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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.

Hodge theory is ehe study of relations between the topology of a smooth manifold as encoded in the cohomology groups and solutions to the Laplace operator on differential forms relative to some Riemannian metric on the manifold.

348 questions
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Counterexample to decomposition of Harmonic Forms

It is well known that if $X$ is a Kahler manifold, then $$\bigoplus_{p+q=k}\mathcal{H}^{p,q}=\mathcal{H}^{k}(X,g)_{\mathbb{C}}=\mathcal{H}^{k}_{\overline{\partial}}(X,g)=\mathcal{H}_{\partial}^{k}(X,g)$$ If one loosens the restriction that $X$ is…
pleasantpheasant
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How do I compute the harmonic component of a given differential form?

The Hodge decomposition theorem tells us that any $r$-form $\omega$ on a Riemannian manifold $M$ (without boundary and compact) may be uniquely decomposed as $$ \omega = d \gamma_1 + d^\dagger \gamma_2 + \gamma_3 $$ where…
arovai
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The codifferential δ of a k -form on an n -dimensional Riemannian manifold

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 $$ d^*: \Omega^k(M) \to \Omega^{k-1}(M),$$ defined by $ d^* = (-1)^{n(k+1)+1} * d…
Member1434
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What is $H^{p,p}(X,\mathbb{Q})$?

Edited for presentation. How to define it with words or/and quantifiers? Been struggling with a too big number of informations now, for hours. $p\in \mathbb{N}$. X is a $C^{\infty}$-manifold.
someone
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Hodge star/ Technical question

If we have an equation that looks like $$H=Y$$ and we want to multiply $H$ by either $ReM_{IJ}$ or $ImM_{IJ}$ where $M_{IJ}$ is a complex matrix. But the thing is that $$Y=\star(...)$$ where $\star$ is hodge star and (...) is set of complex…
PhilosophicalPhysics
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