All Questions
Tagged with hodge-theory manifolds
10 questions
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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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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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Why is the Hodge Decomposition profound?
The Hodge Decomposition is the following result:
Let $X$ be a smooth projective variety over $\mathbf{C}$. For every integer $k \geq 0$, we have a direct sum decomposition $$H^k(X, \mathbf{C}) \cong \...
- 2,510
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What is an example of harmonic form on a compact manifold?
I'm doing some self-study on Hodge theory and elliptic operators right now. I'm trying to come up with an example of a harmonic $p$-form $\omega$ on a compact manifold, i.e. a form such that $d\omega =...
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Laplacian on 1-forms on $(\mathbb R^m,\delta)$
I'm trying to show that if $\omega$ is a 1-form on $(\mathbb R^m,\delta)$, the action of the Laplacian is given by
$$\Delta\omega=-\sum_{\mu=1}^m\frac{\partial^2\omega_\nu}{\partial x^\mu\partial x^\...
4
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Let $M$ is compact Riemann surface, if $\omega$ is a 2-form and $\int_{M} \omega =0$ then there exists a smooth function $f$ such that $\omega=d*df$
I want to show that:
$(*)$If $\omega \in \Omega^{2}(M)$, which $M$ is compact Riemann surface and $\Omega^{2}(M)$ means 2-form, and $\int_{M} \omega =0$, then there exists a smooth function $f$(i.e....
user469065
9
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1
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What is the meaning and importance of the Hodge codiferential?
In differential geometry given a smooth manifold $M$ we can define the exterior derivative $d$ acting on $k$ forms giving back $k+1$ forms. It is a map $d : \Omega^k(M)\to \Omega^{k+1}(M)$ which is in ...
- 27.2k
7
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Calabi-Yau $3$-fold given as elliptically fibered manifold over $\mathbb{C}P^1 \times \mathbb{C}P^1$
Consider a Calabi-Yau three-fold given as an elliptically fibered manifold over $\mathbb{C}P^1 \times \mathbb{C}P^1$$$y^2 = x^3 + f(z_1, z_2)x + g(z_1, z_2),$$where $z_1$, $z_2$ represent the two $\...
user356687
9
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0
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Could you explain the failure of the Hodge decomposition to exist for non-compact manifolds?
I'm a physicist and the mathematics around the Hodge Decomposition is way formal than I can currently follow (I'm trying to better myself but it'll take a while).
Specifically what I'm (pragmatically)...
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Hodge decomposition on a manifold with a nontrivial connection
I am familiar with the notion of Hodge decomposition of an arbitrary differential form into an exact form, a co-exact form, and a harmonic form. Given a curved space with a connection, could you ...
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