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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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 \...
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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 ...
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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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Cohomology of local systems and cohomology of vector bundles
Let $X$ be a compact Kähler manifold, $L$ be a local system of finite dimensional $\mathbb{C}$-vector spaces on $X$. Let $E=L\otimes_{\mathbb{C}}O_X$ be the induced holomorphic vector bundle on $X$. ...
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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 ...
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Kernel for the Hodge chain homotopy
Let $M$ be a Riemannian manifold of dimension $n$. For $p\in \mathbb Z$, let
$C^p=A^p(M;\mathbb R)$,
$d:C^p\to C^{p+1}$ be the differential,
$*:C^p\to C^{n-p}$ be the Hodge star, satisfying $**=(-1)^{...
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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 $M$...
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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 $X$...
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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 ...
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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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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 ...
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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 ...
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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 $g = \sum dx_i^2+dy_i^2$, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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\...
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fibres varying non-holomorphically
I'm currently reading Claire Voisin's book Hodge theory and complex manifolds. Here is an extract from it:
in the last paragraph she writes that we cannot chose the trivialisation $T$ to be ...
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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 ...
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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\...
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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 ...
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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 \...
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Definition of Hodge star operator for general pseudo-riemannian manifolds
Let $M$ be a smooth $n$-dimensional real manifold and denote by $\bigwedge^rT^\star M$ the exterior vector bundle of $r$-forms. Let us denote $i$ the interior product of $\bigwedge^rT^\star M$, that ...
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Holomorphic forms are closed on compact manifold $X$ if $\dim(X)=2$.
Let $X$ be a compact complex manifold and $\dim(X)=2$, $\eta$ is a holomorphic form on $X$. Prove that d$\eta=0$.
I know when $X$ is a compact complex Kähler manifold, holomorphic forms are closed.
In ...
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Bochner's formula and harmonic 1-forms on 2-manifold
For a harmonic 1-form $\omega$ (we'll denote its dual vector by $X$), we have from Bochner's formula: $\frac{1}{2}\triangle ||X||^2 = Ric(X , X) + g(X, \nabla div X) + ||\nabla X||^2 = Ric(X , X) + ||\...
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Dimension of the space of harmonic one-forms on compact manifold
It is shown by Hodge theory that the space of k-harmonic forms are isomorphic to the space of k-cohomology class, and for harmonic one-forms on compact manifold we also know that they must be parallel ...
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Complex conjugation of bundle-valued forms
Consider a holomorphic vector bundle $V$ over an $n$_dim complex manifold $M$. I am interested in the notion of an inner product between complex bundle-valued forms $\phi^a, \psi^b \in \Omega^{(p,q)}(...
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Is a finite covering of a $\partial\bar{\partial}$-manifold still $\partial\bar{\partial}$-manifold?
A compact complex manifold is called a $\partial\bar{\partial}$-manifold if for every pair $p,q\in \mathbb{N}$, every smooth $d$-closed $(p,q)$-form $\eta$ on $X$, $\eta$ is $d$-exact iff $\partial$-...
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Is the Hodge decomposition on a simplicial complex canonical?
Motivation
The question arises because in the paper arXiv:1005.2405 the authors pull back the Hodge decomposition on a simplicial complex from $C^1$ to $(C^0)^N$ for some integer $N$ via some ...
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Equivalent Definitions of Hodge Structure
I have read some materials on Hodge structures and all of them state the equivalence of definitions from the following perspectives: (suppose that we are considering a Hodge structure of weight $n\in \...
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Relations between Galois representations and complex geometry
We know that the Hodge structures of rational singular cohomology of complex projective manifolds are semisimple. The Galois representations attached to etale cohomology of smooth projective varieties ...
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Computation involving the Hodge star and exterior covariant derivative on an almost complex manifold
Let $M$ be an almost complex manifold of complex dimension $n$. Let $E$ be a complex vector bundle over $M$ with a Hermitian metric $h$. Let $A$ be a Hermitian connection with respect to $h$. Let $d_A$...
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On the construction of the de Rham period ring
I'm currently reading "CMI Summer School Notes on $p$-adic Hodge Theory"(https://math.stanford.edu/~conrad/papers/notes.pdf) p.51 and could not understand why there exists a canonical ...
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Polarization of Hodge structure
The Polarization of Hodge structure is defined as a bilinear form $S$ on rational Hodge cohomology $H_{\mathbb{Q}}$ of given n dimensional projective variety $X$, more explicitly, defined by the Hodge-...
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Functional analysis of forms on Riemannian manifolds
May I have some suggestions about advanced books on functional analysis on Riemannian manifolds, especially regarding forms? I am in particular interested in the theory of Hodge operators with ...
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The cohomology of the Jacobian and its subvariety
I read the following in a paper:
Let $G\subset J$ a subgroup of the Jacobian $J$ which is a countable union of Zariski closed subsets in the abelian variety $J$, so the irredundant decomposition of ...
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Multiplying the volume form by the codifferential of a one form yields an exact form
Let $(M, g)$ be an oriented Riemannian manifold with volume form $dv$. Let $*$ be the Hodge star operator defined by the relation $\omega \wedge *\tau = g(\omega, \tau) dv$ where $g$ is extended to ...
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Could a differential form be both exact and co-exact?
If my understanding is correct, on a closed manifold, the exact forms and co-exact forms are disjoint, but I'm not sure about the differential forms on manifolds with boundary. Could there be a ...
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Pull-back of harmonic form in the universal covering
Let $(X, \omega)$ be a compact complex manifold of dimension n. Let $\pi:Y\to X$ be the universal cover of $X$. Let $\phi$ be a $\delta_\omega$ - harmonic 2-form on $X$ such that $\pi*(\phi)$ is a d-...
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$d^+d^*\omega=0$ implies $dd^*\omega=0$ for a 2-form $\omega$ on a 4-manifold
Let $\omega$ be a smooth 2-form on a closed oriented riemannian 4-manifold $X$. How can we show that $d^+d^*\omega=0$ (i.e. $dd^*\omega$ is anti-self-dual) implies $dd^*\omega=0$?
This statement is ...
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Constructing a Hodge structure from a representation of a torus
I will follow Chapter 2 of this paper by Valloni. Let $V$ be a finitely generated free abelian group. An integral Hodge structure of weight $k$ on $V$ is a decomposition
$$
V \otimes_\mathbb{Z} \...
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Is there a $p$-adic analogue of the Hodge index theorem and Hodge--Riemann relations?
This question may be quite vague and weird, but let me try to make the motivation clear.
Let $X$ be a compact Kähler manifold of dimension $n$. The Hodge index theorem says that
The signature of the ...
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First chern class of a line bundle and curvature
Let $L\to X$ be a complex line bundle ($U(1)$-bundle) over a 4-manifold $X$, $A$ a connection on $L$, and $F_A\in \Omega^2(X;i\Bbb R)$ its curvature form. I am reading Morgan's book on
Seiberg-Witten ...
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Integral of Hodge Star of Differential Form over Submanifold of Complementary Dimension which Intersects Transversely
Let $([M]^n, g)$ be a closed, smooth, oriented, Riemannian manifold, and let $[S]^k$ and $[T]^{n-k}$ be closed, smooth, oriented submanifolds of $M$ which intersect transversely. Let $\omega$ be a ...
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