# Hodge Theory II

*1971*

#### Translator’s note

*This page is a translation into English of the following:*

Deligne, P. “Théorie de Hodge II.” *Publ. math. IHÉS* **40** (1971), 5–57. publications.ias.edu/node/361.

*The translator (Tim Hosgood) takes full responsibility for any errors introduced, and claims no rights to any of the mathematical content herein.*

Version: `375ad31`

# Introduction

*Work presented as a doctoral thesis at l’Université d’Orsay.*

By Hodge, the cohomology space

The reader will find an explanation in [5] of the yoga that underlies this construction.

The proof, which is essentially algebraic, relies on one hand on Hodge theory, and on the other on Hironaka’s resolution of singularities, which allows us, via a spectral sequence, “to express” the cohomology of a non-singular quasi-projective algebraic variety in terms of the cohomology of non-singular projective varieties.

Section 1 contains, apart from reminders on filtrations gathered together for the ease of the reader, two key results:

- Theorem 1.2.10, which will only be used via its corollary, Theorem 2.3.5, which gives the fundamental properties of “mixed Hodge structures.”
- The “two filtrations lemma,” Lemma 1.3.16.

Section 2 recalls Hodge theory and introduces mixed Hodge structures.

The heart of this work is §3.2, which defines the mixed Hodge structure of

Section 4 gives diverse applications, all following from Theorem 4.1.1 and the theory of the

# 1 Filtrations

## 1.1 Filtered objects

Let

We will be considering

A *decreasing* (resp. *increasing*) *filtration*

A *filtered object* is an object endowed with a filtration.
When there is no chance of confusion, we often denote by the same letter filtrations on different objects of

If

The *shifted filtrations* of a decreasing filtration

If *unless otherwise explicitly mentioned, when we say “filtration” we always mean “decreasing filtration”*.

A filtration *finite* if there exist

A *morphism* from a filtered object

Filtered objects (resp. finite filtered objects) of

A morphism *strict*, or *strictly compatible with the filtrations*, if the canonical arrows from

Let *dual* filtration on

The double dual of

If *associated graded* is the object of **!!TO-DO: diagram (1.1.7.1)!!**

Let *filtration induced by F* (or simply

*induced filtration*) on

Dually, the *quotient filtration* on

If **!!TO-DO: diagram!!**
the arrows are strict.

We call the filtration (1.1.9) on *filtration induced by that of A* (or simply the

*induced filtration*). By (1.1.9), its definition is self-dual.

In particular, if **!!TO-DO: o-suite?!!** sequence, and if

The reader can show that:

—

Let

f\colon(A,F)\to(B,F) be a morphism of filtered objects with finite filtrations. Forf to be strict, it is necessary and sufficient that the sequence0 \to \operatorname{Gr}(\operatorname{Ker}(f)) \to \operatorname{Gr}(A) \to \operatorname{Gr}(B) \to \operatorname{Gr}(\operatorname{Coker}(f)) \to 0 be exact.Let

\sigma\colon(A,F)\to(B,F)\to(C,F) be a**!!TO-DO: o-suite?!!**sequence of strict morphisms. We then have\operatorname{H}(\operatorname{Gr}(\Sigma)) \cong \operatorname{Gr}(\operatorname{H}(\Sigma)) canonically. In particular, if\Sigma is exact in{\mathscr{A}} , then\operatorname{Gr}(\Sigma) is exact in{\mathscr{A}}^\mathbb{Z} .

In a category of modules, to say that a morphism

If

Dually, if

If

We extend these definitions to contravariant functors in certain variables by (1.1.6).
In particular, for the left-exact functor

Under the above hypotheses, we have obvious morphisms

These constructions are compatible with composition of functors, in a sense whose details we leave to the reader.

## 1.2 Opposite filtrations

Let

Let

Two *finite* filtrations * n-opposite* if

If

A^{p,q}=0 except for a finite number of pairs(p,q) , andA^{p,q}=0 forp+q\neq n

then we define two

Conversely:

—

Let

F and\overline{F} be finite filtrations onA . ForF and\overline{F} to ben -opposite, it is necessary and sufficient that, for allp,q ,[p+q=n+1] \implies [F^p(A)\oplus\overline{F}^q(A) \xrightarrow{\sim}A]. If

F and\overline{F} aren -opposite, and if we set\begin{cases} A^{p,q} = 0 &\text{for }p+q\neq n \\A^{p,q} = F^p(A)\cap\overline{F}^q(A) &\text{for }p+q=n \end{cases} thenA is the direct sum of theA^{p,q} , andF and\overline{F} come from the bigradingA^{p,q} ofA by the procedure of (1.2.4).

