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Results tagged with complex-geometry
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user 73622
Complex geometry is the study of complex manifolds, complex algebraic varieties, complex analytic spaces, and, by extension, of almost complex structures. It is a part of differential geometry, algebraic geometry and analytic geometry.
2
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1
answer
289
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Resolving complexes of coherent analytic sheaves
Background
Throughout, let $X$ be a smooth complex manifold.
It is a classical fact that a coherent analytic sheaf admits a local resolution by locally free sheaves (also known as a local syzygy). Gr …
- 1,131
10
votes
0
answers
234
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Čech representatives for Chern classes in holomorphic Deligne cohomology
Let $X$ be a complex-analytic manifold with "nice" (e.g. Stein) cover $\mathcal{U}=\{U_\alpha\}$, and $E$ a holomorphic vector bundle on $X$ defined by transition functions $\{g_{\alpha\beta}\}$.
For …
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2
votes
1
answer
75
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When is the pullback of a coherent analytic sheaf again coherent?
Let $f\colon X\to Y$ be a morphism of complex analytic spaces (though I'm very happy to restrict to complex manifolds).
Theorem (Grauert).
The pushforward $f_*\colon\mathcal{O}_X\text{-mod}\to\mathcal …
- 1,131
7
votes
1
answer
603
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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 subse …
- 1,131
3
votes
2
answers
126
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Vector bundles over a Stein space are projective
It is a "well known" fact that
locally free sheaves over a Stein space $X$ are projective as $\mathcal{O}_X$-modules
(see e.g. just after Lemma 1.6 in O'Brian-Toledo-Tong's "The trace map and charac …
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