11
votes
4answers
1k views

The prerequisites for Deligne's Théorie de Hodge I, II, III

I am an undergraduate student. I am not sure if it's OK to ask this question here. I want to learn Hodge theory. But I do not know how to start it, and how much mathematics I should need before I ...
8
votes
3answers
798 views

Primitive Cohomology Useful?

In her book, after proving the hodge decomposition, Voisin spends time discussing primitive cohomology $H^r(X, \mathbb{C})_{prim} = \ker L^{n-r+1} \subset H^r(X, \mathbb{C})$ (where $L$ is the ...
3
votes
3answers
309 views

When can Hodge filtrations (decompositions?) be described explicitly in terms of periods?

It seems that there is no chance to explain the Hodge theory (to students) in an hour or so. Yet do there exist any cases when the Hodge filtration (or the Hodge decomposition) of the cohomology of a ...
3
votes
1answer
319 views

comparing Hodge structures on cohomology of conjugate varieties

What can one say about the relation between the Hodge decompositions of $H^\*(X,C)$ and $H^{*}(X_\sigma,C)$ for a complex algebraic smooth projective variety $X$ and $\sigma$ an automorphism of the ...
4
votes
2answers
597 views

motivating examples of family of Hodge structure

Let $\phi: \chi \to B$ be a proper holomophic submersion with smooth fiber $X$. Then, we get local systems $R^i \phi_* \mathbb C$ and corresponding flat connection $(B, \bigtriangledown)$ In this ...
6
votes
1answer
517 views

Mumford-Tate group and Galois representations

Could someone please point me towards a proof of why the image of a Galois representation on the Tate-module of an abelian variety is limited by its Mumford-Tate group?
4
votes
2answers
773 views

Hodge decomposition in Betti cohomology

The broad, generic and badly posed question may be formulated in this way: Let $X$ be a compact Kälher manifold (or even a projective one). If one considers the Hodge decomposition $H^k(X, ...
11
votes
1answer
752 views

Hodge structures on algebraic spaces

Let $X$ be a proper smooth algebraic space over $\mathbb C$ (which amounts, due to Artin, to giving a Moishezon space: a compact complex manifold whose dimension equals the transcendence degree of its ...
7
votes
1answer
607 views

Reference for the Hodge Bundle

For the purposes of this question, let the Hodge bundle $\lambda$ be the bundle on a fibration of abelian varieties $X\to B$ with fiber over $b\in B$ the space of 1-forms on $X_b$, or the pullback to ...
5
votes
3answers
2k views

Where can we find Deligne's paper “ Theorie de Hodge I”?

Where can we find Deligne's paper " Theorie de Hodge I"?
6
votes
1answer
510 views

Weight filtration for smooth analytic manifolds

In his ICM 2002 talk (Topology of singular algebraic varieties, available also on arXiv) B. Totaro says on p. 3 (of the arXiv version): "Using the method of Guillen and Navarro Aznar I was able to ...
18
votes
3answers
2k views

Exercises in Hodge Theory

I was wondering: is there a good place to find exercises in Hodge theory? Mostly computations and proving small (preferably nifty) theorems, is what I have in mind. Something roughly like the ...