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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.
3
votes
An example of a complex manifold without a finite open cover
Blow-up $\mathbb C^2$ simultaneously at all points of $\mathbb Z \times \{0\}$.
Part a) is evident for the blown-up manifold $X$ since it contains one-dimensional compact submanifolds. As for …
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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
5
votes
Accepted
Does isomorphisms of sheaf of holomorphic sections implies isomorphisms of two holomorphic v...
No. On $\mathbb P^1=\mathbb P^1(\mathbb C)$ we have $\Gamma(\mathbb P^1,\mathcal O_{\mathbb P^1}(-1))=\Gamma(\mathbb P^1,\mathcal O_{\mathbb P^1}(-2)=0$, but $O_{\mathbb P^1}(-1)$ and $O_{\mathbb P^1 …
- 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.
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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. …
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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 …
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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 …
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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
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 …
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2
votes
Complex analytic vs algebraic families of manifolds
Dear Gunnar, let $f:X\to Y$ be a proper flat morphism of smooth varieties.
This does not imply that $f$ has smooth fibres either in the algebraic or in the analytic case. For example, any non-constant …
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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 $f_x\in \mathcal O_{X,x}$ …
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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 …
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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 …
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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
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 …
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