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Results tagged with complex-geometry
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user 943
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.
42
votes
1
answer
3k
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Complex vector bundles that are not holomorphic
Is there an example of a complex bundle on $\mathbb CP^n$ or on a Fano variety (defined over complex numbers), that does not admit a holomorphic structure? We require that the Chern classes of the bun …
- 28.9k
18
votes
0
answers
871
views
Almost complex 4-manifolds with a "holomorphic" vector field
Main question. What is the class of smooth orientable 4-dimensional manifolds that admit an almost complex structure $J$ and a vector field v, that preserves $J$?
The following sub question is rewri …
- 28.9k
17
votes
4
answers
2k
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Comparing fundamental groups of a complex orbifolds and their resolutions.
Let $X$ be a complex manifold with quotient singularities, and let $\tilde X$ be its resolution (that exists, for example, by Hironaka). Then I am pretty sure that $\pi_1(X)\cong \pi_1(\tilde X)$.
Q …
- 28.9k
6
votes
1
answer
438
views
Smooth algebraic varieties with smooth Kahler quotients.
Let $V$ be a smooth algebraic variety defined over complex numbers. Suppose that $G$ admits a free discrete action on $V$ so that $V/G$ is compact and Kahler (or algebraic). Is it ture that $G$ is vi …
- 28.9k
6
votes
2
answers
981
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Deformations of sheaves via automorphisms. How to express $Ext^1$?
Let $X$ be a complex manifold (for example $\mathbb CP^n$), let $v$ be a holomorphic vector field on $X$, and let $F$ be a coherent sheaf (for example a vector bundle or a structure sheaf of a point). …
- 28.9k
5
votes
2
answers
847
views
Topology of the preimage of a point for degree one holomorphic maps
Let $M^n$ and $N^n$ be two compact complex (or complex projective) manifolds. Let $f: M\to N$
be a holomorphic map of degree one. How to prove that for each $x\in N$ the set
$f^{-1}(x)$ is simply-con …
- 28.9k
4
votes
1
answer
261
views
Stein manifolds isomorphic at infinity
Suppose $M$ and $N$ are two Stein manifolds of dimension at least $3$ with compact subsets $U$ and $V$ such that $M\setminus U$ is biholomorphic to $N \setminus V$. It it true that $M$ is biholomorphi …
- 28.9k