The hodge-theory tag has no usage guidance.

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### Basis for hodge decomposition of Elliptic K3 Surfaces

We know the hodge numbers of K3 Surfaces. To work out some ideas, I'd like to know an explicit basis for the hodge decomposition $H^{p,q}$ of a smooth Elliptic K3 Surface over $\mathbb{C}$ (for all ...

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### Fontaine-Mazur for Hodge Structures

Is there a conjecture, or known result, describing which integral Hodge structures are composition factors in the Hodge structure on the cohomology groups of smooth proper algebraic varieties over ...

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### on variable and primitive cohomology of a hypersurface in a projective space

I have a smooth hypersurface D in $\mathbb{P}^n$: in many books about Hodge theory (as the ones of Voisin and Carlson) they take for granted that the primitive cohomology of D is equal the variable ...

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274 views

### octic K3s inside cubic 4-folds

From the Thesis of B.Hassett I seem to understand that a smooth cubic 4-fold $X$ containing a $\mathbb{P}^2$ should contain also a octic K3, but I cannot see a natural way by which this K3 octic could ...

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### Polarizable variations of (mixed) Hodge structures

I am trying to come to grips with Saito's theory of mixed Hodge modules (slightly) beyond just the basic axiomatic formalism. I will take my Hodge structures and sheaves to be rational, but I would be ...

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### Mixed structures on Hom spaces induced by mixed sheaves

Let $D^b_m(X)$ (resp $D^b(X)$) denote the derived category of mixed Hodge modules (resp. constructible sheaves) on a complex variety $X$. Let
$rat\colon D^b_m(X)\to D^b(X)$
be the `forgetful' ...

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### 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 ...

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257 views

### Hilbert function of a Hilbert scheme

Given a Hilbert scheme $H$ of curves in $\mathbb{P}^3$ satisfying certain Hilbert polynomial, is there any way of understanding the degree or arithmetic genus of an irreducible component of the ...

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### Are complex varieties Kahler? - Algebraic, non-projective complex manifolds

Let $X/\mathbb{C}$ a nonsingular proper variety and $X_{an}$ it's associated analytic space. Is $X_{an}$ necessarily Kahler? Certainly we know this if $X$ is projective.
A complex torus is algebraic ...

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275 views

### Inclusion of logarithmic de-Rham complex into differentials

Let $X$ be a complex manifold and $D$ a normal crossing divisor. Let $U = X - D$ and $j: U \rightarrow X$ the natural map. Voisen observes that there is a natural inclusion $$\Omega^k_X(\log D) ...

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### How do Hodge classes for Calabi-Yau 4-folds compare with the classes for tori?

Let $X$ be a Calabi-Yau 4-fold, i.e., a connected 4-dimensional compact Kahler manifold with $K_{X} \cong \mathscr{O}_{X}$ and $h^{i} (X,O_{X} )= 0$ for $0 \lt i \lt 4$.
Given a general ...

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### Leray degeneration for smooth projective morphisms and formality of families of compact Kähler manifolds

Let $\pi \colon X \to S$ be a smooth projective morphism of algebraic varieties, say over $\mathbf C$. By Deligne's argument ("Théorème de Lefschetz...", 1968), there is for each $i$ an injection
$$ ...

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### Second lowest weight piece of the cohomology of an algebraic variety

If $U$ is a smooth algebraic variety, then one can give a simple description of the lowest weight part of its cohomology: if $X$ is a smooth compactification and $j \colon U \to X$ the inclusion, then ...

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### 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 ...

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593 views

### Equivalence between statements of Hodge conjecture

Dear everyone,
I was unable to obtain the equivalence between the two statements of the Hodge conjecture. I searched for some previous questions that others asked here, to check whether someone has ...

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258 views

### Leray spectral sequence for lowest weight part of a smooth morphism

Let me assume everything in sight is as nice as possible, probably if the result I want is true then these conditions are too restrictive. All spaces will be smooth algebraic varieties over the ...

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### Heuristics for the Hodge Conjecture

W. V. D. Hodge is famous for his Hodge conjecture, one of the Millennium prize problems. Hodge might have had some rough heuristics or ideas that led him to the formulation of the conjecture.
I am ...

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### Cattani-Deligne-Kaplan Cattani-Deligne-Kaplan for non-integral cycles

Can one can omit the integrality condition in the Cattani-Deligne-Kaplan theorem?
That is, in Corollary 1.2 in the Cattani-Deligne-Kaplan paper, is the condition "$u\in V_s$ is integral" in "$u\in ...

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### Is it written anywhere that open subvarieties of affine spaces have “completely impure” cohomology?

Consider complex affine space $\mathbb{C}^n$ and let $U$ be a Zariski open subset of $\mathbb{C}^n$. By a celebrated result of Deligne, the cohomology $H^i(U)$ has a canonical Hodge structure. In ...

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### Adjoint of a Connection Using the Hodge Map?

For a Riemannian manifold $(M,g)$ with exterior derivative d, the codifferential d$^\ast$ is defined to be the unique map for which
$$
g(\omega,d\omega') = g(d^* \omega,\omega'), ~~~ \omega,\omega' ...

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### Definition of the $L^2$-metric for the Determinant of Cohomology of a Vector Bundle on a Riemann surface

I start describing my setup. $X$ is a Riemann surface with a metric which can have a finite number of singularities, $E$ is a vector bundle on $X$ equipped with an Hermitian structure.
In an article ...

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### 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 ...

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### 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 ...

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### 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 $G_{B\times C}$, ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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?

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### 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.

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### 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 ...

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### 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, ...

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### 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) ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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, ...

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### 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 ...

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### 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,..)
...

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### 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 $g$ ...

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### 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 ...

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### 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, ...