizable variation of MHS, as a condition of admissibility. The results on the degeneration of a VHS on the complement of a NCD are based on [Sc 73], [CaKSc 86] and [CaKSc 87], see also ([Gr 74] Chapter V, [CEGL 14] Chapters 7, 8).

A filtration $W$ of $\mathbf{L}$ over a punctured disc extends to a filtration by sub-bundles $W \subset \mathcal{L}_X$. The local monodromy $T_i$ around a component $Y_i$ of $Y$, defines a nilpotent endomorphism $N_i := \text{Log } T_i^u$, logarithm of the unipotent part of $T$ which acts on the restrictions to $Y_i$ of the sub-bundles $W$. The definition of admissibility is based on the introduction by Deligne of the relative monodromy filtration $M(\sum_i N_i, W)$ on the limit vector space $L$ in ([De 80], I.8.15).

The required properties are proved in the case of geometric variation of MHS over a punctured disc in [E 83] and studied axiomatically as conditions of admissibility in [StZ 85]. In [Ka 86], the definition of admissibility along a NCD is reduced to the case over a punctured disc.

To state the degeneration of a variation of MHS locally, Kashiwara introduced the abelian category of infinitesimal MHS (IMHS) called also infinitesimal mixed Hodge modules in reference to $D_X$-modules. A proof of the decomposition theorem is given in terms of distinguished pairs of IMHS.
### 2.3.0.1 The monodromy filtration

On the cohomology group of a complex projective non singular variety $X$ of dimension $n$: $L := \oplus_{k \in \mathbb{Z}} H^k(X, \mathbb{Q}[n])$ centered at 0, an increasing filtration $W$ on $L$ is defined by $W_i = \oplus_{j \ge -i} H^j(X, \mathbb{Q}[n])$ ($W_n = L$ and $W_{-n} = H^n(X, \mathbb{Q}[n])$). The Lefschetz structure, defined by the cup product with the class of an hyperplane section is a linear morphism $\eta: W^i \to W^{i-2}$ satisfying for each $i$ an isomorphism $\eta^i: \mathrm{Gr}_i^W L \xrightarrow{\sim} \mathrm{Gr}_{-i}^W L$.

To state conjecturally the properties of the degeneration of a VHS, Deligne defines in general on an object $L$ of an abelian category and a nilpotent endomorphism $N$ on $L$, a unique increasing filtration $W$ satisfying the following conditions ([De 80] 1.6)

(i) $\forall j \in \mathbb{N}: NW_j \subset W_{j-2}$.

(ii) The induced morphism $N^i: \mathrm{Gr}_{-i}^{W(N)} L \xrightarrow{\sim} \mathrm{Gr}_i^{W(N)} L$ is an isomorphism for all $i \in \mathbb{N}$.

Such filtration $W$ is denoted by $W(N)$ and called the monodromy filtration. In fact $W_{i-1} = (N^i)^{-1}W_{-i-1}$ and $W_{-i} = N^i W_i$ for $i \ge 1$.

If $N^{k+1} = 0$ and $N^k \ne 0$ set $W_i = L$ for $i \ge k$ and $W_i = 0$ for $i < -k$, then defines $W_i$ inductively for $i \in [-k, k-1]$, to start with $W_{-k} = N^k L$, $W_{k-1} = \mathrm{Ker} N^k$. It is surprising that the defining properties of $W$, will set various decomposition properties through the whole chapter.

For $i \ge 0$, let $P_i L := \mathrm{Ker} N^{i+1} : \mathrm{Gr}_i^W L \to \mathrm{Gr}_{-i-2}^W L$ denote the subspace called the primitive subspace, then $\mathrm{Gr}_i^W L = P_i L \oplus N \mathrm{Gr}_{i+2}^W L = \oplus_{j \ge 0} P_{i+2j} L$.

### 2.3.0.2 The relative monodromy filtration

Let $N: (L, W) \to (L, W)$ be a nilpotent endomorphism compatible with $W$ ($NW_i \subset W_i$) in an abelian category $\mathfrak{A}$ where $W$ is an increasing filtration of an object $L$ of $\mathfrak{A}$.

There exists at most a unique filtration $M$ satisfying ([De 80], I.8.15)

$$NM_j \subset M_{j-2}, \text{ and } \mathrm{Gr}_{i+k}^M \mathrm{Gr}_i^W L \xrightarrow{N^k} \mathrm{Gr}_{i-k}^M \mathrm{Gr}_i^W L$$

If $M$ exists, it is called the relative monodromy filtration. In particular, if $N_i$ is the endomorphism induced by $N$ on $\mathrm{Gr}_i^W L$, then $M$ induces on $\mathrm{Gr}_i^W L$ the monodromy filtration $W(N_i)[i]$ shifted by $i ((W(N_i)[i])_{i+j} := W(N_i)_j)$. Properties of the filtration are given in [StZ 85] and [E 83].
### 2.3.1 Nilpotent orbit

Let $(L, F, S, N_1, \dots, N_n, w')$ be defined by a finite dimensional complex vector space $L$ with a $\mathbb{Q}$-structure, a decreasing filtration $F$ by complex vector subspaces, an integer $w'$, a non-degenerate rational bilinear form $S$ satisfying

$$S(x, y) = (-1)^{w'} S(y, x) \text{ for } x, y \in L \text{ and } S(F^p, F^q) = 0 \text{ for } p+q > w',$$

and pairwise commuting nilpotent endomorphisms $N_1, \dots, N_n$ defined over $\mathbb{Q}$, such that $S(N_i x, y) + S(x, N_i y) = 0$ and $N_i F^p \subset F^{p-1}$.

**Definition 2.12 (Nilpotent orbit).** ([Sc 73], [CaKSc 86], [Gr 74], [Ka 86], 4.1) The above data is called a (polarized) nilpotent orbit of weight $w'$ if:

(i) The monodromy filtration $W(N)$ of the endomorphism $N = \sum_{j \in [1, n]} t_j N_j$ is independent of $t_j$ if $t_j > 0$ for all $j$.

(ii) Let $M := W(N)[w']$ denote the filtration shifted to right ($M_{w'+k} := W(N)_k$). The data $(L, M, F)$ is a MHS on $L$.

(iii) The bilinear form $S_k$ such that $S_k(x, y) = S(x, N^k y)$ polarizes the primitive subspace $P_k = \text{Ker} (N^{k+1} : \text{Gr}_{w'+k}^M \to \text{Gr}_{w'-k-2}^M)$ with its induced HS of weight $w' + k$.

The various definitions in this chapter are based on the following result applied throughout this volume to Deligne's extension $\mathcal{L}_X$ (Equation 3.9):

**Theorem** (Cattani, Kaplan and Schmid [CaKSc 86]) Let $\mathbf{L}$ be polarized VHS on the complement of a NCD $Y \subset X$. At each point $x \in Y$, the extension of the Hodge filtration $\mathcal{F}$ on $\mathcal{L}_{X\setminus Y}$ defines a nilpotent orbit with local monodromy actions $(N_i, i \in I := [1, n])$ on the limit fiber $L := \mathcal{L}_X(x)$.

Let $N_K := W(\sum_{i \in K \subset I} N_i)$, for each partition $K \cup J = I, W(N_I)$ coincides with the relative monodromy filtration $M(N_K, W(N_J))$.

*Remark 2.8.* i) The results mentioned above, along a period starting around 1966, are difficult to prove but well covered by the references.

i) The proof of the theorem is based on a generalization of the nilpotent and $SL_2$ orbit approximations of the period matrix to higher dimension [Sc 73, GrSc 73, CaKSc 86].

In general, for each $J \subset [1, n]$ and fixed $t_j > 0$, $N_J := \sum_{j \in J} t_j N_j$, is a nilpotent endomorphism of $L$ such that the increasing monodromy filtration $W(N_J)$ is independent of the positive real integers $t_j$ and we have a variation of limit MHS at points of $Y_J^*$ depending on $J$.

ii) The concept of splitting of a MHS is necessary to the proof: there exists a real number $c$ such that $(L, \exp(i \sum_j t_j N_j)F)$ is a HS of weight $w'$, polarized by $S$ whenever $t_j > c$.

iii) The filtration $W(N)$ in the definition above is also said to define a MHS on $L$ of weight $w'$.

iv) In the text, we refer to the summary of the results in ([KaK 87] §1.2).
### 2.3.2 Infinitesimal mixed Hodge structure (IMHS)

We consider a data $(L, W, F, N_1, \dots, N_n)$, where $L$ is a finite dimensional complex vector space with a $\mathbb{Q}$-structure, $W$ is an increasing $\mathbb{Q}$-filtration, $F$ is a decreasing $\mathbb{C}$-filtration and $N_1, \dots, N_n$ are commuting nilpotent endomorphisms defined over $\mathbb{Q}$ such that: $N_j F^p \subset F^{p-1}$ and $N_j W_k \subset W_k$.

**Definition 2.13 (Mixed nilpotent orbit).** The above data is called a (graded polarizable) mixed nilpotent orbit if for each integer $i$ the data with restricted structures

$$(\mathrm{Gr}_i^W L, F|, (N_1)|, \dots, (N_n)|)$$

is a nilpotent orbit of weight $i$ for some polarization $S_i$.

We consider only graded polarized mixed nilpotent orbits (called pre-infinitesimal mixed Hodge module in [Ka 86], 4.2).

**Definition 2.14 (IMHS).** ([Ka 86], 4.3)

A mixed nilpotent orbit $(L, W, F, N_1, \dots, N_n)$ is called an infinitesimal mixed Hodge structure (IMHS) if the following conditions are satisfied:

(i) For each $J \subset I = \{1, \dots, n\}$, the monodromy filtration $M(J)$ of $\sum_{j \in J} N_j$ relative to $W$ exists and satisfies $N_j M(J)_i \subset M(J)_{i-2}$ for all $j \in J$ and $i \in \mathbb{Z}$.

(ii) The filtrations $M(I)$ and $F$ define a graded polarizable MHS on $L$ such that the filtrations $W$ and $M(J)$ are compatible with this MHS (filtrations by sub-MHS and the $N_i$ are morphisms of type $(-1, -1)$).

We refer to the filtration $M(\sum_{i \in I} N_i, W)$ as the limit MHS.

**Remark 2.9.** (i) The filtration $M(J)$ is the weight filtration relative to $W$ of any $N \in C(J) := \{\sum_{j \in J} t_j N_j; t_j > 0\}$.

