# 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.

339
questions

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### Applications of Hodge-Riemann bilinear relations [closed]

I am wondering if the Hodge-Riemann bilinear relations have any further applications/ developments in Kahler or algebraic geometry.
Let me briefly remind the statement.
Given a compact Kahler ...

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186 views

### Every homotopy class contains at least a harmonic representative

Let $(M^3,g)$ be a closed, connected and oriented Riemannian $3$-manifold. A circle-valued map $v : M \to S^1$ is harmonic iff the gradient $1$-form $\omega_v = v^* d\theta \in \Omega_1(M)$ is ...

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161 views

### Middle cohomology of very general hyperplane sections

Let $X$ be a smooth, projective variety over $\mathbb{C}$ of dimension $n$ satisfying the property that for every $i \ge 0$, $H^{i,i}(X,\mathbb{C}) \cap H^{2i}(X,\mathbb{Q})=\mathbb{Q}c_1(\mathcal{O}...

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### $\mathbb{Z}_2$-grading by Hodge star operator (for signature theorem)

This question may be a bit low level for MO but I have not received any attention from the SE post.
Consider the algebra of exterior forms $\bigwedge T^*M$ on an even dimensional $n$-manifold $M$. We ...

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117 views

### non-singular divisors of the jacobian variety

Let $X$ be a smooth, projective curve of genus at least $4$. The well-known divisor $\theta$ of the associated Jacobian variety is $\mathrm{Jac}(X)$ is singular and also ample. The $\theta$ divisor ...

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### Understanding the $\text{SL}_{2}$-orbit theorem

I am trying to understand a meaning of Schmid's $\text{SL}_{2}$-orbit theorem. The nilpotent orbit theorem now seems quite clear to me (both its proof and why one would want to consider such theorem), ...

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85 views

### (Implemented) algorithm for Hodge numbers

Let $X$ be a smooth projective toric variety. Do any of the math computer algebra systems have an algorithm implemented to compute the Hodge numbers of a generic complete intersection in $X$? Say in ...

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90 views

### Hodge theorem for cohomology group on holomorphic vector bundles and harmonic forms

Consider the holomorphic vector bundle $\pi: E \rightarrow M $ where $M$ is a complex manifold of dimension $\dim_{\mathbb{C}}M = m$. Denote by $\Omega^{p,q} (M)$ the bundle for bidegree $p,q$ ...

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136 views

### cohomology of dual intersection complex of a k3 surface

Let $\Delta \subset \mathbb{C}$ be a small disc and let $f: X \to \Delta$ be a flat morphism of complex manifolds such that $X_t$ is a smooth K3 surface for $t \neq 0$, and $X_0$ is an snc divisor. ...

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175 views

### Hodge decomposition and Kunneth formula on product manifold

Let $M$ be a compact oriented Riemannian manifold. Then we have the famous Hodge decomposition theorem:
$$
\Omega^*(M)= im(d)\oplus \mathcal H^*(M) \oplus im(d^*)
$$
Now, we want to consider the ...

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123 views

### Elliptic boundary value problem for vector valued forms

Let $U \subset R^n$ be a regular bounded domain having the topology of a ball. Then, the boundary value problem for $\omega\in \Omega^2(U)$,
$$
d\omega = 0
\qquad
\delta\omega = \sigma
\qquad
\...

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276 views

### Is $h^{0,k}$ a topological invariant?

Let $X$ and $Y$ be two smooth projective varieties over $\mathbb{C}$ such that $X(\mathbb{C})$ is homeomorphic to $Y(\mathbb{C})$. Is it true that $\dim_{\mathbb{C}} H^k(X,\mathcal{O}_X)=\dim_{\mathbb{...

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283 views

### Non-compact hard Lefschetz theorem

For a compact Kaehler manifold $M$, a basic structural result for its de Rham cohomology is the hard Lefschetz theorem. See here or here for an overview of the result.
What happens in the non-...

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323 views

### How to understand the proof of Proposition 2.1 in the paper 'Nodes and the Hodge conjecture'?

In the Proposition 2.1 of the paper 'Nodes and the Hodge conjecture', R.P.THOMAS gives a proof to descending the Hodge conjecture into showing that every (n,n)-Hodge class in a $2n$-dimensional smooth ...

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114 views

### Hypersurfaces with maximal Picard rank

Is it true that for any $d \ge 4$, there exists a smooth, degree $d$ surface $X$ in $\mathbb{P}^3$ with maximal Picard rank i.e., Picard rank of $X$ equals $h^{1,1}(X)$?

