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Questions tagged [hodge-theory]

For question about Hodge theory, which is a method for studying the cohomology groups of a smooth manifold using partial differential equations.

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de Rham cohomology is isomorphic to the harmonic forms on compact Kähler manifolds [duplicate]

Corollary 3.2.12 in Huybrechts's Complex Geometry states that the de Rham cohomology on compact Kähler manifolds can be decomposed in terms of the Dolbeault cohomology: $$ H^k(X, \mathbb{C}) \cong \...
領域展開's user avatar
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Is $\delta \Delta^{-1} d$ the identity operator?

Let $d, \delta, \Delta = (d\delta + \delta d)$ be the exterior derivative, codifferential and Laplace-de-Rham operator. Let $\omega$ be a closed $k$-form, one can then say $\Delta \omega = d \delta \...
Theo Diamantakis's user avatar
2 votes
1 answer
87 views

Definition of Polarised Hodge Structure

I'm reading section 7.1.2 of Voison's book. For a compact Kahler manifold with a Kahler form $\omega$, we have intersection pairing $$Q(\alpha,\beta) = \int_X\omega^{n-k}\wedge \alpha\wedge \beta$$ ...
Hydrogen's user avatar
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Hodge decomposition on a riemannian manifold for another measure?

Let $M$ be a closed riemannian manifold, we denote $dx$ the canonic volume measure on $M$. If we take $\mu$ a probability measure such that $d\mu = \rho dx$ with $\rho > 0$ we can define $d_\mu^*$ ...
Aymeric Martin's user avatar
2 votes
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abelian differentials are $L^2$ integrable

Let $C$ be a smooth algebraic curve over $\mathbb{C}$ (we can think for instance of a compact Riemann surface). Let $A$ be the subset of classes of holomorphic forms $\omega$ in $H^1(C,\mathbb{C})$ ...
Conjecture's user avatar
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What on earth is a topological cavity in a certain simplicial complex? [closed]

I am dealing with basic algebraic topology for my research, and have been confused about topological cavities for a long time. Until now, I have already understood the concepts about k-dimensional ...
dhliu's user avatar
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Prove that $d_3d_3^*+d_3^*d_3=-\nabla^2$

Consider the geometry in $\mathbb R^3$, define $$d_3=dx\frac{\partial}{\partial x}+dy\frac{\partial}{\partial y}+dz\frac{\partial}{\partial z}.$$ We then define the Hodge star operator $*_3:\Omega^p(\...
Ho-Oh's user avatar
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Morphism of varieties is continuous between analytic varieties.

I'm seeking clarification on the significance of commutative diagrams in understanding the analytic topology of smooth varieties. In Aleksander Horawa's notes, a commutative diagram is used to ...
ben huni's user avatar
  • 173
3 votes
0 answers
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Hodge decomposition on vector bundle-valued differential forms

Let $M$ be a compact Reimannian manifold and let $(E,h)$ be a Hermitian vector bundle over $M$. Let $A^k(M,E)$ denote the space of $E$-valued $k$-forms, i.e. smooth section of the bundle $\bigwedge^kT^...
Kanae Shinjo's user avatar
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The codifferential δ of a k -form on an n -dimensional Riemannian manifold [closed]

The Hodge dual (or formal adjoint) to the exterior derivative $d: \Omega^k(M) \to \Omega^{k+1}(M)$ on a smooth manifold $M$ is the codifferential $ d^* $, a linear map $$ d^*: \Omega^k(M) \to \Omega^{...
Member1434's user avatar
3 votes
1 answer
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Standard counterexample in Hodge decomposition

I am studying Hodge theory on complex manifolds; Höring's notes (https://math.univ-cotedazur.fr/~hoering/hodge/hodge.pdf p. 87) suggest, as an exercise (4.38) and I guess as a counterexample to Hodge ...
user720386's user avatar
7 votes
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Counterexample to decomposition of Harmonic Forms

It is well known that if $X$ is a Kahler manifold, then $$\bigoplus_{p+q=k}\mathcal{H}^{p,q}=\mathcal{H}^{k}(X,g)_{\mathbb{C}}=\mathcal{H}^{k}_{\overline{\partial}}(X,g)=\mathcal{H}_{\partial}^{k}(X,g)...
pleasantpheasant's user avatar
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Terminology for Complex Algebraic Geometry with Complex Conjugation

Semialgebraic geometry is essentially real algebraic geometry but with the defining polynomial relations allowed to be inequalities rather than just equalities. This doesn't make sense over $\mathbb{C}...
Harry Wilson's user avatar
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A nice description for $t^{k-1}B_{cr}^+/t^kB_{cr}^+$?