(ii) In the case of an admissible VMHS $(L, W, F)$ on the polydisc $(D^*)^n = D^n \setminus Y$ where $Y \subset D^n$ is the natural NCD, an IMHS is attached to $L$, at each $x \in Y_J^*$, $J \subset [1, n]$, on the fiber of Deligne's extension $\mathcal{L}(x)$. Since $Y_J$ is defined by $z_i$ for $i \in J = \{i_1, \dots, i_j\}$, we introduce the local system on $\psi_J L$ as the restriction to $Y_J^*$ of $\psi_{i_j} \circ \dots \circ \psi_{i_1} L$ (independent of the order in $J$ up to isomorphism; we write $\psi_j$ for $\psi_{z_j}$) to define the VMHS $(\psi_J L, \mathcal{L}_{Y_J^*}, M(J), F(J))$ on $Y_J^*$ where $M(J)$ and $F(J)$ form the limit IMHS at varying $x \in Y_J^*$. Moreover, a combination of $\psi_{z_i}$ and $\varphi_{z_j}$ is necessary to characterize locally a perverse sheaf.

(iii) At $y \in Y_J \setminus Y_J^*$ there exist two IMHS on $\mathcal{L}(y)$, one defined directly by $(L, W, F)$ and one defined by $(\psi_J L, \mathcal{L}_{Y_J^*}, M(J), F(J))$. They coincide since $M(N_1 + \dots + N_i)$ is the relative weight filtration of $N_i$ with respect to $M(N_1 + \dots + N_{i-1})$ ([Ka 86, Prop. 5.2.5]). The IMHS $(M(J), F(J))$ can be viewed as successive IMHS along the various $Y_J^*$.
*Example 2.5.* In the case of two variables, a remarkable result states that the limit $(W(N_1+N_2), F)_0$ is relative to the limit at 0 of the limit $(W(N_i), F)_x$ at $x_j, j \neq i$

<figure><img src="image_4.png" /></figure>

A morphism of two IMHS $f : (L, W, F, N_1, \dots, N_n) \to (L', W, F, N'_1, \dots, N'_n)$ is a linear morphism $L \to L'$ respecting $W, F$, commuting with $N_i$ and $N'_i$ inducing a morphism of a graded nilpotent orbits with respect to $\mathrm{Gr}_*^W$.

A graded polarized MHS is an IMHS when identified with the object with $N_i = 0$ for $i \in [1, n]$. If $(L, N, F)$ is a nilpotent orbit, then $(L, W(N), F)$ is an IMHS with $N_i = 0$ and $N : (L, W(N), F) \to (L, W(N)[2], F(-1))$ is a morphism of IMHS with $N_i = 0$ on both sides.

**Proposition 2.7.** ([Ka 86] Prop. 5.2.6) The category of IMHS is abelian. The filtrations $W, F$, and the limit MHS $M(J)$ for any set $J$ of indices, define exact functors on this category.

A morphism $f : L \to L'$ of two IMHS is necessarily strict for $W$, since by the nilpotent orbit theorem $W$ is the weight filtration and $\exp(i \sum_j t_j N_j F)$ the Hodge filtration of a MHS on $L$ (resp. $L'$) for $t_i$ large enough. Similarly $M(N_1, W)$ is the weight of a MHS on $L$ (resp. $L'$).

To prove that the restriction of $M(N_1, W)$ to $L'' := \mathrm{Ker} f$ is the monodromy filtration relative to $W \cap L''$, let $W''$ and $M''$ denote the restrictions to $L''$, then we deduce from the exact sequence $0 \to \mathrm{Gr}_{k+i}^{M''} \mathrm{Gr}_k^{W''} \to \mathrm{Gr}_{k+i}^M \mathrm{Gr}_k^W \to \mathrm{Gr}_{k+}^{M'} \mathrm{Gr}_k^{W'}$ and the corresponding exact sequence for $\mathrm{Gr}_{k-i}$ that $N_1^i : \mathrm{Gr}_{k+i}^{M''} \mathrm{Gr}_k^{W''} \to \mathrm{Gr}_{k-i}^{M''} \mathrm{Gr}_k^{W''}$ is an induced isomorphism.

**Definition 2.15 (Preadmissible VMHS).** A variation of mixed Hodge structure $(\mathbf{L}, W, F)$ graded polarized over the punctured unit disc $D^*$ with local monodromy $T$ and associated filtration $W$ by subbundles of Deligne's extension $\mathcal{L}_{\mathcal{O}_D}$, is called *preadmissible* if the following conditions are satisfied:

i) The Hodge filtration $\mathcal{F} \subset \mathcal{L}_{\mathcal{O}_D}$ extends to a filtration $\mathcal{F}$ on $\mathcal{L}_{\mathcal{O}_D}$ by subbundles inducing for each $k$ on the graded sheaf with respect to the filtration $W$ defined by $W \mathrm{Gr}_k^W \mathcal{L}_{\mathcal{O}_D}$, Schmid's extension of the Hodge filtration.

ii) Let $W^0 := W(0)$, $F_0 := \mathcal{F}(0)$ denote the filtrations on the fiber $L := \mathcal{L}_{\mathcal{O}_D}(0)$ at $0 \in D$ and $T$ the local monodromy at $0$, $N = \log T^u$; then the following two conditions must be satisfied:

1) $NF_0^p \subset F_0^{p-1}$ for all $p \in \mathbb{Z}$.

2) The weight filtration $M(N, W^0)$ relative to $W^0$ exists.
Notice that the extension of the filtration $\mathcal{F}$ to $\mathcal{L}_{\mathcal{O}_D}$ cannot be deduced in general from the various Schmid extensions to $Gr_k^W \mathcal{L}_{\mathcal{O}_D}$. We remark that the filtrations $M := M(N, W^0)$ and $F_0$ at the origin define a MHS.

**Definition 2.16 (Admissible VMHS).** ([Ka 86], §1.8) Let $X$ be a complex analytic space and $U$ a nonsingular open subset, complement of a closed analytic subset of $X$. A graded polarized variation of mixed Hodge structure $(\mathbf{L}, W, F)$ on $U$ is called *admissible* if, for every analytic morphism $f : D \to X$ on a unit disc which maps $D^*$ to $U$, the inverse $(f|_{D^*})^*(\mathbf{L}, W, F)$ is a preadmissible variation on $D^*$.

In the case of locally unipotent VMHS, Kashiwara remarked that preadmissible VMHS on the unit disc are necessarily admissible [Ka 86].

*Remark 2.10 (IMHS defined by an admissible VMHS).* i) Let $Y \subset X$ be a NCD and $x \in Y_M^*$ for $M \subset Y$. In the text we introduce Deligne’s extension $\mathcal{L}_X$ associated to a local system $\mathcal{L}$ on $X \setminus Y$ and the vector space $L := \mathcal{L}_X(x)$ (Definition 3.1). An admissible VMHS $(\mathbf{L}, W^0, F)$ on $X \setminus Y$ defines at $x$ an

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

where the filtration $W^0$ of $L$, preserved by $N_i$, is the limit of the filtration $W^0$ of $\mathcal{L}$. In this case, the relative monodromy filtration $M(\sum_{i \in M} N_i, W^0)$ defines the limit MHS on $L$.

ii) Let $Z := X \setminus U$ and let $\pi : X' \to X$ be a desingularization of $X$ such that $Z' := \pi^{-1}Z$ is a NCD, then $(\pi^*\mathbf{L}, W, F)$ is admissible along $Z'$. In particular $(\pi^*\mathbf{L}, W, F)|_{D^*}$ defines an IMHS at any point in $Z'$.

In the text, we define an induced Hodge filtration $F$ on the intermediate extension of a local system on $X \setminus Y$ (see §2.2.1.3) which reflects the behaviour of the limit filtration $F$ on $\mathcal{L}$ at ‘infinity’ along the NCD $Y$.

## 2.4 Induced filtrations on the perverse cohomology

The category of bifiltered complexes is defined in ([De 72] and [De 75]). Let $(K', W, F) \in D^b F_2(X, \mathbb{C})$ be a bifiltered complex of abelian sheaves on the variety $X$. A morphism $\phi : (K_1, W, F) \to (K_2, W, F)$ in the category $D^b F_2(X, \mathbb{C})$ is an isomorphism if the morphism $Gr_*^W Gr_F^* K_1 \to Gr_*^W Gr_F^* K_2$ is a quasi-isomorphism. If $W$ is the weight filtration defined with $\mathbb{Q}$-coefficients, a precise diagram of compatibility with the $\mathbb{Q}$-structure is satisfied.

Let $f : X \to V$. The derived direct image $(K'', W, F) := (f_*K', f_*W, f_*F)$ is defined in $D^b F_2(V, \mathbb{C})$ by taking $f_*$-acyclic bi-filtered resolutions of the complex $(K', W, F)$ ([De 72] 1.4).

The perverse filtration $p_{\tau_i}$ on a complex $K$ on a variety $V$ is defined by a family of morphisms $\varphi_i : p_{\tau_i} \to K$. Given a filtration $F$ of $K$, the problem is to define the induced filtration $F$ on the perverse cohomology.
It is possible to define the resticron of $F$ to $p_{\tau_i}$ by $\varphi_i^{-1}(F)$, however by a remark of Deligne, it is possible to suppose $p_{\tau}$ a filtration by sub-complexes of $K$ which has the advantage to apply directly the results of ([De 72]).

### 2.4.0.1 Embedding of the perverse filtration

For any morphism of complexes $f : L \to K$, let $L'$ denote the acyclic cone $L' := \mathrm{cone}(\mathrm{id} : L \to L)$. We consider after ([BBD 83] 1.1.2) the following embedding $\sigma_f$ defined by:

$$ \sigma_f : L \to K' := K \oplus L' : \sigma_f : L^i \to K^i \oplus L^{i+1} \oplus L^i : v \mapsto (f^i(v) + 0 + v) $$

The injection $u : K \to K \oplus L'$ is a quasi-isomorphism such that $u \circ f$ is homotopic to $\sigma_f$, while the projection $p : K \oplus L' \to K$ defines a quasi-isomorphism $K' \xrightarrow{\sim} K$ such that $p \circ \sigma_f = f$.

Given a filtration $F$ by sub-complexes of $K$, we define the filtration $F' := F \oplus L' \subset K'$ such that the restriction of $F'$ to $\sigma_f(L)$ is isomorphic to the inverse image $f^{-1}F$ on $L$. We deduce:

**Lemma 2.8.** Let $K$ be a bounded constructible complex on a variety $V$, $F_1$ and $F_2$ two filtrations of $K$ by sub-complexes (not necessarily constructibles) and let $p_{\tau}$ denote the perverse filtration of $K$.

