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4
votes
2answers
632 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 ...
11
votes
2answers
776 views

Can one find the hodge number by counting points over finite fields?

Given a proper smooth variety $X$ of dimension $n$ over $\mathbb{C}$, assume it has a model over a DVR of mixed characteristic $(0,p)$ with residue field $\mathbb{F}_q$, and assume the closed fiber ...
1
vote
1answer
311 views

Adjoint groups of Mumford-Tate groups

Let $F$ be a sub-field of $\mathbb{C}$ and $B/F$ and $C/F$ be abelian varieties, with $C$ of CM type. Denote the Mumford-Tate groups of $B$, $C$ and $B\times_F C$ by $G_B$, $G_C$ and $G_{B\times C}$, ...
2
votes
1answer
367 views

Poincare pairing and polarization of Hodge structure. Kuga-Satake construction.

If $X$ is a K3 surface over the complex (algebraic) then I wonder if the poincare pairing induces a polarization on the Hodge structure $H^2(X,\mathbb Z)$? The point is that I see that when ...
7
votes
1answer
691 views

Is there an integral version of Faltings' isomorphism in p-adic Hodge theory between etale and Hodge cohomologies

Let $K$ be a $p$-adic field, that is a complete discrete valuation ring of characteristic $0$ with a perfect residue field $k$ of characteristic $p > 0$ (to simplify one could also take $K$ to be a ...
10
votes
1answer
601 views

Hodge numbers in a family

Let $X \to Y$ be a smooth projective morphism of smooth varieties over $\mathbb{C}$. Then the fibers $X_y, y \in Y$ have locally constant Hodge numbers $H^q(X_y, \Omega^p_{X_y})$. Namely, one can ...
1
vote
0answers
163 views

Question about specifying complex 1-motives

A 1-motive over a field $k$ is an algebraic torus $T$, an abelian variety $A$, a group scheme $G$ that's an extension of $A$ by $T$, a finitely generated free abelian group $L$, and a group ...
5
votes
1answer
387 views

Mumford-Tate groups of abelian varieties with potentially good reduction everywhere

Let $A$ be an abelian variety defined over a number field, and let $MT(A)$ be its Mumford-Tate group. It is a conjecture of Morita that if $MT(A)$ is anisotropic-mod-center (that is, it has no ...
2
votes
0answers
233 views

Does Artin's vanishing hold for '$E_2$-weight pieces' for (torsion) cohomology of affine varieties?

Let $X$ be an affine variety of dimension $n$ (say, over complex numbers). Then Artin's vanishing theorem yields: both singular and etale cohomology $H^i(X):=H^i(X,\mathbb{Z}/l\mathbb{Z})=0$ for ...
3
votes
1answer
307 views

Does the (torsion) Zariski cohomology of a (singular) hyperplane section of a smooth projective variety vanish (in small degrees)?

Let $P$ be a smooth connected projective variety (say, over complex numbers); $H$ is its smooth hyperplane section. What can be said about the Zariski cohomology of $H$ with constant coefficients? It ...
6
votes
1answer
539 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?
2
votes
1answer
173 views

Non-uniruled variety with level one Hodge structure.

I wonder if there exists one example of non-uniruled algebraic variety with level one Hodge structure.
4
votes
1answer
558 views

Serre's Analogue of the Weil Conjectures for Non-Compact Kahler Manifolds

The classical Riemann Hypothesis concerns the locations of zeroes of the Riemann zeta-function, or more generally the Dedekind zeta-functions of number fields. Its analogue for varieties defined over ...
3
votes
0answers
311 views

Which properties of p-adic representations can be recovered from (\varphi,\Gamma)-modules?

In Berger's "An introduction to the Theory of $p$-adic Representations", he mentions that due the the equivalence of categories between etale $(\varphi,\Gamma)$-modules and $p$-adic representations, ...
6
votes
1answer
482 views

Kähler Structure for Projective Varieties over a Finite Field

(i) In 1960 Serre proved a famous analogue of the Weil conjectures for Kähler manifolds. This poses an obvious question: Does there exist an analogue of a Kähler structure for (non-singular) ...
4
votes
2answers
513 views

Hodge theory and varieties defined over subfields of the complex numbers

This question is related to the question: Is there a $k$-structure for Hodge modules over a $k$-variety?. Suppose $K$ is a subfield of $\mathbb{C}$ and $M$ is a holonomic $D$-module "of geometric ...
3
votes
1answer
256 views

de Rham cohomology class of diagonal

I post again a question I asked in the post by Descartes: Since this is the topic on diagonal, I like to ask a question: Let $X$ be a compact Kahler manifold of complex dimension $n$, and let $\Delta ...
9
votes
2answers
1k views

What exactly does the weight filtration in Hodge theory have to do with the Weil conjectures?

Let $X$ be a variety over $\mathbb{C}$, say separated. According to Deligne's results, there is a "mixed Hodge structure" on the total cohomology $H^\bullet(X(\mathbb{C}), \mathbb{Z})$. One component ...
1
vote
2answers
170 views

Why does the divisor $Z$ homologous to $0$ in projective mainfold satisfy that every irreducible hypersurface appears in $Z$ with multiplicity $1$?

In Voisin's book "Hodge theory and complex algebriac geometry I", the proof of proposition 12.7 (page 296) says that if $X$ is projective, then every divisor $Z$ homologous to $0$ can be written as ...
3
votes
2answers
350 views

Why do the subgroups of $Hdg^{2k}(X)$ generated by the cycle classes and Chern classes of vector bundles coincide in algebraic variety $X$?

In Voinsin's book [1], Theorem 11.32 (page 280) says: "If X is an algebraic variety, these subgroups of $Hdg^{2k}(X) coincide." However, the proof did not show that the subgroup generated by cycle ...
6
votes
2answers
718 views

How does one view the De Rham spectral sequence as a Grothendieck spectral sequence?

I was rereading basic results on de Rham cohomology, and this led me inevitably to the fact that $H^q(X,\Omega^p)$ converges to $H^*(X)$ for any smooth proper variety (over any field). How does one ...
0
votes
0answers
207 views

Weight filtration of MHSs

This is probably a very stupid question, but could someone explain to me where the weight filtration of mixed Hodge structures come from and why we actually need it? If the Hodge-to-de Rham spectral ...
3
votes
0answers
293 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
457 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
445 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
583 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
697 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 $g$ ...
59
votes
2answers
11k 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
801 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
876 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
612 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
776 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 ...
6
votes
0answers
506 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 ...
23
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
660 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
391 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
166 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
423 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
651 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
345 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
130 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
196 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
967 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
414 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
298 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
921 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
296 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
704 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, ...