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.
365 questions
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Apparent contradiction in the definition of harmonic differential forms
Setting
Consider a closed Riemannian manifold $(M,g)$, and the space of differential $k-$forms $\Omega^k(M)$ equipped with the exterior derivative $d$. The Riemannian metric allows also for the ...
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Question about representations of $s l(2,\mathbb{C})$
I'm studying Griffiths and Harris Principles of Algebraic Geometry. In particular I'm studying the Lefschetz Decomposition of a Kahler Manifold (pag 118-126).
Before talking about Kahler manifolds, ...
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Hodge-Riemann form is non-degenerate
I'm studying Griffiths and Harris Principles of Algebraic Geometry. In particular I'm studying the Lefschetz Decomposition of a Kahler Manifold and its applications (pages 118-126).
After the proof of ...
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Period Mapping in Wells' "Differential Analysis on Complex Manifolds"
In Wells' "Differential Analysis on Complex Manifolds", he defines a period mapping in p. 209, but frankly, I'm unsure what he is trying to do. Either I somehow fail to understand it, or the ...
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Hodge structure and Complex Structure of Riemann surfaces
Let $\Sigma$ be a closed oriented $2$-dimensional topological manifold of genus $g$. We know $H^1(\Sigma,\mathbb{Z}) \simeq H_1(\Sigma,\mathbb{Z}) \simeq \mathbb{Z}^{2g}$. Fix a set of $2g$ 1-cycles
$$...
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proper surjection of compact Kahlers implies injection on rational Hodge structures
Suppose $X$ and $Y$ are compact Kahler manifolds, and let $f: X \to Y$ be a proper surjective morphism. Why is
$$
f^*: H^i(Y; \mathbb{Q}) \to H^i(X; \mathbb{Q})
$$
an injection on Hodge structures?
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What is the precise definition of "defined over $\mathbb{Q}$" in the definition of a Hodge structure?
In Definition 1 of this note https://webspace.science.uu.nl/~looij101/trentorendicontifinal.pdf : "a Hodge structure on V is a decomposition $\mathrm{V}=\oplus_{\mathfrak{p}, \mathrm{q}} \mathrm{...
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Are mixed hodge structures sheaves on some stack?
For p-adic motives, it seems one should view F-crystals on $S$ as an approximation of motives over $S$. F-crystals on $S$ are sheaves on $S^{\Delta}$
For characteristic zero, mixed hodge are our ...
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Property of the Hodge-Star operator
I am following the book "Riemannian Geometry and Geometric Analysis" by Jost, and at some point there is written the following property of the Hodge-Star operator without proof. Let $V$ be a ...
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PDE with Neumann conditions on compact Kahler manifold with boundary
I am searching for a reference on the PDE system
\begin{align*}
\partial \omega & = \alpha \\
\overline{\partial} \omega & = \beta
\end{align*}
with Neumann boundary conditions on a compact ...
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Relation between wedge products of Kähler forms and Ricci forms and the volume form
Given a Kähler manifold $(M,g)$ with Kähler form $\omega$ with $\textrm{dim}_{\mathbb{C}}M = n$, the volume form is given as $\frac{\omega^n}{n!}$.
As any (n,n)-form on $M$ will be a multiple of the ...
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1
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Proving that $R^if_*\mathbb{C} \otimes_{\mathbb{C}} \mathcal{O}_S \simeq R^if_*(f^{-1}\mathcal{O}_S)$
I'm reading "Introduction to Hodge Theory", by Bertin, Demailly, Illusie and Peters and I'm stuck trying to understad this identification.
Let $f: X \rightarrow S$ be a smooth and proper map ...
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How to prove that $\Pi_{\substack{1\leq i\leq m}\\{m+1\leq j\leq n}}\dfrac{1}{i-j}=\Pi_{\substack{1\leq i\leq m}\\{m+1\leq j\leq n}}\dfrac{1}{j-i-n}$?
While attempting to prove that the Hodge *-operator an an oriented real inner product vector space is involutive up to sign, I came across the following problem which I needed to complete the proof. ...
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Relationship between the spectra of the zero and first Hodge Laplacian on 2 dimensional manifolds.
Considering an oriented and compact surface embedded in $\mathbb{R}^3$. I would like to know if there exisits a particular relationship between the spectrum of the Laplace-Beltrami operator $\Delta_0 =...
1
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1
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For a $q$-form $\varphi$, $\nabla\varphi=0$ if and only if $\nabla(*\varphi)=0$, where $*$ is the Hodge star operator.
I know the formula that connects differentiation and covariant differentiation is
$$d \varphi\left(X_{1}, \cdots, X_{q+1}\right)=\sum_{i=1}^{q+1}(-1)^{i-1}\left(\nabla_{X_{i}} \varphi\right)\left(X_{1}...
- 3,095
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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:
$$
H^k(X, \mathbb{C}) \cong \...
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Is $\delta \Delta^{-1} d$ the identity operator?
Let $d, \delta, \Delta = (d\delta + \delta d)$ be the exterior derivative, codifferential and Laplace-de-Rham operator.
Let $\omega$ be a closed $k$-form, one can then say $\Delta \omega = d \delta \...
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Definition of Polarised Hodge Structure
I'm reading section 7.1.2 of Voison's book.
For a compact Kahler manifold with a Kahler form $\omega$, we have intersection pairing
$$Q(\alpha,\beta) = \int_X\omega^{n-k}\wedge \alpha\wedge \beta$$
...
