Questions tagged [hodge-theory]

The study of harmonic differential forms on complex projective varieties, their invariantly defined filtrations, their integrals over topological cycles, especially over subvarieties, the deformations of these integrals and filtrations in families, and a multitude of generalizations.

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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 ...
Sándor Kovács's user avatar
14 votes
1 answer
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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, ...
Mikhail Bondarko's user avatar
115 votes
2 answers
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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 ...
Fitzcarraldo's user avatar
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37 votes
4 answers
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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 ...
Konrad Voelkel's user avatar
32 votes
7 answers
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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 (http://en.wikipedia.org/wiki/...
Eric Zaslow's user avatar
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19 votes
1 answer
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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 ...
Charles Siegel's user avatar
17 votes
2 answers
3k views

Hodge decomposition in Minkowski space

This question is motivated by the physical description of magnetic monopoles. I will give the motivation, but you can also jump to the last section. Let us recall Maxwell’s equations: Given a semi-...
The User's user avatar
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10 votes
2 answers
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When do flat holomorphic connections exist?

Let $X$ be a smooth projective variety over $\mathbb{C}$. I know that a vector bundle $\mathcal{E}$ on $X$ admits a holomorphic/algebraic connection iff its Atiyah class vanishes, $A(\mathcal{E}) = 0$....
Ben C's user avatar
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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 $\...
Will Sawin's user avatar
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7 votes
1 answer
948 views

How to think about $\mathbf{Z}(n)_{\mathcal{M}}$

One definition of motivic cohomology for smooth schemes $X$ over a field, is via Friedlander-Suslin complexes. A refresher (you may skip to the question at the bottom) One defines (1) $z_n(X,d) :=$...
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Can Hodge symmetry and invariance of Hodge numbers in smooth families be proven purely algebraically?

Let $k$ be an algebraically closed field of characteristic 0. I am wondering if there are proofs of the following facts that do not use the analytic topology over $\mathbb{C}$: Let $X$ be a smooth ...
William C. Newman's user avatar
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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 ...
Jun Lu's user avatar
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Is the Leray projection continuous with respect to the Frechet topology of smooth periodic vector fields in $3$ dimensions?

Let $\mathbb{T}^3:=(\mathbb{R}/\mathbb{Z})^3$ be the $3$-torus and $C^\infty(\mathbb{T}^3,\mathbb{R}^3)$ be the Frechet space of smooth periodic vector fields on $\mathbb{T}^3$. By Helmholtz ...
Isaac's user avatar
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39 votes
1 answer
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Clausen's modified Hodge Conjecture

In a recent talk at the University of Geneve, Dustin Clausen presented a "modified Hodge Conjecture". I found the abstract intriguing but couldn't find videos or notes available online. If I'...
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26 votes
0 answers
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Is there a proof of Hodge theory using condensed mathematics?

As is well known, many results in complex geometry "feel" algebraic (and often have statements which are "completely algebraic") but only have "transcendental" proofs (i....
Gabriel's user avatar
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26 votes
4 answers
6k 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 ...
Charles Siegel's user avatar
22 votes
2 answers
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 $X_{\mathbb C}$...
Alexander Braverman's user avatar
15 votes
3 answers
776 views

Log-concavity of matroids: characterization of equality?

Let $M$ be a (loopless) matroid of rank $r$. The characteristic polynomial $\chi_M(x)$ is defined by $\chi_M(x)=\sum_{F \in \mathcal{L}(M)}\mu(\hat{0},F) \cdot x^{\mathrm{rk}(F)}$, where $ \mathcal{L}(...
Sam Hopkins's user avatar
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15 votes
1 answer
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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 ...
Akhil Mathew's user avatar
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12 votes
3 answers
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Motivic vs Deligne cohomology

Where can I find the construction of the cycle class map from motivic cohomology to Deligne cohomology of smooth projective varieties over the complex numbers? It should be a construction by Bloch ...
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11 votes
1 answer
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How to think about infinite generatedness of motivic cohomology

In this question I previously asked how to think about the motivic complex $\mathbf{Z}(1)_{\mathcal{M}}$, whose Zariski hypercohomology should morally be the "singular cohomology" $H^*((-)\wedge S^{2n}...
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11 votes
0 answers
360 views