When $t$ is a uniformizer of the integral de Rham period ring, $B_{dR}^+$, there is an isomorphism $t^{k-1}B_{dR}^+/t^{k}B_{dR}^+\cong \mathbb{C}_p(k-1)$. Is there a nice description for $t^{k-1}B_{cr}...
kindasorta's user avatar
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Cohomology group induced by $d^*$

It is well known that for a smooth manifold $M$, the de Rham cohomology group is defined by $$H_{dR}^k(M):=\frac{A^k(M)\cap \ker d}{A^k(M)\cap \text{im }d}.$$ Similarly, if we assume that $M$ being a ...
Tom's user avatar
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The complexification of Hodge class is properly contained in the space of type $V^{k,k}$

I was reading Professor Huybrechts's Lectures on K3 surfaces, there is a statement about Hodge class that I can't figure out. Let's consider the (integral or rational) Hodge structure $V$ with the ...
yi li's user avatar
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Hodge structure on $\operatorname{Sym}(\mathbb{R}[1])$

I am at best superficially acquainted with the intricacies of Hodge theory. The following question comes from my study of the paper On the $\Gamma$-factors attached to motives by Christopher Deninger. ...
The Thin Whistler's user avatar
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What is $H^{p,p}(X,\mathbb{Q})$?

Edited for presentation. How to define it with words or/and quantifiers? Been struggling with a too big number of informations now, for hours. $p\in \mathbb{N}$. X is a $C^{\infty}$-manifold.
someone's user avatar
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Cohomology basics: relation between de Rham duality and Hodge duality.

Consider a product manifold $X=P \times P_{\perp}$, with $P$ and $P_{\perp}$ cycles of $X$, and let $\omega$ be the associated differential form of $P$ in the sense of de Rham duality (I suppose the ...
math_lover's user avatar
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How is the Hodge star operator defined for vector-valued forms?

Let $M$ be an oriented Riemannian manifold of dimension $n$. For any $\omega \in \Omega^k(M)$, we define the Hodge star operator $\star$ of a $\omega$ as the unique $n-k$ form $\star\omega$ that ...
CBBAM's user avatar
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Hodge decomposition metric dependency

I’m currently studying Complex Geometry by Daniel Huybrechts and I can’t understand why it is crucial to prove that Hodge Decomposition does not depend on the chosen metric. I’m talking about ...
Matteo Rossi's user avatar
2 votes
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112 views

Interpretation of eigenvalues and eigenvectors of combinatorial Hodge Laplacian in algebraic topology

Let $\Sigma$ be an abstract simplicial complex. Do the eigenvectors and eigenvalues of the combinatorial Hodge Laplacian $\Delta_k$, $$\Delta_k^\Sigma = (\partial_k^\Sigma)^\dagger \partial_k^\Sigma + ...
incud's user avatar
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Is there an easy direction for the Higgs correspondence?

There is a deep famous correspondence between analytic and algebraic properties. For a complex curve $X$, representations $Hom(\pi_1(X), U(n))$ correspond to degree $0$ semistable bundles. This is a ...
user135743's user avatar
1 vote
0 answers
29 views

Hodge polygon of tensor filtration

Let $V$ and $W$ be finite-dimensional vector spaces over a field $k$ with (exhaustive, separated, finite, descending) filtrations $F^\bullet$ and $G^\bullet$, respectively. On $V \otimes_k W$, we can ...
gimothytowers's user avatar
1 vote
0 answers
108 views

Is it possible to construct geometrically the ($\phi$, $\Gamma$)-module corresponding to a $p$-adic representation coming from geometry?

The $p$-adic étale cohomology of algebraic varieties over $p$-adic fields is a fundamental subject in the study of $p$-adic representations. Moreover, thanks to the comparison theorems in $p$-adic ...
Hiroyuki Sunata's user avatar
-3 votes
1 answer
150 views

Hodge conjecture [closed]

Hello I'm trying to understand the idea behind Hodge conjecture and I have naive approach but what does it mean Hodge classes statement: $H^{2k}(X,\mathbb{Q}) \cap H^{k,k}(X)$? these symbols? My ...
Anonim Mors's user avatar
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27 views

If $v_n : \mathbb{T}^3 \to \mathbb{R}^3$ is a sequence of uniformly convergent divergence-free vector fields, does its "antiderivative" converge?

