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2
votes
0answers
131 views

On 'graded polarizable' triangulated categories; are there any mixed Galois module analogues known? Also on mixed realizations

There is a construction by Beilinson (in section 3 of "Notes on absolute Hodge cohomology") of the derived category of graded polarizable mixed Hodge complexes; he also proved that this category is ...
5
votes
0answers
197 views

Which pure categories related with Weil cohomology theories are semi-simple?

As far as I understand, the category of pure polarizable Hodge modules is semi-simple, whereas the cohomology of the corresponding schemes is graded polarizable. Is it true that one doesn't have any ...
6
votes
3answers
1k views

Betti number and harmonic forms

Dear Experts, On a compact, boundless, Riemannian manifold, the dimension of the space of harmonic k-forms is equal to the k-th Betti number. Is this correct (by Hodge theory)? For example, on the ...
4
votes
1answer
423 views

Special fiber of the Neron Model of an Abelian scheme in terms of Limit Hodge Structure

Let $\mathcal{A}$ be an Abelian scheme over a smooth curve $S^*\subset S$ and let $\mathcal{A_S}$ be the Neron model of $\mathcal{A}$ over $S$. Is it possible to describe the special fiber of the ...
6
votes
0answers
301 views

Riemann-Roch as an index theorem [closed]

I am sorry to make this a new question. I would have liked to leave a comment, but I suppose I don't have enough rep for that.... So, in the accepted answer to this question I don't understand why in ...
19
votes
1answer
1k views

Diffeomorphic Kähler manifolds with different Hodge numbers

This question made me wonder about the following: Are there orientedly diffeomorphic Kähler manifolds with different Hodge numbers? It seems that this would require that those manifolds are not ...
3
votes
3answers
1k views

Can the adjoint of the exterior derivative in semi-Riemannian geometry be defined without the Hodge * operator?

The adjoint of the exterior derivarive is defined by $\delta:=(-1)^k\ast^{-1}d\ast$, but I need a way which avoids the Hodge $\ast$ operator. Is there another definition? For example, for ...
5
votes
0answers
304 views

Generalized Hodge conjecture for cohomology of smooth non-proper varieties?

Is there such a statement known i.e. does there exist a conjectural description of the coniveau filtration for singular cohomology of a smooth non-proper variety over the field of complex numbers (in ...
8
votes
1answer
747 views

On polarized (pure) Hodge structures

Some simple questions, for which I know no precise reference (and would be deeply grateful for a nice one!): Is it true that the category of (pure) polarized Hodge structures is abelian semi-simple, ...
4
votes
0answers
466 views

Torelli theorem for cubic hypersurfaces

I'm reading the paper of Claire Voisin on the Torelli theorem for cubic hypersurfaces of $\mathbb{P}^5$ (in french : http://www.springerlink.com/content/j8675gn214l17523/). Are there lecture notes, or ...
3
votes
1answer
401 views

Groups of Hodge type, hodge structure on Lie algebra

Hi, Let $W$ be a real algebraic group, and $G$ the associated complex group. Then $W$ is of Hodge type if there is a $\mathbb{C}^*$ action on $G$ such that $U(1)$ preserves $W$ and the action of $-1$ ...
18
votes
2answers
1k views

Comparing cohomology over ${\mathbb C}$ and over ${\mathbb F}_q$

I have the following (probably well-known) question: let $X$ be a regular scheme over $\mathbb Z$. Let $p$ be a prime and Let us denote the reduction of $X$ mod $p$ by $X_p$. Let also $X_{\mathbb ...
1
vote
0answers
226 views

Schubert Cells for the Projective Line

I am trying at the moment to understand Schubert calculus, and have taken the simple example of the complex projective line ${\mathbb CP}^1$ as a guide. Now in the simplest formulation I know, we have ...
6
votes
0answers
268 views

Cohomology of Zariski neighborhoods

Do there exist smooth compact (=complete) connected complex algebraic varieties $X\subset Y$ and a Zariski neighborhood $U$ of $X$ in $Y$ such that the image of $H^{\ast}(U,\mathbf{Z})$ in ...
8
votes
1answer
708 views

Cubical cohomology and de Rham cohomology

Qiaochu's question on a discrete analogue of harmonic function theory reminded me of some thoughts I had a long time ago about the relationship between cubical cohomology and de Rham cohomology. The ...
9
votes
0answers
784 views

Weight filtration over the integers

This is a follow up question to Weight filtration for smooth analytic manifolds As mentioned in that question, the integral cohomology of some smooth complex analytic manifolds is equipped with a ...
9
votes
2answers
625 views

Hodge numbers of compactifications

Let $X$ be a smooth complex quasi-projective variety. We can find good compactification: a smooth proper variety $\bar{X}$ such that ${\bar X} \setminus X$ is a divisor with normal crossing. The ...
24
votes
4answers
4k views

Proving Hodge decomposition without using the theory of elliptic operators?

