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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What are the Hodge and log Hodge groups of $M_{g,n}$?

I would like to know, ideally with a reference, what the Hodge and log Hodge numbers of the moduli space of stable curves $\bar M_{g, n}$ are. At the very least I'd like to know the genus zero case $g ...
Leo Herr's user avatar
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Original proof of Lefschetz's theorem on $(1,1)$ classes

Is there a "modern" account of Lefschetz proof of his theorem about $(1,1)$ classes for projective surfaces ? I believe that would be very interesting to understand the original arguments ...
Nicolas Hemelsoet's user avatar
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Using connection form for unknown frame field

I have a way to calculate the connection 1-form $\alpha$ associated to a compact simply connected parallelizable Riemannian surface $(M,g)$ (so, $M$ is topologically a disk) and a special orthonormal ...
yak's user avatar
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Hodge coniveaux of Calabi-Yau manifolds

Let $X$ be a strict compact Calabi-Yau manifold of dimension $n$. By this, I mean that $X$ is a simply connected projective manifold whose holomorphic forms are generated by a nowhere zero top degree ...
Pène Papin's user avatar
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Fiedler vector of an abstract simplicial complex and partitioning

Let $G=(V,E)$ be a connected graph, and $L = A - D$ the corresponding graph Laplacian. The second smallest eigenvalue of $L$, $\lambda_1$, is Fiedler's value, and the associated eigenvector, $\phi_1$, ...
incud'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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On Simpson's motivicity conjecture

Simpson's motivicity conjecture says that for any rigid, flat irreducible connection $(V,\nabla)$ on a smooth complex variety $M$, there exists a proper smooth morphism $f:X \to M$ s.t. $(V,\nabla)$ ...
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Torelli theorem for veronese double cone(reference needed)

Let $Y$ be a smooth Veronese double cone, which is a smooth del Pezzo threefold of degree one, which can be regarded as a weighted hypersurface of degree 6 in $\mathbb{P}(1,1,1,2,3)$. I was wondering ...
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Simpson correspondence for perverse sheaves

Let $X$ be a projective complex manifold. Then Simpson's correspondence from nonabelian Hodge theory shows that the category of semisimple local systems on $X$ is equivalent to the category of ...
Doug Liu's user avatar
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Fourier transform and Hodge-$*$ operator

Suppose I have a full-rank lattice $\Lambda\subset\mathbf{C}$. Then the classical Poisson summation formula says $$\sum_{\lambda\in\Lambda}f(\lambda)=\sum_{\lambda\in\Lambda'}\widehat{f}(\lambda)$$ ...
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Hodge conjecture as the equality of arithmetic and algebraic weights of motivic L-functions

Recently I became aware of the following statement given on page 13 of this paper. First, let us recall the following definitions: Definition 4.1. Suppose $L(s)$ is an analytic $L$-function with ...
KStarGamer's user avatar
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189 views

Hodge symmetry without $\mathbb{C}$ [duplicate]

If $k$ is a field of characteristic zero and $X$ is a smooth irreducible projective variety over $k$, then $X$ satisfy Hodge symmetry, meaning that $$\dim H^p(X, \Omega_{X/k}^q) = \dim H^q(X, \Omega_{...
Antoine Labelle's user avatar
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118 views

Intermediate Jacobian for small resolution of a singular Fano threefold?

I am mainly interested in the nodal Gushel-Mukai threefold. Let $X$ be a Gushel-Mukai threefold with one node, then by page 21 of the paper https://arxiv.org/pdf/1004.4724.pdf there is a short exact ...
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Is $H^*($vanishing cycles$)$ computed by the twisted de Rham complex?

In notes by Sabbah (Theorem 3), it is stated that the cohomology $$\text{H}^*(X,\varphi_f)$$ of the vanishing cycle sheaf of a function $f:X\to \mathbf{A}^1$ for certain $X$ is expected to be the same ...
Pulcinella's user avatar
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Eigenforms of the Laplacian on Lie groups

I am a bit rusty in my differential geometry and I would like to confirm that my reasoning below holds, and I have some related questions (and all references to related concepts are of interest to me)....
Daniel Robert-Nicoud's user avatar
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263 views

Cohomology of singular curves

Suppose $X$ is a singular quasi-projective curve over the complex numbers, and $X'$ is a good nonsingular compactification of a resolution of singularities $Y\to X$. Let $D$ be the complement of $Y$ ...
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2 votes
1 answer
241 views

Geometric Interpretation of absolute Hodge cohomology

$\quad$Let $\mathcal{Sch}/\mathbb C$ denote the category of schemes over $\mathbb C$. For an arbitrary $X\in\mathcal{Ob}(\mathcal{Sch}/\mathbb C)$, Deligne in his Article defined a polarizable Hodge ...
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12 votes
2 answers
1k views

Does Poincaré duality preserve algebraic cycles?