Let $\mathbb{T}^3:=[\mathbb{R}/\mathbb{Z}]^3$ be the $3$-dimensional torus and $v_n : \mathbb{T}^3 \to \mathbb{R}^3$ be a sequence of smooth vector fields with the following property: $\sup_{x \in \...
Keith's user avatar
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1 vote
1 answer
132 views

Definition of Hodge tensor

The following is the definition of Hodge structures as given in [Milne]: Let $R$ be one of $\mathbb{R}, \mathbb{Q}$ or $\mathbb{Z}$. And, let $(V,h)$ be an $R$-Hodge structure of weight $n$. Then, ...
Coherent Sheaf's user avatar
1 vote
0 answers
61 views

Primitive cohomology and intersection with a hyperplane

Let $X$ be a smooth complex projective variety of dimension $n$, and $E\in H^{n-k}(X,\mathbb{Q})$. We have the operation of intersecting with a hyperplane class $H$, i.e. $- \cup H \colon H^i(X, \...
Conic3264's user avatar
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1 answer
95 views

The existence and uniqueness of the curvature of a Yang-Mills connection.

I am reading Jost's Riemannian Geometry and Geometric Analysis (7-Ed) and having a question about the Yang-Mills functional (page 183). Let $M$ be a compact manifold and $E$ be a metric bundle with ...
Justin Lien's user avatar
4 votes
2 answers
156 views

Coclosed form is sum of coexact and harmonic form.

Let $(\mathcal{M},g)$ be a compact and connected Riemannian manifold, $\mathrm{d}$ and $\delta$ differential and codifferential, respectively, and $\Delta:=\delta\mathrm{d}+\mathrm{d}\delta$ the ...
B.Hueber's user avatar
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0 answers
44 views

Hodge star operator and heat propagator

I am currently studying the Laplacian on a Riemaniann Manifold: An introdcution to analysis on manifolds by S. Rosenberg. I am solving some of the exercises and one of them (ex.3, ch 4.1 page 113) is ...
Bigalos's user avatar
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4 votes
0 answers
103 views

How to compute the exterior derivative of a 1-form on projectivization of a vector space

Let $V$ be a complex vector space, and let $\mathbb{P}(V)$ denote the projectivization of $V$ (i.e. space of 1-dimensional subspaces, i.e. 1st Grassmanian). Suppose further that $V$ is endowed with a ...
Milo Moses's user avatar
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3 votes
0 answers
218 views

What can we say about the divergence of Hamiltonian vector fields?

Let $M$ be a smooth $n$-dimensioanl manifold. To set some notations $C^\infty(M)$ denote smooth functions $M \to \mathbb{R}$ $\Omega^k(M)$ denote $k$-forms on $M$ $\tau(M)$ denote vector fields on $M$...
DavideL's user avatar
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1 vote
1 answer
173 views

Applications of the Lefschetz Hyperplane Theoren

There are a couple of applications of the Lefschetz Hyperplane Theorem I am struggling to wrap my head around. Hopefully someone knows how these facts are deduced directly from the theorem. Suppose $X$...
Shrugs's user avatar
  • 1,591
3 votes
1 answer
259 views

Equivalence of two definitions of Laplace-Beltrami on differential forms

I know of two ways of defining the (negative - depending on your convention) Laplace-Beltrami operator on the differential forms of a compact, orientable Riemannian manifold $M$. The Levi-Civita ...
Daniel Robert-Nicoud's user avatar
2 votes
0 answers
130 views

Laplacian comparison with Lefschetz decomposition

Set-up: Let $X$ be a complex manifold. Let $A^k$ be the sheaf of sections of the differential $k$-forms on a differentiable manifold, and let $A_{\mathbb{C}}^k$ be the sheaf of sections of $\Omega^k_{...
asking's user avatar
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3 votes
0 answers
107 views

What is the definition of a fiber of a vector bundle in algebraic geometry?