There exists a bifiltered complex $(K', F_1', F_2')$ where $K'$ is bounded and constructible with filtrations by sub-complexes of $K'$ and a bifiltered quasi-isomorphism $\pi : (K, F_1, F_2) \to (K', F_1', F_2')$ such that the perverse filtration of $K'$ is defined by sub-complexes of $K'$.

In the case of a finite sequence of complexes $g_i : p_{\tau_i} \to K$ we repeat the above process by a descending induction on the indices $i$.

**Corollary 2.2.** i) A well defined filtration $F$ on the perverse cohomology $\mathcal{P}\mathcal{H}^i(K)$ is obtained by inducing the filtration $F$ on $p_{\tau_i}K$ followed by the projection on $\mathcal{P}\mathcal{H}^i(K)[-i] := p_{\tau_i}K/p_{\tau_{i-1}}K$.

In the text, we suppose the filtrations such as $p_{\tau}, W, F$ defined by sub-complexes of a complex $K$ on a variety $V$.

**Remark 2.11.** i) There is no perverse filtration defined by a sub-complex $F^i \subset K$ if $F$ is not constructible.

The bifiltered complex $(K, p_{\tau}, F) \in D^b F_2(V, \mathbb{C})$ is defined in the bifiltered category of sheaves of complex vector spaces (forgetting constructibility).

ii) Given a functor $T$, we consider acyclic $T$-resolutions of $Gr_p^{p_{\tau}}Gr_F^q K$. In the case of the global section functor $\Gamma$, we have the canonical acyclic bi-filtered Godement resolution $\mathcal{C}^*(K, p_{\tau}, F)$ of $(K, p_{\tau}, F)$ to define $R(\Gamma, *)$. Since the resolution $\mathcal{C}^*$ is canonical, the complex $Gr_F^iGr_j^{p_{\tau}}\mathcal{C}^*$ is also an acyclic resolution of $Gr_F^iGr_j^{p_{\tau}}K$. In particular, we deduce in this way the correct Hodge filtration on $\mathbb{H}^k(V, \mathcal{P}\mathcal{H}^i(K))$.
iii) In presence of Hodge filtration $F$ or weight filtration $W$ we deduce induced filtrations on the perverse cohomology. The construction is similar to Deligne's definition of induced filtrations filtration such that we can apply [De 72]. Since the filtration $W$ is perverse, we consider the perverse weight spectral sequence and the induced filtration $F$ on $\mathcal{P}\mathcal{H}^{p+q}(Gr_{-p}^W K)$.

The image filtration $\text{Im}: \mathcal{P}\mathcal{H}^{p+q}(W_{-p}K) \to \mathcal{P}\mathcal{H}^{p+q}(K)$ in the abelian category of perverse sheaves defines a filtration $W$ on $\mathcal{P}\mathcal{H}^{p+q}(K)$ with induced filtration $F$ on the limit of the spectral. By abuse of language, we say the filtered perverse spectral sequence or perverse filtration to refer to such induced filtrations.

### 2.4.0.2 Abbreviations and conventions

1. Polarized VHS (resp. VMHS) stands for polarized variation of Hodge structure (resp. mixed Hodge structure). A polarized VHS $(\mathbf{L}, F)$ of weight $w'$ is given by a local system $\mathbf{L}$ on a smooth algebraic open subset $U \subset V$ and a decreasing filtration $F$ of $\mathcal{L} = \mathbf{L} \otimes \mathcal{O}_U$. The complex $\mathcal{P}\mathbf{L} := \mathbf{L}[\dim V]$ shifted by the dimension of $V$, is a perverse sheaf and $(\mathcal{P}\mathbf{L}, F)$ is a complex of weight $\mathcal{P}w = w' + \dim V$. We say that the perverse sheaf $\mathcal{P}\mathbf{L}$ is a shifted polarized VHS.

2. Given a functor $T: \mathfrak{A} \to \mathfrak{B}$ of abelian categories, its derived functor is also denoted by $T: D^+\mathfrak{A} \to D^+\mathfrak{B}$ after chapter 3 instead of $RT$ in chapter 2 [Ve 77] (with exceptions if confusion may arise such $R\Gamma$ for example). However, we still denote the $i$-th cohomology of the derived image by $T$ by $R^i T(K) := H^i(T(K))$.

3. On a complex variety $V$, the topological middle perversity truncations $p_{\tau \le i}$ on $K$ ([BBD 83] section 2 and Proposition 2.1.17) define an increasing filtration denoted $p_{\tau}$ on the hypercohomology:

$$p_{\tau_i} \mathbb{H}^k(V, K) := \text{Im} \{ \mathbb{H}^k(V, p_{\tau_i} K) \to \mathbb{H}^k(V, K) \}.$$

4. An intermediate extension is denoted by $j_* \mathcal{P}\mathbf{L}$, while $\mathcal{P}\mathcal{H}^i(K)$ denotes the $i$-th perverse cohomology of a complex $K$.
<figure><img src="image_5.png" /></figure>
Chapter 3

Perverse sub-sheaves of the
logarithmic complex Ω*PΛ

In [De 72] Deligne introduced the concept of Hodge complex (HC) of sheaves on a compact complex variety X as a filtered complex (K, F) such that its hypercohomology (RΓ(X, K), F) is a Hodge structure (HS). The de Rham complex of a non singular complex variety with its Hodge filtration is a HC. Originally Deligne distinguished between the cohomological HC of sheaves and the HC of its hypercohomology. By abuse of language, only the terminology of HC survived.

If X is not compact, a bifiltered mixed Hodge complex (MHC) (K, W, F) is constructed on a compactification X such that (RΓ(X̄, K), W, F) is a mixed Hodge structure (MHS).

The introduction of the intersection complex by Goresky and MacPherson
[GMacP 88] lead to a new type of HC.

Let $\mathbf{L}$ be a local system on the complement of a NCD $Y \subset X$ in a complex non singular variety $X$ and let $j := (X \setminus Y) \to X$. We describe in this chapter the well known logarithmic complex $\Omega^*\mathcal{L} := \Omega_X^*(\text{Log}Y) \otimes \mathcal{L}_X \simeq j_*\mathbf{L}$ with coefficients in Deligne's extension $\mathcal{L}_X$ [De 70] in order to define the Intersection complex $\text{IC}^*\mathcal{L} \subset \Omega^*\mathcal{L}$ embedded as a sub-complex with its Hodge filtration (see §3.1).

Let $\mathcal{P}\mathbf{L} := \mathbf{L}[\dim X]$, the complex $\Omega^*\mathcal{P}\mathcal{L} := \Omega^*\mathcal{L}[\dim X]$ is the associated
perverse sheaf with the conventional shifted degree. It is common to say
perverse logarithmic complex with or without shift.

Let $i_Z : Z \to X$ be the embedding of a NCD sub-divisor of $Y \subset X$ and
$j_Z := (X \setminus Z) \to X$. We construct an embedded logarithmic complex isomor-
phic to $j_Z * j_Z^*(j_* \mathcal{P} \mathbf{L})$ denoted by $\mathrm{IC}^* \mathcal{P} \mathcal{L}(\mathrm{Log} Z) \subset \Omega^* \mathcal{P} \mathcal{L}$ (cf §3.2).

To motivate the constructions are motivated by the application of Kashi-
bara's criteria for a decomposition of a perverse sheaf into a direct sum of
intermediate extensions (cf §3.2.6).

We recall the behavior of perverse sheaves along locally principal divisor
(see §3.2.4), to construct later more bifiltered logarithmic complexes, such as

$$
j_Z! j_Z^*(j_* \mathcal{P} \mathbf{L}), i_Z^* j_Z * j_Z^*(j_* \mathcal{P} \mathbf{L}), i_Z^! j_Z * \mathcal{P} \mathbf{L} \text{ and } i_Z^* j_* \mathcal{P} \mathbf{L}
$$

We refer to such complexes as bifiltered logarithmic complexes.
## 3.1 Perverse logarithmic sub-complexes of $\Omega^{*p}\mathcal{L} \simeq j_*^p\mathcal{L}$

Let $\mathbf{L}$ be a locally constant sheaf (finite dimensional local system of vector spaces) on the complement $U$ of a NCD $Y$ in a smooth complex algebraic variety $X$ and $\mathcal{L}_U := \mathbf{L} \otimes \mathcal{O}_U$ the analytic locally free sheaf of modules defined by $\mathbf{L}$ on $U$.

There is a correspondence between local systems $\mathbf{L}$ and connections $(\mathcal{L}_U, \nabla)$ with vanishing curvature and horizontal sheaf of sections of $\nabla$.

Locally at each point $x \in Y$, a ball $B_x$ with center $x$ and $B_x^* := B_x \cap U$ are Stein. The restriction of $\mathcal{L}$ to $B_x^*$ is free. Each trivialization gives rise to a free extension of $\mathcal{L}|_{B_x^*}$.

Moreover, since the local system $\mathbf{L}$ is defined on $U$ ($\mathcal{L}_U$ is flat), there exists a locally free global extension of $\mathcal{L}_U$ with a regular singular connection (or with logarithmic singularities), that is a fibre bundle $\mathcal{L}_X$ extending $\mathcal{L}_U$ and a connection $\nabla: \mathcal{L}_X \to \Omega_X^1(\mathrm{Log}Y) \otimes \mathcal{L}_X$ ([De 70], [Ma 87] definition 3.1). The residue of $\nabla$ is defined along a component $Y_i$ of the NCD $Y$ as an endomorphism of the restriction $\mathrm{Res}_{Y_i} \nabla: \mathcal{L}_{Y_i} \to \mathcal{L}_{Y_i}$, of $\mathcal{L}_X$ to $Y_i$.

The eigenvalues of the residue are constant along a connected component of $Y_j$ and related to the local monodromy $T_j$ of $\mathbf{L}$ at a general point of $Y_j$ by the formula ([De 70], theorem 1.17, proposition 3.11):

$$ \mathrm{Log} T_j = -2i\pi \mathrm{Res}_{Y_j} \nabla \qquad (3.1) $$

Deligne's idea is to fix the choice of the residues of the connection by the condition that the eigenvalues of the residue belong to the image of a section of the projection $\mathbb{C} \to \mathbb{C}/\mathbb{Z}$, determined by fixing the real part of $z: m \le \mathcal{R}(z) < m+1$, hence fixing the determination of the logarithm $\mathrm{Log}: \mathbb{C}^* \to \mathbb{C}$, which forces the uniqueness of the local extensions. Hence the local constructions glue into a global bundle on $X$. The extension in the case $m=0$ is called Deligne's extension.

