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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.
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
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
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
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 …
- 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
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
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