Carleson's Theorem on Manifolds

Let $M$ be an oriented, compact, differentiable manifold with some Riemmanian metric $g$, so that $(M,g)$ has a nice volume form and one can define $L^2(M,g)$ as the completion of $C^\infty(M)$ under ...
Greg Zitelli's user avatar
11 votes
1 answer
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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 ...
Shanmukha_Srinivasan's user avatar
10 votes
2 answers
3k views

Torsion in cohomology of smooth manifolds

I've been interested in the possible (singular) cohomology groups of complex projective algebraic varieties, and there are lots of theorems that give various restrictions on these (Hodge decomposition,...
LMN's user avatar
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9 votes
1 answer
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Betti numbers of Proper nonprojective varieties

This is a question about pathologies. Let $X/\mathbb{C}$ be an irreducible projective variety smooth over $\mathbb{C}$. Then, the singular cohomology groups $H^i(X, \mathbb{C})$ have a hodge ...
LMN's user avatar
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8 votes
1 answer
777 views

A Naive Question on Mixed Motives and Mixed Hodge Structures

As a physicist, I have some naive questions about mixed motives and its mixed Hodge structure (MHS) realization. Any references, comments, answers will be appreciated! The category of mixed motives ...
Wenzhe's user avatar
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8 votes
1 answer
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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 ...
LMN's user avatar
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7 votes
0 answers
719 views

Examples of Maximal degeneration of Deligne on Calabi-Yau degeneration

Pierre Deligne in his celebrated paper entitling "Local behavior of Hodge structures at infinity" introduced Maximal degenerations of Calabi-Yau manifolds. Let $\pi:X\to \mathbb C^*$ be a family of ...
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7 votes
1 answer
798 views

Relation between the cohomology group of a curve and the cohomology group of its jacobian

Let $J_C$ be the Jacobian of a smooth projective curve $C$ over $\mathbb{C}$. I would like understand the isomorphism between $H^1(J_C,\mathbb{C})$ and $H^1(C,\mathbb{C})$. I read in a paper that ...
Roxana's user avatar
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6 votes
1 answer
976 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?
Adam Harris's user avatar
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6 votes
0 answers
128 views

The metric gives the optimal element in a class

In geometry there is plenty of examples in which the following happens: Some elements are considered equivalent, in some topological or algebraic sense We take the quotient The metric is usually not ...
geodude's user avatar
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5 votes
0 answers
356 views

multiplicative structure on the monodromy weight filtration spectral sequence

Let $f: X \to \Delta \subset \mathbb{C}$ be a projective morphism of a complex manifold to a small disc, smooth away from 0, and such that $Y=f^{-1}(0)$ is a strictly normal crossing divisor, and let $...
Dima Sustretov's user avatar
5 votes
2 answers
521 views

Alternative construction of the first Chern class map

Let $X$ be a compact Kähler manifold. Consider the exponential sequence $0 \to \mathbf Z \to \mathscr O_X \to \mathscr O_X^* \to 0$. The boundary map gives a map $H^1(X, \mathscr O_X^*) \to H^2(X, \...
Shubhodip Mondal's user avatar
3 votes
0 answers
165 views

Cycle maps as edge maps

Given a smooth projective algebraic variety over $\mathcal{C}$, let $X$ be its associated complex analytic space. The exponential sequence on $X$: $$0\to\mathbf{Z}(1)\to\mathcal{O}_X\to\mathcal{O}_X^...
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3 votes
1 answer
416 views

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 ...
Ben Webster's user avatar
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2 votes
0 answers
133 views

Can logarithmic connection operate on currents?

Let $(X,\omega)$ be a compact Kähler manifold of dimension $n$, and let $D=\sum_{i=1}^r D_i$ be a simple normal crossing divisor on it, i.e., a divisor with smooth components $D_i$ intersecting each ...
Invariance's user avatar
1 vote
0 answers
179 views

Can logarithmic connection on holomorphic vector bundle induce logarithmic connection on dual bundle?

Let $(X,\omega)$ be a compact K"ahler manifold of dimension $n$, and let $D=\sum_{i=1}^r D_i$ be a simple normal crossing divisor on it, i.e., a divisor with smooth components $D_i$ intersecting ...
Invariance's user avatar