I am learning the variations of Hodge structure, yet getting stuck at the very first beginning. Let $S$ be a projective nonsingular variety over $\mathbb{C}$. I have seen that a variation of Hodge ...
Hetong Xu's user avatar
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1 vote
2 answers
196 views

Compactly supported harmonic forms

If $(\mathcal{M},g)$ is a compact Riemannian manifold (without boundary), then it is well-known that a $k$-form $\alpha$ is harmonic, i.e. $\Delta\alpha=0$, where $\Delta$ is the Laplace-Beltrami ...
B.Hueber's user avatar
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0 votes
0 answers
59 views

Laplacian of standard hermitian inner product on $C^n$.

Treat $C^n$ as a complex manifold, and consider the standard Hermitian metric $h = \frac{1}{2} \sum_{i=1}^n dz_i d\bar{z}_i$, so that the corresponding J-invariant metric is $g = \sum dx_i^2+dy_i^2$, ...
Steven Mai's user avatar
2 votes
0 answers
70 views

Can we infer Betti numbers of a family degenerating to a variety from the variety?

I'm noticing I know positive results to the following question in a number of special cases, but I don't know the general situation. Let $S$ be a complete trait, to make sure I'm not misspeaking ...
Curious's user avatar
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1 vote
0 answers
50 views

Does a co-exact 1-form generate a Hamiltonian vector field?

Let $M$ be an $n$-dimensional Riemannian orientable manifold. Denote the space of vector fields on $M$ by $V(M)$ and of $k$-forms on $M$ by $\Omega(M)$. Let $\xi = \delta \beta$ be a co-exact 1-form ...
DavideL's user avatar
  • 523
4 votes
1 answer
94 views

Products of antiharmonic forms (or functions) with harmonic forms

Let $X$ be a compact Kähler manifold, with fixed Kähler form $\Omega$. Then, the wedge product of two harmonic forms is not necessarily harmonic, as explained for instance here. This prompts the ...
Riccardo Pengo's user avatar
1 vote
1 answer
46 views

Why should analytic classes sit inside $H^{p,p}(X)$ for various values of $p>0$?

I am reading these notes notes by Popa. I did some reading of this post but I didn't exactly answer my question. It is claimed in example 4.5 that analytic classes sit inside $H^{p,p}(X)$ for various ...
Shrugs's user avatar
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3 votes
1 answer
123 views

What does it mean for a 1-form to be orthogonal to a 2-form?

In Baez & Munian's book Gauge Fields, Knots, and Gravity, when introducing the Hodge star operator, they say At any point $p$ in a 3-dimensional Riemannian manifold $M$, the Hodge star operator ...
CBBAM's user avatar
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2 votes
0 answers
45 views

For $\alpha\in A^{p,q}(X)\cap\ker d$, is $\partial\mathcal H_{\bar\partial}\alpha=0?$

Let $X$ be a compact complex manifold with a Hermitian metric $h$, then we can define $\bar\partial^*$ as the adjoint of $\bar\partial$, define $\Delta_{\bar\partial}:=\bar\partial\bar\partial^*+\bar\...
Tom's user avatar
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0 votes
1 answer
80 views

Generalization of Helmholtz theorem to differential forms?

In vector calculus, Helmholtz theorem says the divergence and curl of some vector field uniquely determines the vector field itself (with appropriate boundary conditions). Can this be generalized to ...
Aiden's user avatar
  • 123
1 vote
0 answers
97 views

Space of self-dual connections

Let $(M,g)$ be a 4-dimensional Riemannian manifold and $(P,M,\pi)$ a principal $G$-bundle over $M$. Denote by $ad (P)$ the adjoint bundle of $P$, that is, the vector bundle obtained by dividing $P\...
FUUNK1000's user avatar
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1 vote
1 answer
139 views

Definition of a Hodge structure

What is a pure Hodge structure of integer weight $n$? Wikipedia defines a pure Hodge structure of integer weight $n$ to be an abelian group $H_\mathbb{Z}$ equipped with a direct sum decomposition (as ...
tcamps's user avatar
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3 votes
1 answer
115 views

Correspondence between $H^1_{dR}$ and harmonic 1-forms

Let $S$ be a compact oriented Riemannian surface. And let $\mathcal{H}^1(S)$ be the space of the harmonic $1$-forms of $S$. I’m trying to prove that there is a linear isomorphism: $$H^1_{dR}(S)\to \...
Kandinskij's user avatar
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