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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.
12
votes
Accepted
Is the cup product of holomorphic $n$-forms with a fixed class injective?
The answer to this question is negative in dimensions $\ge 3$. For example, take a quintic in $\mathbb CP^4$ and consider its blow up $X$ in $10^{100}$ points (just to be safe). Then the space $H^1(X …
- 28.9k
11
votes
What do intermediate Jacobians do?
Recently I learned from a talk of Nick Addington one beautiful classical example where intermediate Jacobians contain all information about the variety. Namely, if we consider an intersection of two …
- 28.9k
11
votes
Two definitions of Calabi-Yau manifolds
You can prove that the canonical bundle is torsion without using Yau's theorem. This is contained the following work of Bogomolov, Theorem 3'
F. A. Bogomolov, “Kähler manifolds with trivial canonica …
- 28.9k
3
votes
Dolbeault cohomology of Hopf manifolds
Maybe the following article by Soenke Rollenske (and refferences their), could be usefull for you:
Some very non-Kähler manifolds: the Frölicher spectral sequence can be arbitrarily non degenerate
h …
- 28.9k
7
votes
Accepted
Is the complex moduli of Quintic Calabi-Yau toric?
The complex moduli space does not admit a toric strucutre, since the orbifold fundamental group of a toric orbifold must be abelian. Indeed, $\pi_1(\mathbb C^*)^n$ surjects on the orbifold fundamental …
- 28.9k
5
votes
Accepted
Smooth algebraic varieties with smooth Kahler quotients.
The following article http://arxiv.org/abs/1102.2762 of BENOÎT CLAUDON, ANDREAS HÖRING, AND JÁNOS KOLLÁR answers positively the question, provided $V/G$ is projective, $V$ is quasiprojective and $\pi_ …
- 28.9k
6
votes
Accepted
symplectic classes on rational surfaces.
This answer is rewritten and include more details
First of all I highly recommend you the article of Paul Biran From Symplectic Packing to Algebraic Geometry and Back available on the page http://ww …
- 28.9k
3
votes
"monotone" versus "symplectic Fano"
Edited. For the definitions that you mention "Simplectic Fano" can be non-montone. For example, you can take a $4$-dimensional Kahler non-agebraic torus that does not have complex curves at all. Such …
- 28.9k
4
votes
Algebraic Geometry versus Complex Geometry
1) I don't know if this qualifies, but it seems to me that there is no non-analytic proof of the statement that a surface of general type with $c_1^2=3c_2$ is a quotient of a unit two-dimensional comp …
7
votes
Holomorphic map from a neighborhood in $\mathbb C$ to S^3
Yes, such a map exists, and any such map has to be a map to a point in $\mathbb S^3$, because has to sends any holomorphic curve in the neighbourhood of $\mathbb C$ to a holomorphic curve in $S^3$, bu …
- 28.9k
8
votes
Accepted
Biholomorphism between neighborhood of a complex submanifold and a neighborhood of zero sect...
The existence of such a biholomorphism is rather rare. For example, as a simple exercise you can check that already for a conic in $\mathbb CP^2$ such a biholomorphism does not exist. At the same time …
- 28.9k
5
votes
Accepted
non-trivial locus of a holomorphic vector bundle
If your manifold is complex projective, then the answer is yes. Otherwise it is no. You can take a $K3$ surface without complex curves and just consider its tangent bundle. Of curse it will stay holom …
- 28.9k
1
vote
Accepted
Submersion to $ T^{2}$
Note first that $\Lambda\subset \mathbb C$ is a subgroup generated by a finite collection of numbers (periods) of the type $\alpha+i\beta$, where $\alpha,\, \beta\in \mathbb Q$. Such a subgroup is cl …
- 28.9k
1
vote
Concise expression for a specific holomorphic map $f:D\times D\longrightarrow \mathbb{B}^{2}$
Unfortunately, it doesn't look like the condition $\det(df)\subset(z_1z_2=0)$ is strong enough. Consider, for example, the map
$$f:(z_1,z_2)\to (z_1, z_2^3+z_1z_2).$$
Clearly, the differential of t …
- 28.9k
4
votes
Accepted
Non-constant holomorphic map onto a smooth curve
Let me show that such a map usually doesn't exist (even if we don't remove any additional curves). Consider the case when $\Gamma$ is a curve of degree $\ge 4$. Then one can slightly perturb $\Gamma$ …
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