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### (Higher) extensions of mixed Hodge structures

Mixed Hodge structure is introduced by Deligne and it's very useful for studying complex algebraic varieties. We know $\text{MHS}$, the category of mixed Hodge structures is an abelian category. Where ...

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### Quantum cup product and Dolbeault cohomology

Let $X$ be a smooth projective variety over $\mathbb{C}$. We consider the small quantum cup product $\star$ on the deRham cohomology ring $\displaystyle H^*(X;\mathbb{C})=\bigoplus_{p,q}H^{p,q}(X)$. ...

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193 views

### Intrinsicness of Hodge-theoretic properties of Galois representations in a general reductive group

In the paper "The conjectural connections between automorphic representations and Galois representations" by Buzzard and Gee, it is said
"We say that
$\rho$ is crystalline/de Rham/Hodge–Tate if ...

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151 views

### An archimedean analogue of the non-canonicity of Hodge--Tate decomposition

For smooth proper schemes over $\mathbb{C}_p$, there is no canonical Hodge--Tate decomposition (but there is something close). Is there an analogue of this on the archimedean side? I thought about ...

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237 views

### How to show that Hodge filtration of CM type is algebraic?

I'm asking for a proof or references of the following claim:
Let $V$ be a rational Hodge structure having CM in the sense that its Mumford-Tate group is abelian. Then there is a filtration $F^{\...

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459 views

### Hodge structure not coming from the cohomology of a manifold

What is an explicit example a pure polarizable finite-dimensional $\mathbb{Q}$-Hodge structure that is not a subquotient of the cohomology of a scheme smooth proper over $\mathbb{C}$? What if replace "...

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71 views

### Period mapping in non-abelian Hodge theory

Given a family of Kaehler manifolds, by looking at the cohomology we can construct the period mapping. What should be the analogue of the period mapping in non-abelian Hodge theory (i.e. if we look at ...

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161 views

### Hodge-theoretic criterion for smoothness

Let $k$ be an algebraically closed field of any characteristic. Is it possible to give an equivalent condition for a $k$-variety to be smooth using only the cohomology of the variety (and whatever ...

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109 views

### Is there a Hodge structure on $\text{Hom}(V,W)$?

Let $V, W$ be real (pure) Hodge structures of weight $m, n$. Is there a natural Hodge structure on $\text{Hom}(V,W)$?
As I understand, there is one in the case $V = W$, although the definition I ...

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271 views

### An integral of the Hodge-Neumann Laplacian on a Riemannian manifold

Background
Let $M$ be a compact, oriented Riemannian manifold with boundary, with $\text{vol}$ the Riemannian volume form. Let $\nu$ be the outwards unit normal vector field on $\partial M$ and $\nu^\...

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### Flattening a connection on a Kähler manifold

Say $M$ is a closed Kähler manifold and $(V, \nabla)$ is a (say) constant Hermitian bundle on $V$ with (say) trivial flat connection. Now $M$ Kähler gives several distinguished classes of closed one-...

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256 views

### Nearby cycles and extension by zero

Let $f: X\to \text{Spec}(R)$ be a proper and smooth morphism, with $R$ a strictly henselian dvr. Call $s = \overline{s}$ the closed point and $\eta$ the geometric point of $\text{Spec}(R)$.
Call $i_s ...

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214 views

### p-adic Hodge theory for singular projective varieties

In p-adic Hodge theory, one has comparison theorems relating, for example, the crystalline cohomology of the special fiber of a smooth proper family with the etale cohomology of the rigid-analytic ...

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207 views

### Regular functions vs holomorphic functions

Let $X$ be an affine smooth variety over the complex numbers, $X^{an}$ its associated smooth complex analytic space, and $\mathcal{O}$, resp. $\mathcal{O}^{an}$ the respective structure sheaves.
Is ...

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153 views

### Mixed Hodge Polynomial for Algebraic Stacks

Let $X$ be a complex algebraic variety. The numerical invariants associated with the Mixed Hodge Structure of $X$ can be encoded in a polynomial in three variables called the mixed Hodge polynomial $H(...

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### Constancy of Hodge numbers in a family of compact complex manifolds

Does there exist a family of compact complex manifolds over unit disk such that the Hodge numbers are not constant in the family?
The answer is manifestly positive in complex dimension 1.
It is ...

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126 views

### Family of Hodge decomposition

It is known that a metric $g$ gives a Hodge decomposition:
$$
\Omega^*(M)=\mathcal H^*(M)\oplus d\Omega^*(M) \oplus \delta_g \Omega^*(M)
$$
Note that the usual differential restricts to an isomorphism ...