The connection $\nabla$ extends naturally into a complex, called the logarithmic complex $\Omega_X^*(\mathrm{Log}Y) \otimes \mathcal{L}_X$ and denoted by $\Omega^*\mathcal{L}$.

### 3.1.1 Local definitions along a NCD $Y \subset X$

Let $Y = \cup_{i \in I} Y_i$ denote the decomposition of $Y$ into smooth irreducible components with finite index $I$.

For $M \subset I$, set $Y_M := \cap_{i \in M} Y_i$, $Y_M^* := Y_M \setminus \cup_{i \notin M} (Y_i \cap Y_M)$ ($Y_0^* := X \setminus Y$). We denote uniformly the various embeddings by $j: Y_M^* \to X$.

To describe $\mathcal{L}_X$ in local coordinates, let $n := |M|$, $x \in Y_M^*$ and $B_x$ a small polydisc in $X$ with center $x$, isomorphic to $D^{n+\ell}$ with coordinates $y_i$. It is not restrictive to suppose $\ell=0$, such that $Y \cap B_x$ is defined by $y_1 \cdots y_n = 0$.
and $B_x^* := B_x \setminus (Y \cap B_x)$ is isomorphic to $(D^*)^n$. We identify the ball $B_x$ with $D^n \subset \mathbb{C}^n$ the product of complex discs.

The restriction of the VHS $\mathbf{L}_{|B_x \setminus (B_x \cap Y)}$ corresponds to a VHS $\mathbf{L}$ on $B^* = (D^*)^n$ defined by a set $(L, T_i, i \in [1, n])$ including a rational vector space $L$ isomorphic to the multivalued horizontal sections of $\mathbf{L}$ and $n$-commuting linear automorphisms called the monodromy transformations of $L$.

Let $\pi: \tilde{B} \to B^*$ denote a universal covering and $z = (z_1, \dots, z_n)$ a set of coordinates on $\tilde{B} \subset \mathbb{C}^n$ such that $\pi(z) = y$ has components $y_j := \exp 2\pi i z_j \in (D^*)^n$. For each $j$, the monodromy automorphism $T_j$ on $L$ along a closed path around a component $Y_j$ of $Y$, is defined over $\mathbb{Q}$ and decomposes into a semi-simple $T_j^s$ and a unipotent $T_j^u$ automorphisms: $T_j = T_j^s T_j^u$. On the complex vector space $L$, all automorphisms $T_j^s$ are simultaneously diagonalizable and conjugate to the diagonal matrix of their eigenvalues.

We define the logarithm of $T_j^u$ by:

$$N_j := \log T_j^u := \sum_{k \ge 1} (-1)^{k+1} (T_j^u - I)^k / k \quad \text{for } j = 1, \dots, n. \quad (3.2)$$

Along any closed path in $B^*$ the logarithm of the monodromy is an integral linear combination of the nilpotent endomorphisms $N_j$.

Let $(a.) = (a_1, \dots, a_n)$, where $a_j$ is an eigenvalue of $T_j$, then $L = \oplus_{(a.)} L^a$, where $L^a = \bigcap_{i \in [1, n]} \ker(T_i^s - a_i I)$. Let $\alpha_j \in \mathbb{C}$ with real part in $[0, 1[$ such that $\mathbf{e}(\alpha_j) := \exp(-2i\pi\alpha_j) = a_j$. For $v \in L^a$:

$$T_j^s(v) = a_i v, \quad T_j^u(v) = (\exp N_j)v \quad \text{and} \quad T_j(v) = \exp(-2i\pi\alpha_j + N_j)v.$$

Set $L_\infty := \Gamma(\tilde{B}, \pi^{-1}\mathbf{L}) \simeq L$, $T_j$ acts on $L_\infty$ by translation $z_j \mapsto z_j + 1$:

$$\forall v \in L_\infty := \Gamma(\tilde{B}, \pi^{-1}\mathbf{L}), v(z_1, \dots, z_j + 1, \dots, z_n) = T_j v(z_1, \dots, z_j, \dots, z_n).$$

Let $\mathcal{L}_{B^*} := \mathbf{L} \otimes_{\mathbb{Q}} \mathcal{O}_{B^*}$ denote the locally free analytic bundle on $B^*$. It is free since $B^*$ is Stein. For each $v \in L^a$, the analytic section $\tilde{v} \in (\tilde{B}, \pi^*\mathcal{L}_{B^*})$

$$\tilde{v} = \exp\left(\sum_{j \in [1, n]} z_j(2i\pi\alpha_j - N_j)\right)v, \quad (a_j := \exp(-2i\pi\alpha_j), \quad N_j := \log T_j) \quad (3.3)$$

([De 70], 5.2.1-3) is the inverse image of a (single valued) analytic section of $\mathcal{L}_{B^*}$ on $B^*$, since $\tilde{v}$ is invariant by each $T_j$:

$$\tilde{v}(z_1, \dots, z_j + 1, \dots, z_n) = \tilde{v}((T_j^s)^{-1}(T_j^u)^{-1}T_j v) = \tilde{v}(z_1, \dots, z_j, \dots, z_n).$$

Since $L = \oplus_{(a.)} L^a$, the above tilde correspondence extends to $L$:

**Definition 3.1 (Tilde correspondence).** There exists a canonical tilde embedding

$$\sim: \Gamma(\tilde{B}, \pi^{-1}\mathbf{L}) \xrightarrow{\sim} \tilde{L} \subset \mathcal{L}_{B,0} \subset \Gamma(B^*, \mathcal{L}), \quad \sim: v \mapsto \tilde{v}.$$

The fibre $\mathcal{L}_{B,0}$ is the $\mathcal{O}_{B,0}$-module generated by the image $\tilde{L}$ of the tilde embedding. It is the fiber of a free analytic sheaf $\mathcal{L}_B$ on $B$.
The vector space $\mathcal{L}_B(0) \simeq L_\infty$ will be denoted also by $L$. A basis $v_a$ of $L$ lifts into a basis $\tilde{v}_a$ of $\mathcal{L}_{B,0}$.

The connection $\nabla$ extends to a differential of the logarithmic de Rham complex $\Omega_X^* \log Y) \otimes_{\mathcal{O}_X} \mathcal{L}_X$ ([De 70] Définition I.2.11, (3.12, b), [Ma 87] §4). The endomorphisms $\alpha_i I$ and $N_i$ correspond to endomorphisms on the image sections $\tilde{v}$ in $\mathcal{L}_{X,x}$ denoted by the same symbols $\alpha_i I$ and $N_i$.

They define the connection ([De 70], Théorème 1.17)

$$ \nabla \tilde{v} = \sum_{j \in J} \left( \widetilde{\alpha_j v} - \frac{1}{2i\pi} \widetilde{N_j v} \right) \otimes \frac{dy_j}{y_j}. $$

Since $2\pi i z_j := \log y_j$, the tilde correspondence is often expressed in terms of the local coordinates $y_j$ as follows

$$ \tilde{v}((y_1, \dots, y_n) = \Pi_{j \in J} y_j^{\alpha_j} \exp(\sum_{j \in J} -1/2i\pi(\log y_j)N_j) \cdot v \quad (3.4) $$

Recall, that there exists a correspondence between representation of $\pi_1(M)$, local systems and integrable connections ([De 70], [Vo 2007]).

### 3.1.2 The complex $\Omega^*L$ and the tilde embedding

The proofs are carried for the transcendental topology although we omit the analytic notation $X^{an}$.

In view of the local description of perverse sheaves on a product of discs (see 3.2.6.1 and 3.2.6), many constructions on the logarithmic complex, such as the weight filtration, are described first locally on a complex of vector spaces $\Omega^*L$, defined at a point $x \in Y$, then carried to the logarithmic complex by an extension of the tilde correspondence $L \hookrightarrow \mathcal{L}_x$ to a tilde embedding into the de Rham complex: $\Omega^*L \to (\Omega_X^*(\log Y) \otimes_{\mathcal{O}_X\mathcal{L}})_x$ (see Lemma 3.1).

First, we give a general definition of the complex $s(L_J, f_i)$ determined by:

1. a vector space $L$, a nonnegative integer $n$ and a linear subspace $L_J$ of $L$ for each $J \subset [1, n]$;

2. a collection of commuting endomorphisms $f_i$, $i = 1, \dots, n$ of $L$.

This data should satisfy the condition that $f_i L_{J \setminus \{i\}} \subset L_J$ for each $i \in J$.

The subsets $J$ of length $k := |J|$ correspond to strictly increasing sequences $i_1 < \dots < i_k \subset [1, n]$). Let $S_k$ denote the set of such strictly increasing sequences. For $J \in S_k$ and $j \in J$, let $\varepsilon_j = (-1)^r$ when $j$ is the $r$-th element of $J$ in the increasing order.

**Definition 3.2** $(s(L_J, f_i))$. The complex $s(L_J, f_i)$ is defined by

$$ s(L_J, f_i)^k = \bigoplus_{J \in S_k} L_J \text{ for } k = 0, \dots, n $$

with differentials $d_{k-1} : s(L_J, f_i)_{k-1} \to s(L_J, f_i)_k$ for $k = 1, \dots, 1$ given by

$$ (d_{k-1}v)_J = \sum_{j \in J} \varepsilon_j f_j v_{J \setminus \{j\}}. $$
Returning to the case where $L$ is a complex vector space with commuting endomorphisms $\alpha_j Id - (1/2i\pi)N_j$ for $j \in [1, n]$ where $\alpha_j \in \mathbb{C}$ and $N_j$ is nilpotent, let $L^u \subset L$ be the subspace on which the action of each monodromy $T_i$ is unipotent. We define $\Omega^*L^u$ and $\Omega^*L$ as follows:

**Definition 3.3** ($\Omega^*L^u \subset \Omega^*L$). i) The complex $\Omega^*L$ is defined by the special case where $L_J = L$ for each $J$ and $f_j = \alpha_j Id - (1/2i\pi)N_j$ for $j \in [1, n]$.

ii) The sub-complex $\Omega^*L^u \subset \Omega^*L$ is defined by the special case where $L_J = L^u$ for each $J$ and $f_j = N_j$ for $j \in [1, n]$.

$$ \Omega^*L^u := 0 \to L^u \to \dots \to \bigoplus_{\{i_1<\dots<i_{k-1}\}} L^u \to \bigoplus_{\{i_1<\dots<i_k\}} L^u \to 0 $$

With the same notations, for each $J \in S_k$ let $f_J := \Pi_{j \in J}(\alpha_j Id - \frac{1}{2i\pi}N_j)$ denote the composition morphism for $j \in J$.