**!!TO-DO: why is the following proof not appearing in the PDF version?!!**

*Proof*. —

The condition

\operatorname{Gr}_F^p\operatorname{Gr}_{\overline{F}}^q(A)=0 forp+q>n implies thatF^p\cap\overline{F}^q=(F^{p+1}\cap\overline{F}^q)+(F^p\cap\overline{F}^{q+1}) forp+q>n . By hypothesis,F^p\cap\overline{F}^q is zero for large enoughp+q ; by decreasing induction, we thus deduce that the condition\operatorname{Gr}_F^p\operatorname{Gr}_{\overline{F}}^q(A)=0 forp+q>n is equivalent to the conditionF^p(A)\cap\overline{F}^q(A)=0 forp+q>n . Dually ((1.1.6), (1.1.7), (1.1.10)), the condition\operatorname{Gr}_F^p\operatorname{Gr}_{\overline{F}}^q(A)=0 forp+q<n is equivalent to the conditionA=F^p(A)+\overline{F}^q(A) for(1-p)+(1-q)>-n , i.e. forp+q\leqslant n+1 , and the claim then follows.If

F and\overline{F} aren -opposite, then we can prove by decreasing induction onp that\bigoplus_{p'\geqslant p} A^{p',q'} \xrightarrow{\sim}F^p(A). \tag{1.2.5.1} - For
F^p(A)=0 , the claim is evident. - The decomposition
A=F^{p+1}(A)\oplus\overline{F}^{n-p}(A) induces onF^p(A)\supset F^{p+1}(A) a decompositionF^p(A) = F^{p+1}(A)\oplus(F^p(A)\cap\overline{F}^{n-p}(A)) and we conclude by induction. Forp small enough, we haveF^p(A)=A . By (1.2.5.1), theA^{p,q} thus form a bigrading ofA , andF satisfies (1.2.4.1). The fact that\overline{F} satisfies (1.2.4.2) then follows by symmetry.

- For

The constructions (1.2.4) and (1.2.5) establish equivalences of categories that are quasi-inverse to one another between objects of

Three *finite* filtrations *opposite* if

This condition is symmetric in

Let

p_i\leqslant p_j andq_i\leqslant q_j fori\geqslant j , andp_i+q_i=p_0+q_0-i+1 fori>0 .

Set

*Proof*. We will prove by induction on

(

For

By (1.2.5), (i) we have

Equation (1.2.8.1) can then be written as

The claim (

Let

*Proof*. To prove surjectivity, we write

To prove injectivity, we write

This finishes the proof of (1.2.8), noting that (1.2.8) is equivalent to (

Let

{\mathscr{A}}' is an abelian category.- The kernel (resp. cokernel) of an arrow
f\colon A\to B in{\mathscr{A}}' is the kernel (resp. cokernel) off in{\mathscr{A}} , endowed with the filtrations induced by those ofA (resp. the quotients of those ofB ). - Every morphism
f\colon A\to B in{\mathscr{A}}' is strictly compatible with the filtrationsW ,F , and\overline{F} ; the morphism\operatorname{Gr}_W(f) is compatible with the bigradings of\operatorname{Gr}_W(A) and\operatorname{Gr}_W(B) ; the morphisms\operatorname{Gr}_F(f) and\operatorname{Gr}_{\overline{F}}(f) are strictly compatible with the filtration induced byW . - The “forget the filtrations” functors,
\operatorname{Gr}_W ,\operatorname{Gr}_F , and\operatorname{Gr}_{\overline{F}} , and\begin{gathered} \operatorname{Gr}_W\operatorname{Gr}_F \simeq \operatorname{Gr}_F\operatorname{Gr}_W \\\simeq \operatorname{Gr}_{\overline{F}}\operatorname{Gr}_F\operatorname{Gr}_W \\\simeq \operatorname{Gr}_{\overline{F}}\operatorname{Gr}_W \simeq \operatorname{Gr}_W\operatorname{Gr}_{\overline{F}} \end{gathered} from{\mathscr{A}}' to{\mathscr{A}} are exact.

Denote by

If

The

*Proof*. By symmetry, it suffices to prove the claims concerning

Equation (1.2.11.1) then says that

We now prove (1.2.10).
Let

We thus deduce that

If

The “forget the filtrations” functor is exact by (ii). The exactness of the other functors in (iv) follows immediately from (ii), (iii), and (1.1.11), (i) or (ii).

Let *increasing* filtration *opposite* if the filtrations

Theorem (1.2.10) translates trivially to this variation.

## 1.3 The two filtrations lemma

Let *biregular* if it induces a finite filtration on each component of

We recall the definition of the terms

We note that the use here of the notation

We have, by definition:

For

For

Let

- The spectral sequence defined by
F degenerates (E_1=E_\infty ). - The morphisms
d\colon K^i\to K^{i+1} are strictly compatible with the filtrations.

