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Results tagged with complex-geometry
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user 1450
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.
38
votes
Accepted
Diffeomorphic Kähler manifolds with different Hodge numbers
This question was debated in another forum a few years ago. The result was a note by Frédéric Campana in which he describes a counterexample as a corollary of another construction. In 1986 Gang Xiao …
- 19.6k
9
votes
Accepted
Small neighborhoods of singularities on varieties
There is a good paper of Goresky, "Triangulation of Stratified Objects", that I think reasonably quickly implies Milnor's result and its generalization to non-isolated singularities. The result is th …
- 56.6k
30
votes
Accepted
Dolbeault cohomology of Hopf manifolds
Even though this question has an accepted answer, the answers so far are not complete or explicit. I kept working on this question, because I have been curious for a long time about the structure of …
- 56.6k
15
votes
Accepted
Number Theory and Geometry/Several Complex Variables
I have heard algebraic number theory called "algebraic geometry in one dimension". (Or maybe you could call it arithmetic geometry in one dimension.) There is a natural emphasis in algebraic number …
- 56.6k
5
votes
Most important domains, extension theorems, and functions in several complex variables
The question seems like something a red herring, because these different types of domains aren't really all that different. A Stein manifold is a holomorphically convex manifold, which also has enoug …
- 56.6k
10
votes
Accepted
The 2-sphere and $\mathbb{CP}^1$
Any projective variety is also a real affine variety, by using the real and imaginary parts of the coordinates $x_{jk} = z_j\overline{z_k}$. You should first normalize the projective coordinates to h …
- 56.6k
14
votes
Accepted
Hodge Index Theorem for Gr(n,k)
The answer is a happy surprise for me: The Hodge index theorem for a Grassmannian matches a special case of John Stembridge's $q=-1$ phenomenon, that I also studied in an old paper.
First, some gene …
- 56.6k
58
votes
Accepted
The Relationship between Complex and Algebraic Geomety
The Wikipedia article is more technical than it should be, and for the reader in a hurry not all that well written. Here is a summary of the main points as best I understand them:
Complex manifolds …
- 56.6k