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