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530 views

### Some basic questions on crystalline cohomology

Let $X_0$ be a smooth projective variety over $\mathbf{F}_q$ and ${X}$ its base change to an algebraic closure $k$ of $\mathbf{F}_q$.
Crystalline cohomology $H^*_{\rm cris}(X) := H^*((X/W(k))_{\rm ...

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### Bounding the denominator in the canonical bundle formula

My question concerns with Theorem 3.1 in the paper "A canonical bundle formula" by Fujino and Mori.
The theorem claims the following:
Suppose $X \to C$ is a fiberation whose general fiber $F$ has ...

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### Functoriality of crystalline cohomology

Let $k$ be a perfect field of characteristic $p>0$, $X$ a smooth projective $k$-variety.
Denote by $(X/W_n(k))_{\rm cris}$ the small crystalline site of $X$.
Let $f : X\to Y$ be a morphism of ...

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### What is Prismatic Cohomology?

Prismatic cohomology is a new theory developed by Bhatt and Scholze; see, for instance, these course notes. For the sake of the community, it would be great if the following question is discussed in ...

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133 views

### Intermediate Jacobian of abelian varieties

Is the intermediate Jacobian of an abelian variety again an abelian variety?

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### Principal bundle analogue for Hodge bundle

Let $X$ be a connected smooth complex projective variety.
A holomorphic Higgs bundle is a pair $(E, \theta)$ consists of a holomorphic vector bundle $E$ on $X$ together with a Higgs field $\theta \...

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### Does local system cohomology come equiped with a mixed hodge structure?

Let $X$ be a quasi-projective variety over $\mathbb{C}$, and let $\mathcal{L}$ be a rank one $\mathbb{C}$ local system on $X$. Does $H^*(X,\mathcal{L})$ come with some mixed hodge structure in general?...

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### Differential construction of mixed Hodge structure on smooth open varieties

Let $\bar{X}$ be a complete smooth variety over $\mathbb{C}$ and $D$ be a simple normal crossing divisor. Denote $X:=\bar{X}\backslash D$. Then it is known that $H^\ast(X,\mathbb{C})$ admits a ...

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### Classes of hyperplane sections in cohomology

Let $X$ be a smooth projective variety over the algebraic closure of a finite field with Galois group $G$.
Is it true that the vector space $H^{2k}(X,\mathbf{Q}_{\ell}(k))^G$ has always positive ...

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### What is Kontsevich's Hodge theory of path integrals?

I was reading about the appearance of Calabi-Yau manifolds in Feynman integrals, and I thought to wonder if there is such a thing as "infinite-dimensional Hodge theory". Googling the phrase turned up ...

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### Hodge theory (after Deligne)

In an interview with Deligne on the Simons Foundation website, I heard Robert MacPherson say that at the time Deligne's papers on Hodge theory were being published, the results seemed absolutely ...

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### Two notions of a “nilpotent orbit”

I am wondering about the equivalence of two notions of a "nilpotent orbit".
The first notion, which I am familiar with, is as follows: given a lie group $G$ and a lie algebra $\frak{g}$, the orbit of ...

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### Hodge decomposition and degeneration of the spectral sequence

I am teaching a course on Hodge theory and I realised that I don't understand something basic.
Let first $X$ be a compact Kahler manifold. Let $H^{p,q}(X)=H^q(X,\Omega^p_X)$ where $\Omega^p_X$ is the ...

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568 views

### Is there an example to show the Hodge decomposition fails on non-compact case?

The theorem of Hodge decomposition is on the compact Kahler manifold, is it generally true for the non-compact kahler manifold or are there examples to show the failure?
Here is my Hodge ...

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293 views

### Degeneration of relative Hodge-de Rham spectral sequence

$$\require{AMScd}$$
$$\newcommand{\CC}{\mathbb{C}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\Hdr}{H_{\mathrm{dRh}}}
\newcommand{\tensor}{\otimes}
\newcommand{\Ohol}{\mathcal{O}}$$
Please excuse that ...

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### Generic Mumford Tate group and algebraic points

I will stick with a concrete example for this question, but it should probably be cast in a more general framework.
Let $Sym_g(\mathbf{C})$ be the space of symmetric matrices of order $g$ with ...

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218 views

### dual to Hodge theory

Let $(M,g)$ be a closed Riemannian manifold.
In my understanding Hodge theory shows that any de Rham cohomology class can be represented uniquely by a harmonic form. Moreover the harmonic form ...

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107 views

### Topological cycles with Lagrangian support

For a compact Kähler manifold of dimension $2n$, is there a classification of the homological $n$-cycles which are supported in a compact Lagrangian submanifold?
The main example for this question ...