Let $X$ be a smooth, projective (complex) variety of dimension $n$ and $Z \subset X$ be a subvariety of codimension $k$ (if necessary assume $Z$ is non-singular). We know the cohomology class $[Z]$ of ...
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Confusion about notations in limit mixed Hodge structure

I am reading the paper Monodromy at infinity and Fourier transform by Claude Sabbah and got some confusions about notations. (note first that I am not specialized in mixed Hodge theory but, I am ...
Alexey Do's user avatar
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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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2 votes
1 answer
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Triangulated structure on complexes of mixed Hodge structures

I'm trying to read parts of the Peters Tata Lectures on "Motivic Aspects of Mixed Hodge structures" One aspect I don't really understand is the construction of the ''mixed cone'' for ...
Aaron Wild's user avatar
5 votes
1 answer
252 views

Formula in non-Abelian Hodge theory - Hodge-Riemann bilinear relations

I am currently reading about the non-Abelian Hodge correspondence. Let $(X,\omega)$ be a compact Kahler manifold. Given a Higgs bundle $(E, D_0)$ on $X$, we want to construct the corresponding flat ...
Will Fisher's user avatar
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1 answer
267 views

Given a smooth hyperplane section Y of a variety X there exists a Lefschetz pencil of hyperplane sections of X containing Y

Let $X$ be a variety contained in $\mathbb{P}^N$ and let $Y$ be a smooth hyperplane section of $X$. I have read in page 54 of Voisin's book "Hodge theory and complex algebraic geometry II" ...
Roxana's user avatar
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Hodge-Helmholtz decomposition for 1-form of strategic game

This question is motivated by the attempt of decomposing (in a direct sum sense) a strategic game into components in the spirit of Hodge decomposition. Preamble Combinatorial setting Candogan et al. (...
DavideL's user avatar
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Balanced manifolds and the $dd^c$-lemma

Let $X$ be a compact complex manifold. A Hermitian metric $\omega$ is balanced if $d\omega^{n-1}=0$, where $n=\dim_{\mathbf{C}} X$. By a theorem of Alessandrini-Basanelli, this class of Hermitian ...
AmorFati's user avatar
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292 views

Hodge decomposition on non-compact manifolds

Let $(\mathcal{M},g)$ be a compact Riemannian manifold without boundary. Then we have the well-known Hodge decomposition $$\Omega^{k}(\mathcal{M})\cong\mathcal{H}^{k}(\mathcal{M})\oplus\mathrm{ran}(\...
B.Hueber's user avatar
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1 answer
138 views

Cohomology classes fixed by algebraic automorphism subgroups

Let $X$ be a smooth projective complex variety and $t\in H^{2p}(X,\mathbf{Q})(p)$ a rational cohomology class. Assume that there exist $$t_1,\ldots,t_N\in H^{2*}(X,\mathbf{Q})(*)$$ algebraic classes (...
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Period calculation for elliptic curve

In the paper "Hodge cycles on Abelian Varieties" (Proposition 1.5), Deligne proves the following theorem: Let $X$ be a smooth projective variety over $\overline{\mathbf{Q}}$ of dimension $n$...
Adithya Chakravarthy's user avatar
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Hodge decomposition for non-elliptic complexes

It is a well-known result that there is a bijective correspondence between harmonic sections and cohomology classes of an elliptic complex in a Riemannian/Hermitian manifold. Now consider Riemannian/...
Arturo's user avatar
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Finding a Hodge theoretic condition to measure the rank of isogeny of product abelian surfaces

Let $A$ be an abelian surface over $\mathbb{C}$, then there is a condition on $H^0\left(\Omega^1_A\right)$ to determine if $A$ contains an elliptic curve $E$ as a subvariety. If $A$ were to contain ...
Stormblessed's user avatar
6 votes
0 answers
221 views

Rank $2$ motivic local systems on a curve

This question is about the article "Motivic local systems on curves and Maeda's conjecture" by Yeuk Hay Joshua Lam. In the proof of Theorem 1.1 it is claimed (on lines 4-5 of p. 7) that any ...
naf's user avatar
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Minimal Betti numbers of simply-connected threefolds with trivial canonical class

By a threefold, I mean a compact complex manifold of dimension three. For a simply-connected threefold with trivial canonical class, its Betti numbers satisfy: $$b_2 \ge 0, b_3 \ge 2.$$ I am wondering ...
Basics's user avatar
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5 votes
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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
1 vote
1 answer
98 views

Intermediate Jacobian under group action

Let $X$ be a smooth Fano threefold with a finite group $G$ action. Assume that the orbit space $X/G$ is smooth. Is it true that $J(X/G)\cong J(X)^G$ As an abelian variety? Here, $J(X)^G$ is the $G$-...
user41650's user avatar
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2 votes
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Hard Lefschetz for perverse sheaves on Kähler manifolds

