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3
votes
0answers
314 views

genus one Gromov-Witten invariants of Calabi-Yau 3-folds

In http://arxiv.org/PS_cache/hep-th/pdf/9302/9302103v1.pdf physicists calculate (predict) genus one GW invariants of quintic (Table 1) and some other cases (Table 2). Can any body explain to me ...
1
vote
1answer
491 views

Harmonic forms for Complex Projective Space.

For complex projective space with the Fubini-Study metric and associated Laplace-de Rham operator $dd^\ast+d^\ast d$. How does one find a concrete description of the space of harmonic forms? That is, ...
4
votes
2answers
461 views

Is the weight filtration a topological invariant?

This question is somehow related to (but different from) the following MO question and the one linked from there Diffeomorphic Kähler manifolds with different Hodge numbers Let $X$ and $X'$ be ...
4
votes
1answer
621 views

Mumford-Tate groups and Hodge structures

It is well known that the Mumford-Tate group of a polarizable pure $Q$-Hodge structure is reductive. (this is proved for instance in Deligne et al LNM 900, Voisin's books on Algebraic Geometry,..) ...
8
votes
4answers
727 views

Does there exist a family of curves (or abelian varieties) on the punctured line with specified monodromy on H^1?

Suppose I have a finite set of points in $\mathbb{P}^1$ (over the complex numbers), and suppose that at each point, I am given a [Edit: quasi-unipotent] conjugacy class in $Sp(2g,\mathbb{Z})$ for $g$ ...
64
votes
2answers
14k views

Why is the Hodge Conjecture so important?

The Hodge Conjecture states that every Hodge class of a non singular projective variety over $\mathbf{C}$ is a rational linear combination of cohomology classes of algebraic cycles: Even though I'm ...
4
votes
2answers
865 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, ...
7
votes
1answer
915 views

If the numerical equivalence of cycles coincides with the homological one, does the Hodge standard conjecture follow?

Suppose that over an algebraically closed field $K$ of finite characteristic the numerical equivalence of cycles relation (for algebraic cycles of smooth projective varieties) coincides with the ...
10
votes
2answers
691 views

Weight filtration and Hodge theory for tropical varieties

Many concepts is algebraic geometry have tropical analogues. Question: Is there an analogue of the weight filtration or Hodge filtration for tropical varieties? A tropical curve ends up being ...
11
votes
1answer
831 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
0answers
521 views

How does complex conjugation act on the Hodge decomposition?

Let $A$ be a principally polarized abelian variety over $\mathbf{Q}$. Let $G$ be the Mumford--Tate group of $A$. The action of complex conjugation on $A(\mathbf{C})$ induces an involution on the de ...
25
votes
7answers
3k views

Examples of Mixed Hodge Structures

Does anyone know a user-friendly, example-laden introduction to mixed Hodge structures? I get from Wikipedia how to calculate for a punctured and pinched curve ...
7
votes
1answer
770 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 ...
1
vote
1answer
1k views

Does the Hodge star operator respect complex structure?

The Hodge star operator $\ast$ acts on the differential forms of a differential manifold sending $\Omega^{k}$ to $\Omega^{N-k}$. If the manifold is complex, then for $p+q=k$, does $\ast$ map ...
6
votes
2answers
407 views

Does there exist a functorial splitting for the weight filtration (of singular cohomology)?

There are plenty of examples of varieties whose singular cohomology with rational coefficients considered as a mixed Hodge structure does not decompose as the direct sum of its pure (weight) factors. ...
2
votes
0answers
173 views

A Hodge substructure with ''nice' weight factors that does not correspond to a mixed submotif?

Suppose that we have a mixed motif $M$ with only two (subsequent) weights, and have a subobject for each weight factor. How we can control whether there exists a submotif of $M$ with these two ...
6
votes
1answer
457 views

Hyperbolicity for Algebraic Varities and relation to curves on them

My question is related to several notions of hyperbolicity, applied to Kahler manifolds (projective, in general). Kahler hyperbolicity was introduced in this paper of Gromov's. He calls a Kahler ...
3
votes
1answer
694 views

Do Deligne's decalage and two filtrations for mixed Hodge complexes have a conceptual explanation?

As far as I understood, there are two way to assign weights to a mixed Hodge complex (a way that does not depend on shifts, and another one that does). There is a clever way to relate these to ways ...
3
votes
2answers
358 views

Is there a $k$-structure for Hodge modules over a $k$-variety?

I am trying to understand (Saito's?) category of mixed Hodge modules as a category (i.e. I am not interested in its construction, just in properties of objects and morphisms). I would be grateful for ...
2
votes
0answers
132 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
198 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
430 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
310 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
313 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
796 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
482 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
413 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
2k 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
227 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
272 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
738 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
838 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
631 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 ...
27
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 ...
6
votes
1answer
732 views

Hodge Standard Conjecture in Positive Characteristic

In the Wikipedia article on the Hodge Standard Conjecture it is written (note: it has since been fixed): In characteristic zero the Hodge standard conjecture holds, being a consequence of Hodge ...
4
votes
2answers
398 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
3k views

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

Where can we find Deligne's paper " Theorie de Hodge I"?
14
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
546 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 ...
16
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
609 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 ...
15
votes
1answer
523 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
324 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
229 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
749 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 ...
19
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 ...