**Definition 3.4** ($IC^*L$). The complex $IC^*L$ is the subcomplex of $\Omega^*L$ defined by $L_J = f_J L := \text{Im}(f_J : L \to L)$ for each $J \subset [1, n]$ and $f_J = \alpha_j Id - \frac{1}{2i\pi}N_j$ for $j \in [1, n]$.

*Remark 3.1.* For each sequence $\alpha. = (\alpha_1, \dots, \alpha_n)$, set $e(\alpha_j) := \exp(-2i\pi\alpha_j)$, such that $e(\alpha.) = e(\alpha_1, \dots, \alpha_n)$ is a sequence of eigenvalues of $T_.$ = $(T_1, \dots, T_n)$. Set $IC^*L e^{(\alpha.)}$ the subcomplex of $\Omega^*L$ where $L_J = f_J L e^{(\alpha.)} := \text{Im}(f_J : L e^{(\alpha.)} \to L e^{(\alpha.)})$ for each $J \subset [1, n]$ and $f_J = \alpha_j Id - \frac{1}{2i\pi}N_j$ for $j \in [1, n]$. We have $IC^*L := \bigoplus_{\alpha.} IC^*L e^{(\alpha.)}$.

We introduce for each sequence $\alpha$. the subset $M(\alpha.) \subset [1, n]$ such that $j \in M(\alpha.)$ if and only if $\alpha_j = 0$. Let $N_{J \cap M(\alpha.)} = \Pi_{j \in J \cap M(\alpha.)} N_j$ (it is the identity if $J \cap M(\alpha.) = \emptyset$). For each $J \subset [1, n]$, we have the equality of the image subspaces: $f_J L e^{(\alpha.)} = N_{J \cap M(\alpha.)} L e^{(\alpha.)}$ since the endomorphism $(\alpha_j Id - \frac{1}{2i\pi}N_j)$ is an isomorphism on $L e^{(\alpha.)}$ whenever $\alpha_j \neq 0$, hence

$$ IC^*L \simeq \bigoplus_{\alpha.} s(N_{J \cap M(\alpha.)} L e^{(\alpha.)})_{J \subset M} \quad (3.5) $$

**Lemma 3.1** (The tilde embedding). Let $x \in Y_M^*$ be a general smooth point on $\cap_{i \in M} Y_i$. The above correspondence $v \mapsto \tilde{v}$ from $L$ to $\mathcal{L}_{X,x}$ (Equation and Definition), extends to an tilde embedding:

$$ \sim: \Omega^*L \to (\Omega_X^*(\log Y) \otimes \mathcal{L}_X)_x \simeq (j_*\mathbf{L})_x $$

$$ v_J \mapsto \tilde{v} \frac{dy_{i_1}}{y_{i_1}} \wedge \dots \wedge \frac{dy_{i_j}}{y_{i_j}} \text{ for each } J = (i_1 < \dots < i_k) \in S_k, v_J = v \in L_J $$

inducing the quasi-isomorphism

$$ \Omega^*L \cong (\Omega_X^*(\log Y) \otimes \mathcal{L}_X)(x). \quad (3.6) $$

The complex $\Omega^*L$ and its sub-complex $IC^*L \subset \Omega^*L$ are acyclic if there exists an index $j$ such that $\alpha_j \neq 0$:
**Definition 3.5 (L unipotent along Y).** The local system L on the complement of a NCD Y = ⋃<sub>i∈I</sub> Y<sub>i</sub> is unipotent along Y if the local monodromies T<sub>i</sub> at x ∈ Y<sub>i</sub> are unipotent for i ∈ I.

*Example 3.1.* Let $M = \{1, 2\}$. Then $\mathrm{IC}^*L$ is isomorphic to the complex

$$L \xrightarrow{N_1 \oplus N_2} N_1 L \oplus N_2 L \xrightarrow{N_2 \oplus -N_1} N_1 N_2 L.$$

**3.1.3 The complex $\Omega^*\mathcal{L} \simeq j_*L \in D_c^b(X, \mathbb{C})**

The following de Rham complex with coefficient represent the complex $j_*L$ in the derived category of constructible sheaves of vector spaces $D_c^b(X, \mathbb{C})$.

**Proposition 3.1 ($\Omega^*\mathcal{L}$).** Let $j : X \setminus Y \to X$. The derived direct image $j_*L \in D_c^b(X, \mathbb{C})$ of L is quasi-isomorphic to the complex of analytic modules ([De 70, Corollaire 3.14]):

$$j_*L \xrightarrow{\sim} \Omega^*\mathcal{L} := \Omega_X^*(\log Y) \otimes_{\mathcal{O}_X} \mathcal{L}_X. \qquad (3.7)$$

In particular, the hyper-cohomology of the complex is isomorphic to the co-homology groups: $H^i(R\Gamma(X, \Omega^*\mathcal{L})) \simeq \mathbb{H}^i(X \setminus Y, \mathcal{L})$.

The quasi-isomorphism $j_*\mathcal{L} \xrightarrow{\sim} \Omega^*\mathcal{L}$ follows from Deligne's generalization to local systems ([De 70], definition 3.1) of Grothendieck's comparison theorem between the topological cohomology and algebraic de Rham cohomology on a compact complex algebraic variety [Gro 58], [Gro 66]. We recall also that if X is projective the complex $\Omega^*\mathcal{L}$ is necessarily algebraic by Serre's comparison theorem.

**Lemma 3.2.** Let $M \subset I$, $x \in Y_M^*$ a general non singular point on $\cap_{i \in M} Y_i$, $L := \mathcal{L}_X(x)$ with monodromy action $T_i$, $i \in M$ and $\Omega^*L^u \subset \Omega^*L$ (Definition 3.3). The tilde correspondence induces a quasi-isomorphism $\Omega^*L^u \simeq (\Omega^*\mathcal{L})_x$.

We prove that for each power of the maximal ideal $\mathfrak{m}^r \subset \mathcal{O}_{Xx}$ at $x$, the subcomplex $\mathfrak{m}^r(\Omega^*\mathcal{L})_x$ is acyclic for $r > 0$, and for $r = 0 : \Omega^*L^u \simeq \Omega^*L$ are quasi-isomorphic with their image by the tilde correspondence.

For each monomial $s := y^{m_1} \cdots y^{m_n}$, $\nabla(s\tilde{v}) = \sum_i (s(m_i + \alpha_i)\tilde{v} - \frac{1}{2i\pi}s(\tilde{N}_i v)) \otimes \frac{dy_i}{y_i}$, hence the image by the tilde correspondence $\tilde{L} := \sim L$ multiplied by $s$ generates in $\mathfrak{m}^r(\Omega^*\mathcal{L})_x$, the sub-complex of $\mathbb{C}$-vector spaces

$$\Omega^* sL := (0 \to sL \to \cdots \to \bigoplus_{\{i_1 < \dots < i_k\}} sL \frac{dy_{i_1}}{y_{i_1}} \wedge \cdots \wedge \frac{dy_{i_k}}{y_{i_k}} \to 0) \quad (3.8)$$

which is quasi-isomorphic to the complex

$$\Omega^*L := \left( 0 \to L \to \bigoplus_{i \in [1,n]} L \to \cdots \to \bigoplus_{\{i_1 < \dots < i_k\}} L \cdots \to 0 \right)$$
with differential in degree $k-1$:

$\forall v_\bullet, d_{k-1}(v_\bullet)_{\{i_1<\dots<i_k\}} = \sum_j (-1)^j ((m_{ij} + \alpha_{ij}) \mathrm{Id} - \frac{1}{2i\pi} N_{ij}) v_{\{i_1<\dots>i_j<\dots<i_k\}}.$
It is acyclic if at least one $m_j + \alpha_j \neq 0$ in which case $(m_j + \alpha_j) \mathrm{Id} - \frac{1}{2i\pi} N_j$ is an isomorphism of $L$ as $N_j$ is nilpotent; indeed, the complex may be written as a cone over such morphism. We deduce the proposition as $\alpha_j \notin \mathbb{Z}$ unless $\alpha_j = 0$, then $m_j + \alpha_j = 0$ iff $m_j = 0$ and $\alpha_j = 0$.

*Remark 3.2.* i) Let $\mathcal{I}_Y$ denote the product of the ideals of the components $Y_i$. The product $\mathcal{I}_Y \Omega^*\mathcal{L}$ is an acyclic sub-complex by the lemma 3.2.

ii) The above description of $(j_*\mathcal{L})_x$ is the model for the description of various perverse sheaves in the rest of the chapter.

For example, to describe the global sub-complex of analytic sheaves $\mathrm{IC}^*\mathcal{L} \subset \Omega_X^*(\log Y) \otimes \mathcal{L}_X$ in the next definition 3.6, we lift locally the subspaces $N_JL$ for various $J$ into subspaces $\widetilde{N_JL} \subset \mathcal{L}_{X,x}$ by the tilde embedding (Lemma 3.1).

iii) The complex $\Omega^*\mathcal{L}$ is defined over $\mathbb{Q}$.

### 3.1.4 The intermediate extension $j_*p\mathcal{L} \simeq \mathrm{IC}^{*p}\mathcal{L} \subset \Omega^{*p}\mathcal{L}$

For convenience, we still say intermediate extension for $j_*\mathcal{L}$ or shifted $j_*p\mathcal{L}$. The intermediate extension $j_*p\mathcal{L}$ on $X$ of $p\mathcal{L}$ on $U$ is defined in the category of constructible sheaves $\mathcal{D}_c^b(X, \mathbb{Q})$ ([GMacP 83], [BBD 83], see 2.2.1.3). We describe below a sub-complex $\mathrm{IC}^*\mathcal{L} \subset \Omega^*\mathcal{L}$ representing $j_*\mathcal{L}$ in $\mathcal{D}_c^b(X, \mathbb{C})$. Let $\mathcal{I}_Y$ denote the ideal product of the ideals of the components $Y_i$. The product $\mathcal{I}_Y \Omega^*\mathcal{L}$ is an acyclic sub-complex (Remark 3.2 i). The complex $\mathrm{IC}^*\mathcal{L} \supset \mathcal{I}_Y \Omega^*\mathcal{L}$ is characterized by the quotient complex supported by $Y$.

