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Results tagged with complex-geometry
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user 450
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.
57
votes
flatness in complex analytic geometry
Instead of trying to say what flatness in analytic geometry means I'll give you some street-fighting tricks for recognizing whether a morphism of analytic spaces ( not necessarily reduced) $f:X\to Y $ …
- 47.5k
43
votes
Simplest examples of nonisomorphic complex algebraic varieties with isomorphic analytifications
Dear Ravi,
maybe the simplest example is one by Serre: the holomorphic Stein surface $\mathbb C^\ast\times \mathbb C^\ast $ underlies two non-isomorphic smooth complex algebraic varieties.
1) $\math …
- 47.5k
35
votes
Accepted
A geometric characterization for arithmetic genus
First let me note that there is an unfortunate clash in terminology: the arithmetic genus of a smooth complex projective variety $X$ of dimension $n$ can mean either
a) The number $\chi (X, \mathca …
- 47.5k
17
votes
Accepted
Uniformization theorem in higher dimensions
There exist infinitely many holomorphically non-isomorphic complex structures on the unit ball of R^4 (or more generally R^2n): this is a beautiful theorem of Burns, Shnider, Wells ( Inventiones Math. …
- 47.5k
16
votes
3
answers
3k
views
When is a holomorphic submersion with isomorphic fibers locally trivial?
A justly celebrated theorem by Ehresmann states that a proper smooth submersion $\pi: X\to S$ between smooth manifolds is locally trivial in the sense that every point $s\in S$ downstairs has a neigh …
- 47.5k
15
votes
Reference request: moduli spaces of vector bundles
Dear Mohammad, there is a rather elementary book Introduction to Moduli Problems and Orbit spaces by P.E. Newstead which will explain to you why stability is important, give you lots of examples (Ch …
- 47.5k
12
votes
Accepted
Most important domains, extension theorems, and functions in several complex variables
Here are a few points to guide you into the beautiful subject you had the good taste to choose.
1) Hartogs extension phenomenon :given two concentric balls in $ \mathbb C^n$, any holomorphic funct …
- 47.5k
11
votes
Accepted
Are spaces of holomorphic maps manifolds?
Let $X,Y$ be analytic spaces with $X$ compact reduced and $Y$ arbitrary (i.e. maybe with nilpotents in its structure sheaf). Then Douady showed in his thesis
that the set of holomorphic maps $Hol(X,Y) …
- 47.5k
11
votes
Accepted
Explicit way to construct simple complex tori/abelian varieties of dimension at least 2
Recall that the Néron-Severi group of a complex manifold $X$ is the subgroup of $NS(X)\subset H^2(X, \mathbb Z) $ consisting of first Chern classes of holomorphic line bundles on $X$.
More algebraical …
- 47.5k
10
votes
Connected complement manifold
Claim: The complement $U=\mathbb C^h\setminus \{F=0\}$ is path-connected and thus connected.
Proof:
Given $a,b\in U$ consider the affine complex line $L_{a,b}=L$ joining $a$ to $b$.
The polynomial $ …
- 47.5k
7
votes
Complex geometry text/research introduction for the analyst
1) There is a great book From Holomorphic Functions to Complex Manifolds by Fritzsche-Grauert.
It is very geometric and gives you the fundamentals on complex manifolds, including specialized topics, f …
- 47.5k
7
votes
Biholomorphism between neighborhood of a complex submanifold and a neighborhood of zero sect...
The answer is "no" because in general there does not even exist a neighbourhood $U$ of your submanifold that holomorphically retracts to the submanifold. In fact Finnur Lárusson has proved the followi …
- 47.5k
6
votes
Kähler metric on projectivised bundle
If $M$ is compact ( the usual assumption in Kähler manifold theory) the answer is "yes". You can look it up in Claire Voisin's book Proposition 3.18, page 78.
- 47.5k
5
votes
Are there compact analogues of Cartan's theorems A and B?
Dear Colin , for $X$ a holomorphic connected manifold, denote by $\mathcal M (X)$ its field of meromorphic functions.
A) It is not true that a germ of holomorphic function $f_x\in \mathcal O_{X,x}$ …
- 47.5k
5
votes
Accepted
Density of holomorphic sections
Dear Benjamin, the statement that holomorphic sections are dense in the smooth sections is false, already for the trivial bundle of rank one $E_1=X\times \mathbb C$ over $X=\mathbb C$. Indeed on any …
- 47.5k