All Questions
Tagged with complex-geometry sheaf-cohomology
11 questions
2
votes
0
answers
60
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Relative Dolbeault cohomology using currents
I need to compute the cohomology groups of some relative holomorphic $i$-forms $H^\bullet(X, \Omega^i_{X/Y})$ for a fibration of complex manifolds $X\to Y$, using a kind of distributional de Rham ...
- 2,054
7
votes
1
answer
607
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Converses to Cartan's Theorem B
Here is a phrasing of some Cartan Theorem B statements:
Consider the following conditions:
$X$ is a {Stein manifold, affine scheme, coherent analytic subvariety of $\mathbb{R}^n$, contractible ...
- 1,131
2
votes
1
answer
186
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Proving that $H^1(X,\mathcal{Hom}(\mathcal{G},\mathcal{E})) \cong \text{Ext}^1(\mathcal{G},\mathcal{E})$ holds for locally free sheaves
The following passage is from a thesis I'm reading:
Suppose we have a short exact sequence of vector bundles $$0 \to \mathcal{E} \to \mathcal{F}\to \mathcal{G} \to 0.$$ Since these sheaves are ...
- 73
4
votes
0
answers
101
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Serre vanishing on one-point blow-ups
This is basically the last step of problem 5.3.7 in Huybrechts' Complex Geometry.
Let $X$ be a complex manifold, $x \in X$, $E$ a holomorphic vector bundle on $X$ and $L$ a positive line bundle. ...
- 1,141
6
votes
1
answer
761
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The Yoneda pairing, hypercohomology, and cup product
Let $\mathcal{F}$ and $\mathcal{G}$ be coherent analytic sheaves on $\mathbb{P}^n$. Let $\mathcal{F}_\bullet$ be a locally free resolution of $\mathcal{F}$. In Principles of Algebraic Geometry by ...
- 294
2
votes
0
answers
251
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Is there a by-hand prove that $\Gamma(\mathbb{C}P^n,E)$ is finite dimensional for a holomorphic vector bundle $E$?
Please let me know whether this question is suitable for Mathoverflow.
Let $E$ be a finite holomorphic vector bundle (or more generally a coherent analytic sheaf) on a compact complex manifold $X$. ...
- 9,019
3
votes
0
answers
285
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Reference for the Koszul--Malgrange Theorem
The Koszul--Malgrange theorem, roughly, identifies holomorphic vectors bundles over a complex manifold, as those finitely generated projective modules admitting a flat $(0,1)$-connection. The ...
- 31
4
votes
0
answers
275
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Dolbeault cohomology of $\text{sl}(2,\mathbb{C})$
Consider the complex Lie group $G=\text{SL}(2,\mathbb{C})$ and let us denote $\Omega$ the sheaf of top holomorphic forms of this group. Are the cohomology spaces $H^{*}(G,\Omega)$ known ? I am ...
- 1,017
6
votes
0
answers
511
views
de Rham isomorphism with holomorphic forms
For a non-compact Riemann surface $X$ there is an isomorphism:
$$\Omega(X)/\mathrm d \mathcal O(X)\simeq H^1(X,\mathbb C)$$
where $\Omega$ is the sheaf of holomorphic forms on $X$. The group on the ...
- 81
4
votes
2
answers
594
views
H. Cartan's "Variétés analytiques complexes et cohomologie"?
Does anyone know where I might find an online version (for free or purchase, translated or in french) of this paper by Henri Cartan from 1953? I know it was published in Colloque sur les fonctions de ...
- 43
5
votes
0
answers
614
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Does the higher cohomology of a quasi-coherent sheaf on a Stein manifold vanish?
It is a well-known result in algebraic geometry that if $X$ is an affine scheme and $\mathcal{F}$ is a quasi-coherent sheaf on $X$, then the higher cohomologies of $\mathcal{F}$ vanish, i.e.
$$
H^i(X,\...
- 9,019