**Definition 3.6** ($\mathrm{IC}^*\mathcal{L} \subset \Omega^*\mathcal{L}$). The complex of sheaves $\mathrm{IC}^*\mathcal{L}$ is the sub-complex of $\Omega^*\mathcal{L}$ whose stalk at a point $x \in Y_M^*$ is defined as the $\mathcal{O}_{X,x}$ sub-module generated by the sections $\widetilde{v}_J \wedge \Omega_{X,x}^* \in \Omega^*\mathcal{L}$ for all $v_J \in f_JL$ (Lemma 3.1) and $J \subset M$ and by $\mathcal{I}_Y \Omega^*\mathcal{L}$.

To check that the complex of sheaves is well defined at a point $x \in Y_M$ of the intersection of $Y_i$ for $i \in M$. Let $z_j = fy_j$ be a change of coordinate equations for $Y_j$ at $x$ for one $j \in M$ at a time with $f$ invertible at $x$. Since $\frac{dz_j}{z_j} = \frac{dy_j}{y_j} + \frac{df}{f}$, where $\frac{df}{f}$ is regular at $x$, the germ of $\mathrm{IC}^*\mathcal{L}$ at $x$ is independent of the choice of coordinates.

Moreover, if $x \in Y_M \cap Y_K$ for $K \cap M = \emptyset$, let $\omega = \wedge_{j \in M} dz_j/z_j$ and $\omega' = \wedge_{j \in K} dz_j/z_j$, then $\omega \wedge \omega'$ restricts to a section of $\mathrm{IC}^*\mathcal{L}$ on $Y_M \setminus Y_M \cap (\cup_{j \in K} Y_j)$ since the restriction of $\omega'$ becomes regular. We say for short that the complex $\mathrm{IC}^*\mathcal{L}$ is generated locally by the tilde embedding of $\mathrm{IC}^*\mathcal{L}$.

**Lemma 3.3.** The complex $j_*\mathcal{L} \in \mathcal{D}_c^b(X, \mathbb{C})$ is quasi-isomorphic to the sub-complex of analytic modules of $\mathrm{IC}^*\mathcal{L} \subset \Omega^*\mathcal{L}$:
$$j_{!*} \mathbf{L} \xrightarrow{\sim} IC^*\mathcal{L}. \qquad (3.9)$$

Let $\mathcal{S}$ denote the stratification defined by the NCD $Y \subset X$ with strata $Y_j^*$ for $J \subset I$. Let $n := \dim X$ (the complex dimension of $X$), $V_k := \cup_{J,|J| \le k} Y_j^*$ ($V_1 = Y, V_{n+1} = \emptyset$) for $k \le n$, $U_k := X \setminus V_k$ ($U_1 = U, U_{n+1} = X$) and $j_k : U_k \to U_{k+1}$ for $k \le n$, then $j_{!*} \mathbf{L}$ is defined by induction on $k$: $j_{!*} \mathbf{L} = \mathbf{L}$ on $U_1$ and $j_{!*} \mathbf{L}|_{U_{k+1}} := \tau_{n-k} j_{k*} j_{!*} \mathbf{L}|_{U_k}$.

The construction of $IC^*\mathcal{L} \subset \Omega^*\mathcal{L}$ is local. If $IC^*\mathcal{L}|_{U_k}$ is defined by coherent analytic sub-complex, the simple direct image $j_{k*} IC^*\mathcal{L}|_{U_k}$ coincides with the derived direct image since $j_k$ is Stein. At a point $x \in Y_j^* \subset (V_k \setminus V_{k+1})$ the local definition of $(IC^*\mathcal{L}|_{U_k})_x$ corresponds to the construction of $IC^*L \subset \Omega^*L$.

**Remark 3.3.** Let $L^u$ denote the subspace of $L$ such that all local monodromies are unipotent, then we have quasi-isomorphisms $IC^*L^u \simeq IC^*L \simeq (IC^*\mathcal{L})(x)$.

## 3.2 The logarithmic complex $j_{Z*}j_Z^*j_{!*}L \simeq IC^*\mathcal{L}(\log Z)$

With the notations of Proposition 3.1, let $Z \subset Y$ be a NCD which is a sub-divisor of $Y$ and $j_Z : X \setminus Z \to X$ the open embedding.

We define a subcomplex $IC^*\mathcal{L}(\log Z) \subset \Omega^*\mathcal{L}$ quasi-isomorphic to the derived direct image $j_{Z*}j_Z^*(j_{!*}\mathcal{L}) \in D_c^b(X, \mathbb{C})$.

Recall that $Y = \cup_{i \in I} Y_i$ is a NCD containing the singularities of $\mathcal{L}$. For any NCD $Z$ such that $Z \cup Y$ is still a NCD, we may always suppose that $\mathcal{L}$ is defined on $X - (Y \cup Z)$ (by enlarging $Y$) and consider $Z$ equal to a union of components of $Y \cup Z$.

Let $Z = \cup_{i \in I_Z} Y_i \subset Y$ with indices in a subset $I_Z$ of $I$. Fix a set of coordinates $y_i$ at $x \in Y_M^*$ such that $Y_M = V(y_i, i \in M)$ near $x$. The local definition of $IC^*\mathcal{L}(\log Z)$ at $x$ in terms of these coordinates is as follows. For each subset $J$ of $M$ let:

$$J_Z := J \cap I_Z, \quad J'_Z = J \setminus J_Z \text{ so } J = J_Z \cup J'_Z, \quad M'_Z := M \setminus M_Z. \qquad (3.10)$$

Let $M = [1, n]$. With the notations of Definition 3.4, for each index $J \subset M$, the subspaces $f_{J'_Z} L \subset L$ form a sub-family of $L$ with induced morphisms $f_j$ for $j \in [1, n]$. Recall that $\Omega^*L = s(L_J, f_j)$ with $L_J = L$ for all $J \subset M$. Replacing $L_J$ by $f_{J'_Z} L$ in this complex we obtain

**Definition 3.7.** Let $f_{J'_Z} L$ denote the image of the composition morphism $f_{J'_Z}$ in $L$. We define $IC^*L(\log Z) := s(f_{J'_Z} L, f_j) \subset \Omega^*L$.

**Example 3.2.** On the 3-dimensional disc $D^3 \subset \mathbb{C}^3$, let $Y = Y_1 \cup Y_2 \cup Y_3$ be the NCD defined by the coordinate equations $y_1 y_2 y_3 = 0$ and $\mathbf{L}$ the unipotent local system defined by a vector space $L$ with the action of 3 nilpotent endomorphisms $N_i$.
Let $Z = Y_1 \cup Y_2$ be defined by $y_1y_2 = 0$, $M_Z = \{1, 2\}$ and $M'_Z = \{3\}$, then $IC^*(\mathrm{Log} Z)$ is defined by the diagram with differentials defined by $\pm N_i$:

$$ \begin{tikzcd} L \arrow[r, "\xrightarrow{N_1,N_2}"] \arrow[d, "N_3"] & L \oplus L \arrow[r, "\xrightarrow{N_2,-N_1}"] \arrow[d, "N_3 \oplus N_3"] & L \arrow[d, "-N_3"] \\ N_3L \arrow[r, "\xrightarrow{-N_2,-N_1}"] & N_3L \oplus N_3L \arrow[r, "\xrightarrow{N_2,-N_1}"] & N_3L \arrow[d, "N_3"] \end{tikzcd} \quad (3.11) $$

We have a quasi-isomorphism $\mathrm{R}\Gamma(D^3 \setminus (D^3 \cap Z), j_*\mathbf{L}) \simeq \mathrm{IC}^*L(\log Z)$.

**Remark 3.4.** We extend (Definition 3.2) in the locally unipotent case, to families of complexes instead of vector spaces, in order to write the complex $\mathrm{IC}^*L(\log Z)$ as the simple complex associated to a double complex in two ways as in the diagram above:

(i) For each fixed subset $J' \subset M'_Z$, we define the complex

$$ \Omega^*(N_{J'}L, N_i)_{i \in M_Z} := s(N_{J'}L, N_i, i \in M_Z)_{J' \subset J \subset M_Z} $$ by summing over $J \subset M_Z$ (the complex on each line in Example ).

Let $N_j : \Omega^*(N_{j'-j}L, N_i)_{i \in M_Z} \to \Omega^*(N_{j'}L, N_i)_{i \in M_Z}$ for $j \in J' \subset M'_Z$.

We sum over $J' \subset M'_Z$ the family of complexes endowed with morphisms $N_j$ for $j \in M'_Z$:

$$ \mathrm{IC}^*(\Omega^*(N_{J'}L, N_i)_{i \in M_Z}, N_i)_{i \in M'_Z} := s(\Omega^*(N_{J'}L, N_i)_{i \in M_Z}, N_i, i \in M'_Z)_{J' \subset M'_Z} $$

$$ \mathrm{IC}^*L(\log Z) = \mathrm{IC}^*(\Omega^*(N_{J'}L, N_i)_{i \in M_Z}, N_i)_{i \in M'_Z} \quad (3.12) $$

(ii) Let $\mathrm{IC}^*(L, N_i)_{i \in M'_Z} := s(N_{J'}L, N_i, i \in M'_Z)_{J' \subset M'_Z}$ be the sum over $J' \subset M'_Z$ (the complex on each column in Example 3.2).

As each $N_i$ for $i \in M_Z$ acts on the complex $\mathrm{IC}^*(L, N_i)_{i \in M'_Z}$, we define:

$$ \Omega^*(\mathrm{IC}^*(L, N_i)_{i \in M'_Z}, N_i)_{i \in M_Z} := s(\mathrm{IC}^*(L, N_i)_{i \in M'_Z}, N_i, i \in M_Z)_{J \subset M_Z} $$

as the sum over $J \subset M_Z$ of the complex independant of $J$, then:

$$ \mathrm{IC}^*L(\log Z) = \Omega^*(\mathrm{IC}^*(L, N_i)_{i \in M'_Z}, N_i)_{i \in M_Z} \quad (3.13) $$

If $Z \neq Y$, the cohomology $H^i(\mathrm{IC}^*L(\log Z))$ vanishes in degree $i \geq |M| - 1$.