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0
answers
47
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Hodge decomposition on a riemannian manifold for another measure?
Let $M$ be a closed riemannian manifold, we denote $dx$ the canonic volume measure on $M$. If we take $\mu$ a probability measure such that $d\mu = \rho dx$ with $\rho > 0$ we can define $d_\mu^*$ ...
2
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abelian differentials are $L^2$ integrable
Let $C$ be a smooth algebraic curve over $\mathbb{C}$ (we can think for instance of a compact Riemann surface). Let $A$ be the subset of classes of holomorphic forms $\omega$ in $H^1(C,\mathbb{C})$ ...
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What on earth is a topological cavity in a certain simplicial complex? [closed]
I am dealing with basic algebraic topology for my research, and have been confused about topological cavities for a long time. Until now, I have already understood the concepts about k-dimensional ...
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2
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Prove that $d_3d_3^*+d_3^*d_3=-\nabla^2$
Consider the geometry in $\mathbb R^3$, define
$$d_3=dx\frac{\partial}{\partial x}+dy\frac{\partial}{\partial y}+dz\frac{\partial}{\partial z}.$$
We then define the Hodge star operator $*_3:\Omega^p(\...
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1
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Morphism of varieties is continuous between analytic varieties.
I'm seeking clarification on the significance of commutative diagrams in understanding the analytic topology of smooth varieties.
In Aleksander Horawa's notes, a commutative diagram is used to ...
- 155
4
votes
0
answers
123
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Hodge decomposition on vector bundle-valued differential forms
Let $M$ be a compact Reimannian manifold and let $(E,h)$ be a Hermitian vector bundle over $M$. Let $A^k(M,E)$ denote the space of $E$-valued $k$-forms, i.e. smooth section of the bundle $\bigwedge^kT^...
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The codifferential δ of a k -form on an n -dimensional Riemannian manifold [closed]
The Hodge dual (or formal adjoint) to the exterior derivative $d: \Omega^k(M) \to \Omega^{k+1}(M)$ on a smooth manifold $M$ is the codifferential $ d^* $, a linear map $$ d^*: \Omega^k(M) \to \Omega^{...
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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 ...
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Counterexample to decomposition of Harmonic Forms
It is well known that if $X$ is a Kahler manifold, then $$\bigoplus_{p+q=k}\mathcal{H}^{p,q}=\mathcal{H}^{k}(X,g)_{\mathbb{C}}=\mathcal{H}^{k}_{\overline{\partial}}(X,g)=\mathcal{H}_{\partial}^{k}(X,g)...
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Terminology for Complex Algebraic Geometry with Complex Conjugation
Semialgebraic geometry is essentially real algebraic geometry but with the defining polynomial relations allowed to be inequalities rather than just equalities.
This doesn't make sense over $\mathbb{C}...
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A nice description for $t^{k-1}B_{cr}^+/t^kB_{cr}^+$?
When $t$ is a uniformizer of the integral de Rham period ring, $B_{dR}^+$, there is an isomorphism $t^{k-1}B_{dR}^+/t^{k}B_{dR}^+\cong \mathbb{C}_p(k-1)$. Is there a nice description for $t^{k-1}B_{cr}...
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Cohomology group induced by $d^*$
It is well known that for a smooth manifold $M$, the de Rham cohomology group is defined by $$H_{dR}^k(M):=\frac{A^k(M)\cap \ker d}{A^k(M)\cap \text{im }d}.$$
Similarly, if we assume that $M$ being a ...
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1
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The complexification of Hodge class is properly contained in the space of type $V^{k,k}$
I was reading Professor Huybrechts's Lectures on K3 surfaces, there is a statement about Hodge class that I can't figure out.
Let's consider the (integral or rational) Hodge structure $V$ with the ...
- 5,190
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Hodge structure on $\operatorname{Sym}(\mathbb{R}[1])$
I am at best superficially acquainted with the intricacies of Hodge theory. The following question comes from my study of the paper On the $\Gamma$-factors attached to motives by Christopher Deninger. ...
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Cohomology basics: relation between de Rham duality and Hodge duality.
Consider a product manifold $X=P \times P_{\perp}$, with $P$ and $P_{\perp}$ cycles of $X$, and let $\omega$ be the associated differential form of $P$ in the sense of de Rham duality (I suppose the ...
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How is the Hodge star operator defined for vector-valued forms?
Let $M$ be an oriented Riemannian manifold of dimension $n$. For any $\omega \in \Omega^k(M)$, we define the Hodge star operator $\star$ of a $\omega$ as the unique $n-k$ form $\star\omega$ that ...
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Interpretation of eigenvalues and eigenvectors of combinatorial Hodge Laplacian in algebraic topology
Let $\Sigma$ be an abstract simplicial complex. Do the eigenvectors and eigenvalues of the combinatorial Hodge Laplacian $\Delta_k$,
$$\Delta_k^\Sigma = (\partial_k^\Sigma)^\dagger \partial_k^\Sigma + ...
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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 ...
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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 ...
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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 ...
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1
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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 ...
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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 \...
- 8,078
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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, ...
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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, \...
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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 ...
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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 ...
- 3,168
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50
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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 ...
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111
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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 ...
- 2,557
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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 $M$...
- 535
1
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1
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251
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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 $X$...
- 1,643
3
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1
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346
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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 ...
- 30.3k
2
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0
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150
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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_{...
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