In the common Hodge theory books, the authors usually cite other sources for the theory of elliptic operators, like in the Book about Hodge Theory of Claire Voisin, where you find on page 128 the ...
5
votes
0answers
549 views

Hodge Standard Conjecture in Positive Characteristic

In the Wikipedia article on the Hodge Standard Conjecture it is written: In characteristic zero the Hodge standard conjecture holds, being a consequence of Hodge theory. In positive characteristic ...
4
votes
2answers
376 views

What kind of line bundles have Chern class of Hodge type (2,0) or (0,2)?

If $L$ is a complex line bundle on a topological space $X$, let $c_1(L)$ denote the image of its Chern class in $H^2(X;C)$. A complex manifold structure on $X$ [ok which is also compact and say ...
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"?
13
votes
2answers
1k views

When does the Torelli Theorem hold?

The Torelli theorem states that the map $\mathcal{M}_g(\mathbb{C})\to \mathcal{A}_g(\mathbb{C})$ taking a curve to its Jacobian is injective. I've seen a couple of proofs, but all seem to rely on the ...
1
vote
1answer
509 views

Abelian varieties of CM type?

Is there any introduction to abelian varieties of CM type?any reference?Like how to construct a abelian varieties given a CM field E?What is the properites of the Mumford Tate group of the abelian ...
14
votes
0answers
1k views

Mixed Hodge structure on the rational homotopy type

A mixed Hodge structure (mHs) on a commutative differential graded algebra (cgda) over $\mathbf{Q}$ is a mixed Hodge structure on the underlying vector space such that the product and the differential ...
6
votes
1answer
587 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 ...
14
votes
1answer
494 views

Semistable filtered vector spaces, a Tannakian category.

Let $k$ be a field (char = 0, perhaps). Let $(V,F)$ be a pair, where $V$ is a finite-dimensional $k$-vector space, and $F$ is a filtration of $V$, indexed by rational numbers, satisfying: $F^i V ...
2
votes
2answers
320 views

The algebraicity of Hodge structure map

Let $\mathbb S$ be $\mathbb C^{\times}$'s restriction of scalar to $\mathbb R$. To give a real Hodge structure on an $\mathbb Q$ vector space $V$ is to give a real representation of $\mathbb S$ on ...
6
votes
0answers
226 views

Positivity properties of virtual Hodge numbers of Calabi-Yaus

Let $X$ be a normal, projective complex variety with an anticanonical divisor $D$. Do the virtual Hodge numbers of the noncompact Calabi-Yau variety $X$ \ $D$ enjoy some sort of positivity property? ...
11
votes
2answers
724 views

When is a homogeneous space a variety?

Let $G$ be a Lie group and let $H$ be a closed subgroup of $G$. Then $G/H$ may not be a group, but it will be a homogeneous space for $G$ with stabilizers conjugate to $H$. Sometimes, this is a ...
18
votes
4answers
3k views

Why do people think that abelian varieties are the hardest case for the Hodge conjecture?

Today, I heard that people think that if you can prove the Hodge conjecture for abelian varieties, then it should be true in general. Apparently this case is important enough (and hard enough) that ...
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 ...
4
votes
0answers
253 views

Rozansky-Witten class associated to the Theta Graph

Suppose I have a holomorphic symplectic manifold, and a smooth $(1,0)$ connection on the tangent bundle which is compatible with both the complex and the symplectic structures. Say that the ...
13
votes
2answers
877 views

Why the similarity between Hodge theory for compact Riemannian and complex manifolds?

I'm aware to varying extents of the existence of certain decompositions of the space of $k$-forms on a compact complex or compact Riemannian manifold that split into closed, co-closed, and harmonic ...