Let $(X,\omega)$ be a compact Kähler manifold, $k\ge0$, $P\in Perv(X)$ be a semisimple object, then do we have the hard Lefschetz isomorphism between perverse cohomology sheaves $\omega^k:{}^p\mathcal{...
Doug Liu's user avatar
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6 votes
1 answer
142 views

Properties of non-integer powers of the Hodge Laplacian

Consider a complete smooth Riemannian manifold $(M,g)$. I think that it is not difficult to prove that the $k$ Hodge Laplacian is essentially selfadjoint in the relevant $L^2$ space of $k$ forms, ...
V. Moretti's user avatar
25 votes
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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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12 votes
0 answers
218 views

Spin 6-fold with signature $\pm 16$

Does there exist a spin (i.e. $\frac{c_{1}}{2} \in H^{2}(M,\mathbb{Z})$) smooth complex projective $6$-fold with signature $\pm 16$? The motivation is the Rochlin-Ochanine theorem, which says that $16$...
Nick L's user avatar
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2 votes
1 answer
489 views

Example motivating mixed Hodge structures

The suggested intuition behind mixed Hodge structures - developed in particular to generalize Hodge decomposition of cohomology groups from complex smooth complete varieties to more general algebraic ...
user267839's user avatar
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4 votes
1 answer
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Hodge conjecture for generic points

I was reading the following paper: "Beilinson’s Hodge Conjecture For Smooth Varieties". They study the cycle class map $cl_{m,r}: H^{2r-m}_{\mathcal{M}}(U, \mathbb{Q}(r))\rightarrow \text{...
user127776's user avatar
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1 vote
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Beilinson-Hodge conjecture and generation of cohomology ring by $H^1$

Beilinson's version of Hodge conjecture has the following form. For any quasi-projective smooth complex variety $X$ the following map is surjective: $$H^i_{\mathcal{M}}(X, \mathbb{Q}(j))\rightarrow \...
user127776's user avatar
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2 votes
0 answers
99 views

The dual of the Lefschetz operator under a perturbation

Let $(X, \omega)$ be a compact Kähler, or more generally, Hermitian manifold. Let $L_{\omega} : \Omega^k(X) \to \Omega^{k+2}(X)$ denote the Lefschetz operator given by $$L_{\omega}(\alpha) : = \omega \...
AshyK's user avatar
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3 votes
1 answer
470 views

What does does the monodromy weight filtration represent?

I'm trying to understand variations of Hodge structure. I understand that this is a very broad field, and that many of the concepts have been extended to algebraic geometry over fields other than $\...
Quaere Verum's user avatar
2 votes
0 answers
170 views

Modular forms and Petersson inner product via De Rham cohomology, Hodge filtration and cup products

I'm looking for an explanation on how and why you can define modular forms through De Rham cohomology via the Hodge filtration and especially how the Petersson inner product is related to the cup ...
Lukas Heger's user avatar
2 votes
1 answer
236 views

Algebraic and homological equivalence relations for $0$-cycles

Let $X$ be a connected smooth projective variety. Let $Z_0(X)_{alg}$ be the group of $0$-cycles algebraically equivalent to $0$ and $Z_0(X)_{\hom}$ be the group of $0$-cycles homologically equivalent ...
Roxana's user avatar
  • 519
1 vote
1 answer
225 views

Hodge theory beyond Riemannian and Kahler manifolds

Recently I read about graph-theoretic Hodge theory, which has uses in graph theory, topological data analysis, and more generally machine learning. I knew only the basics of Riemannian Hodge theory, ...
xuq01's user avatar
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2 votes
1 answer
297 views

Regularity of Gauss Manin connection

I want to understand the "Regularity of Gauss Manin connection" from the most basic example. Suppose we have a family of projective manifold $X\rightarrow \mathbb C^*$ with full rank, then ...
xin fu's user avatar
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3 votes
0 answers
135 views

variation of Hodge structure and singularity of period map

Let $X\rightarrow D^*$ be fiberation of projective manifold, here $D^*$ means a punctured disk. Then it induces a variation of hodge structure, i.e a holomorphic vector bundle $H_{\mathbb C}$ over $D^*...
xin fu's user avatar
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Any manifold in Fujiki class $\mathcal C$ admits a Kähler deformation?

It is known that small deformations of a manifold $X$ in Fujiki class $\mathcal C$ might no longer be in $\mathcal C$, see This paper for example. However, the following question is still open: For ...
Tom's user avatar
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0 answers
41 views

Is the union of Fujiki cones open in $\mathcal H^{1,1}_{\mathbb R}$?

Let $\mathcal X\to B$ be a holomorphic family of compact Kähler manifolds, let $\mathcal K_t$ denote the Kähler cone of the fiber $X_t$, then the union $\cup_{t\in B}\mathcal K_t$ forms an open set in ...
Tom's user avatar
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