ALEX, DARIO. CLASSICAL HODGE THEORY (NOTES BY HIRO,
EDITED BY DARIO)
2011 TALBOT WORKSHOP
1. Dario, Mixed Hodge Structures
A mixed Hodge strcuture (MHS) is a triple
pH, W, F q
where
(1) H is a Z-module of finite type
(2) W is an increasing filtration on HQ : H bZ Q, called the weight filtration
(3) F is a decreasing filtraton on HC : H b C, called the Hodge filtration
such that
GrFp GrFq pGrnW pHC qq 0 ifp
q
n.
Here we are extending W to be a filtration of HC . Notice that this formula makes
sense due to the following observation: Subobjects/quotients of filtered objects are again
filtered. E.g., if A is filtered, then the filtration of a quotient A{B is defined to be
F k pA{B q F k pAq{pB X F k pAqq.
The most boring examples of MHSs are Split MHSs., obtained at follows:
H `k Hk , where Hk is a pure Hodge structure of weight k, with obvious W and F .
Clearly, a MHS is split if and only if W is a grading of HQ , instead of being a filtration.
Some natural questions, the first of which is the more urgent:
(1) Why did I make this definition?
(2) Can we make this more geometric?
(3) Can I assemble mixed hodge structures into a category?—in fact they form a symmetric monoidal category. And is the symm monoidal cat. of MHS a representation of
a group? The answer is yes; this would take us to Tannakian duality.
Let’s answer the first question:
(1) Mixed Hodge Structures appear! For instance, studying cohomology of complex
algebraic varieties. I’ll consider only the non-singular case. So let’s say that X is a
non-singular complex variety, possibly non-compact. We know very well what happens
in the compact case, namely each cohomology H n pX, Zq is a pure HS of weight n.
In the non-compact case, it turns out that, for each degree n, H n pX; Zq carries a
mixed Hodge structure.
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2011 TALBOT WORKSHOP
The idea is to compactify X, you can do it in a particularly nice way using some
algebraic geometry—namely, there exists some smooth compact algebraic variety X
containing X, such that if I define the complement Y : X zX, Y is a normal crossing
divisor.
To fix notation, let me recall that the inclusion Y ãÑ X in analytic topology locally
looks like the inclusion of some coordainate axes tz1 zr 0u into Cn tpz1 , . . . zn qu.
Now it turns out I can relate this to the cohomology of the compact thing.
Fact: Let’s be less ambitious for now and say complex coefficients. H n pX; Cq is
isomorphic to the hypercohomology of X with coefficients in a complex of sheaves
H n pX, Cq H ppX q, Ω pY qX q.
and this has a name—the logarithmic deRham complex. What is the definition fo this?
It is a locally free sheaf generated by holomoprhic forms on X, together with forms
that locally look like dzi {zi . That’s why it’s called logarithmic. Carlos: This is not just
sheaves related to divisor associated to Y —for instance, dz1 {z2 is not allowed.
xY y
ΩX
is generated by ΩX and dzi {zi .
This guy comes with two filtrations.
The first is the obvious filtration: F p is equal to k-forms with k
The second is increasing : it is defined as
Wn pΩX xY y tforms with
¤ n occurrences of
¥ p.
dzi {zi u.
So we have to compute hypercohomology: the coefficient here is a complex that carries
two filtrations. Hence for each of these filtrations, I can write down a spectral sequence
for the cohomology of a filtered complex. So we have two ways to compute the same
thing and then we can compare!
Mike: I thought the F p was the number of dzs showing up. Ans: We’re doing something holmoprhic, so only dz shows up.
p,q
F E1
H 1pX, ΩpX xY yq Ñ H p q pX, Cq
After playing with the spectral sequence associated to W , we get the Leray spectral
sequence:
p,q
p
q
p q
pX, Cq.
W E2 H pX, R j C q Ñ H
where j : X Ñ X.
So, H n pX, Cq comes with two induced filtrations, that I still call F (decreasing) and
W (increasing). However, since the Leray spectral sequence is topological, W is actually
a filtration on the rational cohomology!
Theorem 1 (Deligne). H n pX, Zq, W rns , F q is a mixed Hodge Structure. Moreover
this is independent of thoice of compactification, and it’s functorial in X.