# 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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### 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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### 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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1 vote
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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 ...
1 vote
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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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### 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 ...
1 vote
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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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### 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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### 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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1 vote
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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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### 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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