# Tagged Questions

The study of relations between, one the one hand, the topology of a smooth manifold as encoded in the cohomology groups, and on the other, the set of solutions to the Laplace operator on differential forms relative to some Riemannian metric on the manifold.

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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$\...
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### Is the contraction of an harmonic form harmonic?

As I'm still a beginner in complex differential geometry, as soon as I tried reading an article I got stuck on what (I think) should be a minor detail. I hope someone can help me a bit. Let $M$ be a ...
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### Hodge theory on semi-Riemannian manifolds [reference request]

I need to learn a bit of Hodge theory on manifolds and I am looking for a reference which covers the case where the metric has arbitrary signature $(p,q)$. Most books I have found seem to focus on the ...
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### The cohomology of the Dirac operator $d+d^{*}$

Let $(M,g))$ be a Riemannian manifold with the Hodge dual operator $d^{*}$. Is there a name (and some computation in some reference) for the cohomology of the complex of Harmonic forms with ...
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### Where is the error in this proof of the Hodge theorem?

Let $(M,g)$ be a closed smooth Riemannian manifold. The following is the decomposition part of the Hodge theorem: Theorem The canonical map $\mathscr{H}^k(M)\to H^k(M)$ from harmonic $k$ ...
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### Constructing the Hodge Laplacian from the Laplace-Beltrami one

I know, in principle, how to construct the Hodge Laplacian with the aid of the Hodge star. Is it possible, though, to construct it only from the Laplace-Beltrami operator? I have already found a way, ...
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### Definition of type of a Hodge structure

In the context of mixed Hodge structures what does one mean by "type $(p,q)$"? Surprisingly I was not able to find a definition. In a particular case, what does it mean "$V$ is the maximal $\mathbb{Q}$...
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### Killing vector field for Witten complex?

I am reading a classical paper by Atiyah Bott "The moment map and equivariant cohomology". In paragraph "Relation with Witten complex" (at the very beginning of this paragraph) they claims that “$W_s$...
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### Hodge theory in general

I know a bit of Hodge theory, and I know that there is an analogue in the symplectic case, where instead of inducing the $\star$-product using the metric we use the symplectic form. Is in true in ...
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### The proof of the abc conjecture

I recently heard that there was a workshop on Inter-universal Teichmuller theory in the Clay institute from 7-11 December 2015.This field of mathematics offers a potential proof of the abc conjecture....
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### paper request about mixed hodge

Does any one have the paper?: the obstruction to splitting a mixed hodge structure over the integers I, Preprint, University of Utah, 1979, by James Carlson. I cannot find it on google.
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### Is there a description of variation of (mixed) Hodge Structures in terms of a Deligne operator?

A complex Mixed Hodge Structure is given by a complex vector space $V$ together with a descending filtration $W$ and two ascending filtrations $F,\bar{F}$ that satisfy the condition ...
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### A homological perspective on the Hodge-theorem

Let $M$ be a smooth oriented manifold of dimension $n$. Let $(\mathcal{A} ^*(M),d)$ be the chain complex of differential forms on $M$. Endowing $M$ with an inner product gives us a hodge star operator ...