*Proof*. We will prove this in the case where

If this condition is satisfied for arbitrary

Claim (2) trivially implies (1), and is equivalent to (ii), which proves the proposition.

If *shifted filtration*

This filtration is compatible with the differentials:

Since

—

- The morphisms (1.3.3.2) form a morphism of graded complexes from
E_0(\operatorname{Dec}(K)) toE_1(K) . - This morphism induces an isomorphism on cohomology.
- It induces step-by-step (via (1.3.1.5)) isomorphisms of graded complexes
E_r(\operatorname{Dec}(K))\xrightarrow{\sim}E_{r+1}(K) (forr\geqslant 1 ).

*Proof*. Let

The construction (1.3.3) is not self-dual.
The dual construction consists of defining

Recall that a morphism of complexes is said to be a *quasi-isomorphism* if it induces an isomorphism on cohomology.

—

- A morphism
f\colon(K,F)\to(K',F') of filtered complexes with biregular filtrations is a*filtered quasi-isomorphism*if\operatorname{Gr}_F(f) is a quasi-isomorphism, i.e. if theE_1^{pq}(f) are isomorphisms. - A morphism
f\colon(K,F,W)\to(K,F',W') of biregular bifiltered complexes is a*bifiltered quasi-isomorphism*if\operatorname{Gr}_F\operatorname{Gr}_W(f) is a quasi-isomorphism.

Let

Equation (1.3.1.2) identifies *first direct filtration*.

Dually, Equation (1.3.1.3) identifies *second direct filtration*.

On

*Proof*. For

Equation (1.3.1.5) identifies *recurrent filtration*

- On
E_0^{pq} ,F_r=F_d=F_{d^*} . - On
E_{r+1}^{pq} , the recurrent filtration is that induced by the recurrent filtration ofE_r^{pq} .

Definitions (1.3.8) and (1.3.9) still make sense for

The filtrations

—

- For the first direct filtration, the morphisms
d_r are compatible with the filtrations. IfE_{r+1}^{pq} is considered as a quotient of a sub-object ofE_r^{pq} , then the first direct filtration onE_{r+1}^{pq} is finer than the filtrationF' induced by the first direct filtration onE_r^{pq} Y we haveF_d(E_{r+1}^{pq})\subset F'(E_{r+1}^{pq}) . - Dually, the morphisms
d_r are compatible with the second direct filtration, and the second direct filtration onE_{r+1}^{pq} is less fine than the filtration induced by that ofE_r^{pq} . F_d(E_r^{pq})\subset F_r(E_r^{pq})\subset F_{d^*}(E_r^{pq}) .- On
E_\infty^{pq} , the filtration induced by the filtrationF of\operatorname{H}^\bullet(K) (1.3.12) is finer than the first direct filtration and less fine than the second.

*Proof*. Claim (i) is evident, (ii) is its dual, and (iii) follows by induction.
The first claim of (iv) is easy to verify, and the second is its dual.

We denote by

It is clear by (1.3.4.1) that the isomorphism (1.3.4) sends the first direct filtration on

If the filtration

- The morphism (1.3.3.2) of graded complexes filtered by
F u\colon \operatorname{Gr}_{\operatorname{Dec}(W)}(K) \to E_1(K,W) is a filtered quasi-isomorphism. - Dually, the morphism (1.3.5)
u\colon E_1(K,W) \to \operatorname{Gr}_{\operatorname{Dec}^\bullet(W)}(K) is a filtered quasi-isomorphism.

*Proof*. It suffices, by duality, to prove (i).

By (1.3.3) and (1.3.4), the complex

Each differential **!!TO-DO: diagram!!**
By hypothesis, the morphism

Let

*Proof*. We will prove the theorem by induction on

By (1.3.15), the morphism

On

On

Suppose that

For

By the induction hypothesis, we thus have

Under the general hypotheses of (1.3.16), suppose that, for all

*Proof*. This follows immediately from (1.3.16) and (1.3.13), (iv).

## 1.4 Hypercohomology of filtered complexes

In this section, we recall some standard constructions in hypercohomology. We do not use the language of derived categories, which would be more natural here.

*Throughout this entire section, by “complex” we mean “bounded-below complex.”*

Let *acyclic* for

Let

Let

Let *hypercohomology objects*

- We choose a quasi-isomorphism
i\colon K\to K such that the components ofK' are acyclic forT . For example, we can takeK' to be the simple complex associated to an injective Cartan–Eilenberg resolution ofK . - We set
\operatorname{R}^iT(K) = \operatorname{H}^i(T(K')).

We can show that *isomorphisms*

Let * T-acyclic filtered resolution* of

*hypercohomology spectral sequence of the filtered complex*K . It depends functorially on

The differentials

Let

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