# 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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### Toroidal/polodial decomposition of a solenoid vector field in cartesian coordinates

I am trying to understand some aspects of the toroidal/polodial decomposition of a solenoid vector field in cartesian coordinates. We have a divergence-free ('solenoid') vector field $u$ on a compact ...
1 vote
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### Characterization of harmonic $(1,1)$-forms

Let $(X,\Omega)$ be a compact Kähler manifold. Then there is the "usual" definition of the vector space $\mathcal H^{p,q}_{\bar\partial}$ which is the space of $\bar\partial$-harmonic $(p,q)$...
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### Equivalent condition for 3-form being primitive in 6 dimensions

Let $(V, \omega)$ be a symplectic vector space of dimension 6, with a compatible metric $g$. Let $\varphi$ be a 3-form on $V$. Then $\phi$ is primitive, meaning $\star \varphi \wedge \omega = 0$, if ...
1 vote
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### Characterizing strict morphisms in the category of bifiltered vector spaces

Let $k$ be a field, and let $C$ be the category whose objects are finite dimensional $k$-vector spaces endowed with two finite filtrations $W$ and $F$, the former being ascending and the latter ...
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### What does the $0$ in "Hodge number is the dimension of $H^{0}(V,\Omega^{n})$" mean please?

Here is the page I was reading.. I tried reading more into what Hodge numbers are and it was too complex for the short amount of time I had but I figured the $0$ probably indicated something like a &...
1 vote
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### A coordinate-free criterion for ellipticity of a linear differential operator

In Chapter 6 of Warner's Foundations of Differentiable Manifolds and Lie Groups, he develops a self-contained theory of local elliptic operators to establish the Hodge theorem. I got a bit stuck on a ...
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### The definition of special cubic fourfolds

According to the basic paper "Special Cubic Fourfolds" (https://www.math.brown.edu/bhassett/papers/cubics/cubiclong.pdf, [BH98]) by Brendan Hasset, a special cubic fourfold is defined as ...
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Let $H_{\mathbb{Z}}$ be an integral Hodge structure of weight $n$ with Hodge decomposition $H_{\mathbb{C}} = \sum_{p+q=n}H^{p, q}$. In the definition of a polarized Hodge structure, I have come across ...
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### Hodge star operator commutes with the harmonic projection

the basic setup is a dimension 4 compact oriented Riemannian manifold, consider we have a closed 2-form $a$ on it with the cohomology class $[a]$. Suppose now we have a restriction on the self-dual ...
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### Nowhere vanishing harmonic 1-forms on 3-manifolds

Consider $(S^1 \times \Sigma^2, g)$, where $g$ is any Riemannian metric on the compact and closed $3$-manifold $S^1 \times \Sigma^2$. Question: Does there always exist a nowhere vanishing harmonic $1$...
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### Signature is independent of the Riemannian metric chosen

The signature of a 4-manifold is defined to be the dimension of harmonic self-dual two forms minus the dimension of harmonic anti-self-dual two forms, where the self or anti-self-dual is defined in ...
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### estimation use Hodge theory

I'm confused by how we get the following estimation from Hodge theory It seems to me that it is using the fundamental inequality of elliptic operators, but I could not see what this operator is.
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### Why Hodge decomposition reflect the analytic structure

In the book Principles of Algebraic Geometry by Griffiths and Harris, the authors state that the Hodge decomposition reflects the analytic structure but that the Lefschetz decomposition is essentially ...
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### Counter-example for complex version of Hodge conjecture.

Hodge conjecture claims that every rational Hodge class is in the $\mathbb{Q}$-span of the image of the algebraic cycles in the cohomology. I believe the complexifed version of the conjecture should ...
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### Determinant and trace of Hodge Star operators

The Hodge star operator in differential geometry is a map from $*: \Omega^{k}(\mathcal{M}) \to \Omega^{m-k}(\mathcal{M})$ where $\Omega^{l}(\mathcal{M})$ is the space of $l$-forms on $m$-dimensional ...
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### Condition $\overline{V^{p,q}}= V^{q,p}$ in the definition of a Hodge structure
Suppose that $V$ is a finitely generated $\mathbb Z$-module. A Hodge structure of wight $k$ on $V$ is a decomposition of the complexification of $V$ into complex vector spaces $V^{p,q}$ such that \$\...