**Definition 3.8.** $[\mathrm{IC}^*\mathcal{L}(\log Z) \subset \Omega^*\mathcal{L}]$ Let $I_Z \subset I$ denote the set of indices of the components of $Z$ and for $J \subset M$, let $J_Z = J \cap I_Z$, $J'_Z = J \setminus J_Z$. The subcomplex $\mathrm{IC}^*\mathcal{L}(\log Z) \subset \Omega^*\mathcal{L}$ is generated at $x \subset Y_M^*$ as an $\mathcal{O}_{X,x}$-submodule, by the sections $\tilde{v} \wedge_{j \in J} \frac{dy_j}{y_j} \wedge \Omega_{X,x}^* \in \Omega^*\mathcal{L}$ for $J \subset M$ for all $v \in N_{J'_Z}L$, where $J'_Z = J \setminus J_Z$ and by $\mathcal{I}_Y \Omega^*\mathcal{L}(\log Y)$.

**Proposition 3.2.** Let $L := \mathcal{L}_X(x)$. There exist quasi-isomorphisms

$$ j_{Z^*}j_Z^*j_{*1}^*\mathbf{L} \xrightarrow{\sim} \mathrm{IC}^*\mathcal{L}(\log Z), \quad (j_{Z^*}j_Z^*j_{*1}^*\mathbf{L})_x \xrightarrow{\sim} \mathrm{IC}^*L(\log Z). \quad (3.14) $$
Proof. Let $j' : X \setminus Y \to X \setminus Z$ be the inclusion map such that $j = j_Z \circ j' : X \setminus Y \to X$. There exists a convergent spectral sequence with respect to the truncation filtration on $j'_*\mathcal{L}$:

$$E_2^{p,q}(\mathbf{L}) = (R^p j_{Z*} H^q(j'_* \mathbf{L})) \to (R^{p+q} j_{Z*} j'_* \mathbf{L}) := H^{p+q}(j_{Z*} j'_* \mathbf{L}).$$

We write the complex $\mathrm{IC}^*\mathcal{L}(\mathrm{Log} Z)$ as a double complex, depending on the logarithmic terms, locally as in (Remark 3.4) and then we deduce globally by the tilde correspondence a similar spectral sequence.

Let $I'_Z = I \setminus I_Z$. We introduce the sub-NCD $Z' = \cup_{i \in I'_Z} Y_i$ complement to $Z$, such that $Y = Z \cup Z'$. Then $Z'$ is transversal to $Z$ in the sense that intersections of components of $Z'$, intersects transversally $Z$.

Let $Z'_k = \cup_{J \subset I'_Z, |J| \ge k} Y_J (Z'_0 = X)$, $Z'_k(*) = (Z'_k \setminus Z'_{k+1})$, $j_k : Z'_k(*) \to Z'_k$ and $i_k : Z'_k \to X$. Remark that $Z'_k$ is decreasing and $Z \cap Z'_k(*)$ is a NCD in $Z'_k(*)$.

At a point $x \in Z \cap Z'_s(*)$, let $L := \mathcal{L}_X(x)$. By (Definition 3.8) the tilde correspondence induces a quasi-isomorphism $\mathrm{IC}^*L(\mathrm{Log} Z) \simeq i_x^*\mathrm{IC}^*\mathcal{L}(\mathrm{log} Z)$.

To prove the quasi-isomorphism $i_x^*j_{Z*}j'_Z j'_*\mathbf{L} \simeq \mathrm{IC}^*L(\mathrm{Log} Z)$, we use the structure of double complex on $\mathrm{IC}^*L(\mathrm{Log} Z) = \Omega^*(\mathrm{IC}^*(L, N_i)_{i \in M'_Z}, N_i)_{i \in M_Z}$ (Equation 3.13). The corresponding spectral sequence with respect to the truncation filtration on $\mathrm{IC}^*(L, N_i)_{i \in M'_Z}$, is:

$$E_2^{p,q}(L) = H^q(\Omega^*(H^p(\mathrm{IC}^*(L, N_i)_{i \in M'_Z}), N_i)_{i \in M_Z}) \to H^{p+q}(\mathrm{IC}^*L(\mathrm{Log} Z))$$

We check the isomorphism $i_x^*E_2^{p,q}(\mathbf{L}) \simeq E_2^{p,q}(L)$.

Let $U := D^r \times D^s$ be a product of discs isomorphic to a neighborhood with center $x$ in $X$ with local coordinates $(z, t.) := (z_1, \dots, z_r, t_1, \dots, t_s)$, such that $\mathcal{L}$ induces a locally unipotent local system on $U^* := (D^*)^r \times (D^*)^s$.

Let $p : D^r \times D^s \to D^s$ denote the first projection and $W$ the NCD in $D^r$ defined by the product $z_1 \cdots z_r = 0$ such that $Z = p^{-1}(W)$.

Let $U' := U \setminus Z$, $j' : U^* \to U'$, $j_Z : U' \to U$ such that $j = j_Z \circ j'$.

$$ \begin{array}{ccc} U^* \xrightarrow{j'} U' = (D^r \setminus W) \times D^s \xrightarrow{j_Z} D^r \times D^s & \xrightarrow{p} D^r & \\ \cup & & \cup \\ Z = p^{-1}(W) = W \times D^s & \xrightarrow{p|} W \end{array} $$

The cohomology sheaves $\mathbf{L}^i := R^i(p|_{U'}) j'_* \mathbf{L}$ are locally constant on $(D^*)^r$ with fiber at a point $z = (z.) \in (D^*)^r$ isomorphic to the cohomology $H^i((j'_* \mathbf{L})_z)$, which is in turn isomorphic to the Intersection cohomology of a local system on the sphere in $(D^*)^r$ at the center 0.

Let $j_W : (D^r \setminus W) \to D^r$, since the restriction of $p$ to the family of spheres in the fibers of $p$ is proper, the functor $R p_*$ commute with $R j_{W*}$ and $R j_{Z*}$:

$$ \begin{array}{l} (R j_{W*} \mathbf{L})_0 \simeq \Omega^*(H^p\{\mathrm{IC}^*(L, N_i)_{i \in M'_Z}\}, N_i)_{i \in M_Z}, \\ E_2^{p,q}(L) \simeq (R^q j_{W*} \mathbf{L})_0 \simeq (R^p j_{Z*} H^q(j'_* \mathbf{L}))_x \simeq E_2^{p,q}(\mathbf{L})_x. \end{array} $$
**Lemma 3.4 (Transversality).** Let $j'_Z : X \setminus Y \to X \setminus Z'$, $j_{Z'} : X \setminus Z' \to X$. We have a commutation property: $j_{Z*}j'_{Z*}L \simeq j_{Z'*Z*}L$.

The proof is local at each point of the intersection $Z \cap Z'$ and it is reduced, by induction on the dimension, to the case of a point in the intersection of $n$ components of $Y$ and $Z \neq Y$.

The computation is carried with respect to the stratification defined by $Y$. In terms of (Remark 3.4) where $|M| = n$ and a point $0 \in D^n \subset \mathbb{C}^n$, we check that $(j_{Z'*Z*}L)_0$ is obtained by truncation in degree $\le n-1$ of $(j_{Z*}j'_{Z*}L)_0$. By the remark the complex $(j_{Z*}j'_{Z*}L)_0$ has no cohomology in degrees $\ge n$.

### 3.2.1 Thom isomorphism and Gysin morphism

It is well-known that Thom isomorphism extends to stratified varieties under a normally nonsingular embedding [GMacP 88], ([Max 19] Definition 3.4.5). The following result is cited in ([CaMi 5], Lemma 3.5.4)

**Lemma 3.5.** Let $i: H \to V$ be a normally nonsingular inclusion of complex codimension $d$ of complex varieties transversal to every stratum of a stratification $S$ of $V$ and $K$ a constructible complex with respect to $S$, then $i^1K \simeq i^*K[-2d]$.

With the previous notations $Z \subset Y \subset X$ and $\mathbb{P}L$ defined on $X \setminus Y$, let $i_W: W \to X$ be a closed embedding of a non singular variety of codimension $d$ transverse to $Z$, $Y$ and to the strata of an adequate stratification $S$ of $X$, then we have Thom isomorphism

$$i_W^1 j_{1*} \mathbb{P}L[2d] \simeq i_W^* j_{1*} \mathbb{P}L$$

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

$$\mathbb{H}^i(W^*, j_{1*} \mathbb{P}L) \simeq \mathbb{H}_{W^*}^{i+2d}(X \setminus Z, j_{1*} \mathbb{P}L).$$

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

$$\mathbb{H}^i(H \setminus (Z \cap H), j_{1*}L) \xrightarrow{\sim} \mathbb{H}_{H \setminus (Z \cap H)}^{i+2}(X \setminus Z, j_{1*}L) \quad (3.15)$$

is defined on logarithmic complexes as follows. We have isomorphisms

$$i_H^1 IC^*L(\log Z)[1] \xleftarrow{\sim} IC^*L(\log Z \cup H)/IC^*L(\log Z) \xrightarrow{\sim} i_{H*}IC^*i_H^*L(\log Z \cap H)[-1]$$

where on the right $r$ is induced by the residue map

$$R_H: IC^*L(\log Z \cup H) \to i_{H*}IC^*i_H^*L(\log Z \cap H)[-1]$$
vanishing on $\mathrm{IC}^*\mathcal{L}(\log Z)$ and on the left $c$ is the connecting isomorphism with the cohomology with support. Later, we deduce from this description that the *Gysin morphism* is compatible with MHS.

### 3.2.2 Polarized Variation of HS (VHS)

The complex $\Omega^*\mathcal{L}$ is specifically interesting in the case of a variation of HS (VHS) defined by a Hodge filtration $F$ by sub-bundles of $\mathbf{L} \otimes \mathcal{O}_U$.

Let $x \in Y_M^*$ and a punctured disc $\Delta^* \subset B^* \subset D^n$ with center 0. The monodromy $T$ acting on the restriction $\mathbf{L}|_{\Delta^*}$ generates a group $\Gamma$ acting on the classifying space $\mathbb{D}$ of polarized VHS ([Gr 74] chapter I). The period map $\phi: \Delta^* \to \mathbb{D}/\Gamma$ lifts to the covering space $\tilde{\phi}: \tilde{\Delta}^* \to \mathbb{D}$. Let $N = \log T^u$ (in general we suppose the VHS unipotent along $Y$ by reduction to a finite local covering [Sc 73]). The map $\Psi: \Delta^* \to \tilde{\mathbb{D}}: z \to \exp(-zN\tilde{\phi}(z))$ where $\tilde{\mathbb{D}}$ is the dual space of $D$, extends at 0 and define a filtration $F_0$ on $\mathcal{L}_B(0)$ ([Gr 74] chapter IV, Theorem 2 and Proposition 10).

Globally, since the Hodge filtration $F$ extends to Hodge sub-bundles $\mathcal{F}$ of $\mathcal{L}_X$ satisfying Griffith's transversality $\nabla: \mathcal{F}^p \subset \mathcal{F}^{p-1} \otimes \Omega^1$, we deduce the Hodge filtration by sub-complexes of $\Omega^*\mathcal{L}$

$$F^p = [0 \to \mathcal{F}^p \mathcal{L}_X \cdots \to \Omega_X^i (\log Y) \otimes \mathcal{F}^{p-i} \mathcal{L}_X \to \cdots] \quad (3.16)$$

It should be noted that the existence of a Hodge filtration on $\mathbf{L}$ induces strong properties used in the definition of the weight filtration on $\Omega^*\mathcal{L}$.

### 3.2.3 Duality and Polarization

The topological properties of duality of the intersection cohomology (see [GMacP 83], [Max 19]) must be reinforced by the assumption of polarization of the VHS $\mathbf{L}$ inducing the auto-duality property, in order to prove the decomposition theorem. The polarization of the intermediate extension of a polarized VHS with singularities along a NCD is defined in [KaK 87].

As we reduce the proof to the case of IMHS (Definition 2.14), we use essentially the local definitions of graded polarized limit MHS at infinity in chapter 2.3. As the intermediate extension is canonically defined by the local system on a Zariski open non singular subset, the polarization of the intermediate extension is also canonically defined by the polarization of the VHS on the open subset. Hence, to refer to the polarization of the intermediate extension, it is sufficient to check only the polarization of the VHS on the open subset $U$ and in turn to check the polarization at each point $x \in U$ (§2.3).
Poincaré -Verdier duality. Let $f: X \to V$ a proper morphism of complex algebraic varieties, $K \in D^b(X, \mathbb{Q})$ a complex of sheaves, $\mathbb{D}_X$ the dualizing complex and $D_X$ the dualizing functor $D_X K := \operatorname{Hom}^\bullet(K, \mathbb{D}_X)$ (derived $\operatorname{Hom}^\bullet$), then by (Equation 2.4):

$$ \mathbb{H}^{-i}(X, D_X K) \simeq D(\mathbb{H}^i(X, K)) := \operatorname{Hom}(\mathbb{H}^i(X, K), \mathbb{Q}) $$

If $\mathbf{L}$ is a VHS, to apply this statement of duality with weights, we recall some conventions on duality of filtrations and polarization in Hodge theory.

i) Translation of filtered complexes with weights. The couple of fields $(\mathbb{Q}, \mathbb{C})$ form a HS of weight 0. If $H$ is a HS of weight $w'$, we write $H(r)$ for the HS of weight $w' - 2r$.

A Hodge complex of vector spaces $(K, K_{\otimes \mathbb{Q}}, F)$ defined over $\mathbb{Q}$, is of weight $w$ if its cohomology $H^i(K)$ is a HS of weight $w + i$. We set $\mathbb{Q}[r]$ for the complex with $\mathbb{Q}$ in degree $-r$ and zero otherwise. It is a HC of weight $r$ since $H^{-r}(\mathbb{Q}[r]) = \mathbb{Q}$ is a HS of weight 0.

Let $(\mathbf{L}, F)$ be a VHS of weight $w'$. The filtration $F$ of the shifted VHS $\mathbf{L} := \mathbf{L}[n]$ is defined by: $F_i(\mathbf{L}) := F_i(\mathbf{L})[n]$. Then the weight of $(\mathbf{L}^i, F)$ as a complex of sheaves, is $n + w'$ since the weight of its cohomology satisfy:

$$ w(H^{-n}(\mathbf{L}, F)) = w(H^0(\mathbf{L}, F)) = w' + n - n = w' $$

Let $(\mathbf{L}, W, F)$ be a VMHS. The filtration $W$ and $F$ of the shifted VMHS $\mathbf{L} := \mathbf{L}[n]$ are defined by: $W^i(\mathbf{L}) := W^{i-n}(\mathbf{L})[n]$ and $F_i(\mathbf{L}) := F_i(\mathbf{L})[n]$, then $(\mathbf{L}^i, W, F)$ is called a perverse VMHS.

ii) Dual filtrations. The dual of a bifiltered complex $(K, W, F)$ in the bifiltered derived category of an abelian category with a dualizing functor $D$, is denoted by $(DK, W, F)$ with dual filtrations defined by:

$$ W_{-i}DK := D(K/W_{i-1}), F^{-i}DK := D(K/F^{i+1}) $$

such that: $DGr_i^W K \xrightarrow{\sim} Gr_i^W DK$ and $DGr_F^i K \xrightarrow{\sim} Gr_F^{-i} DK$.

Below, we are interested in various dualities with value in $\mathbb{Q}$ or with value in a dualizing complex $\mathbb{D}_X$ with weights or without.

iii) Dual filtrations with weights. The dual of a variation of MHS $(\mathbf{L}, W, F)$ with value in $\mathbb{Q}$ on a non singular complex algebraic variety $X$ is a variation of MHS $(\mathbf{L}^*, W, F)$ $(\mathbf{L}^* := \operatorname{Hom}_{\mathbb{Q}}(\mathbf{L}, \mathbb{Q}))$ with filtrations dual to $W$ and $F$ on $\mathbf{L}$, such that $Gr_{-i}^W(\mathbf{L}^*) \simeq (Gr_i^W \mathbf{L})^*$ and $Gr_F^{-i}(\mathbf{L}^*) \simeq (Gr_F^i \mathbf{L})^*$.

iv) Verdier dual filtrations with weights (the notations extend to the singular case). In the category of VHS $(\mathbf{L}, F)$ of weight $w'$ on a non singular complex algebraic variety $X$ of dimension $n$, we introduce a non trivial Hodge filtration on the dualizing complex: $\mathbb{D}_X^{w'} = \mathbb{D}_X(n - w')$ (that is $\mathbb{D}_X \simeq \mathbb{Q}_X[2n]$ tensor a HS of weight $2w' - 2n$).

Then, Verdier-Poincaré duality states the following (derived $\operatorname{Hom}^\bullet$):

$$ D_X^{w'}(\mathbf{L}, F) : \operatorname{Hom}^\bullet((\mathbf{L}, F), \mathbb{Q}_X[2n](n - w')) = \operatorname{Hom}^\bullet((\mathbf{L}, F), \mathbb{Q}_X)[2n](n - w') $$

$$ \mathbb{H}^{-i}(X, D_X^{w'}(\mathbf{L}, F)) \simeq D^{w'}(\mathbb{H}^i(X, (\mathbf{L}, F))) := \operatorname{Hom}(\mathbb{H}^i(X, (\mathbf{L}, F)), \mathbb{Q}(-w')) $$
Hence the weight of the left term of the second line, is $w' - i$ since $w' = -w' + 2n - 2n + 2w'$; while the weight of $D^{w'}\mathbb{H}^i(X, (\mathbf{L}, F))$ is also $-w' - i + 2w' = w' - i$ (the same weight as: $w(H^{2n-i}(X, \mathbf{L}^*) = w' - i$). Moreover:

$$D_X^{w'}(\mathbf{L}, F) \text{ is a VHS of weight } w', D_X^{w'} \mathrm{Gr}_F^i(\mathbf{L}, F) \simeq \mathrm{Gr}_F^{w'-i} D_X^{w'}(\mathbf{L}, F) \quad (3.17)$$

v) Polarization of HS.

**Definition 3.9.** i) A polarization of a HS of weight $w'$ is defined by a non degenerate $w'$-symmetric morphism $S: L \otimes L \to \mathbb{Q}$ such that its complex extension $S: L \otimes L \otimes \mathbb{C} \to \mathbb{C}(-w')$ satisfy:

$$1. \forall x \in L^{p,q}, x \neq 0, S(i^{p-q}x, \bar{x}) > 0$$

$$2. \forall x \in L^{p,q}, \forall y \in L^{p' \neq q, q' \neq p}: S(x, y) = 0$$

$$3. \forall p, q: p+q > w', S(F^p, F^q) = 0.$$

Since $\overline{S(x, y)} = S(\bar{x}, \bar{y})$, the last condition is equivalent to $S(F^p, F^q) = 0$ for $p+q < w'$, hence the conditions (2) and (3) are equivalent.

**Corollary 3.1.** A polarized HS is auto-dual.

*Proof.* By definition the isomorphism $S^*: (L, F) \xrightarrow{\sim} \mathrm{Hom}((L, F), \mathbb{C}(-w'))$: $a \mapsto (b \mapsto S(a, b)$ defined by $S$, in the filtered category of HS of weight $w'$, sends $F^p \mapsto F^p$ as $S(F^p, F^{w'-p+1}) = 0$. We deduce from (Equation 3.17) that the dual filtration coincides with the filtration $F$ by transport by $S$. We say that the polarized HS is self dual.

*Polarization of a VHS.* A polarization $S$ of a VHS $(\mathbf{L}, F)$ of weight $w'$ on a non singular complex algebraic variety $X$ induces an isomorphism of HS of equal weight $w'$:

$$(\mathbf{L}, F) \xrightarrow{\sim} (\mathbf{L}, F)^*(-w').$$

The dual of $(\mathbf{L}, F)$ with value in $\mathbb{Q}(-w')$ of weight $2w'$ is a VHS $(\mathbf{L}, F)^*$ of weight $w((\mathbf{L}, F)^*) = -w'$. Then $S$ induces a quasi-isomorphism in the filtered category of VHS

$$(\mathbf{L}, F) \xrightarrow{\sim} \mathrm{Hom}((\mathbf{L}, F), \mathbb{C}(-w')).$$

### 3.2.3.1 Polarization of an intermediate extension and purity

We introduce the concept of non degenerate bilinear pairing in the category of intermediate extensions of VHS of weight $w'$.

Let $(\mathbf{L}, F)$ denote a polarized VHS on an open non singular Zariski open subset $U$ of an algebraic variety $X$ of dimension $n$. In the case where $U$ is the complement of a NCD, the complex $\mathrm{IC}^*\mathcal{L} \subset \Omega^*\mathcal{L}$ (Definition 3.6) with its Hodge filtration induced by $F$ on $\Omega^*\mathcal{L}$ (Equation 3.16), defines a Hodge filtration on the intermediate extension: $(j_*\mathbf{L}, F) := (\mathrm{IC}^*\mathcal{L}, F)$.

The complex $\Omega^*\mathcal{L}_2$ of square summable $\mathcal{C}^\infty$-differential forms with its Hodge filtration $F$ is defined respectively in ([Zu 79], [KaK 87], [CaKSc 87])