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Results tagged with complex-geometry
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user 3377
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.
7
votes
Accepted
Existence of closed manifolds with more than 3 linearly independent complex structures?
A manifold admitting a triple of complex structures satisfying quaternionic
relations also admits a torsion-free connection (called "Obata connection") preserving the quaternionic structure. Such a co …
- 9,237
8
votes
Accepted
Does it make sense to define a holomorphic structure on $\mathbb{C}P^\infty$ and vector bund...
Yes, there is lots of literature on this subject.
However, Tyurin proved that all vector bundles on $CP^\infty$ are
direct sum of line bundles. There are several more recent papers by
Penkov and Tikho …
- 9,237
4
votes
Non Kähler blow-up of a Kähler manifold
When $Y$ is compact, the blow-up is always Kahler;
see e.g. Lemma 3.4 in this paper
(this is a generally known folklore theorem which we
had to use, and hence written down).
For $Y$ non-compact the …
- 9,237
6
votes
Non projective hyperbolic compact complex space
The non-projective Kahler surfaces are either K3, tori or elliptic surfaces, known to be non-hyperbolic. Non-Kahler surfaces are either elliptic or class VII; the elliptic surfaces are obviously non-h …
- 9,237
3
votes
Accepted
Nonpositive curvature of Stein manifolds
Complete, simply connected manifolds of non-positive sectional curvature are diffeomorphic to $R^n$, by Cartan-Hadamard theorem. Conversely, if it's $R^n$, you can put the metric of negative curvature …
- 9,237
2
votes
Accepted
Intuition for holomorphic bisectional curvature
Ngaiming Mok, The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature, J. Differential Geom. Volume 27, Number 2 (1988), 179-214.
Mok improves Mori's …
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8
votes
Accepted
What is the moduli of an algebraic torus
There is just no definition of the moduli for complex structures on non-compact manifolds, but by any reasonable definition, it would be (generally) very bad space, certainly infinite-dimensional. For …
- 9,237
4
votes
Accepted
Kähler Identities: from the untwisted to the twisted case
The most clear proof (through differential operators, graded Jacobi identities and superalgebra) actually works in both twisted and untwisted cases, no need to derive one from another. See http://arxi …
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1
vote
Kahler structure on holomorphic principal bundles
A connected compact complex Lie group is a torus, hence the question is apparently about principal torus bundles. Of course, product of a torus and a compact Kahler manifold is Kahler, giving a trivia …
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25
votes
Accepted
Two definitions of Calabi-Yau manifolds
I have looked for a while for a proof
which does not use the Calabi-Yau theorem
and nobody seems to know it.
Also, there are plenty of non-Kaehler
manifolds with canonical bundle trivial
topologicall …
- 9,237
12
votes
Accepted
Deformations of Kähler manifolds where Hodge decomposition fails?
This is known, for projective (even Moishezon)
manifolds as shown by Dan Popovici in his
paper http://arxiv.org/abs/1003.3605
For general Kaehler manifold, this is conjectured.
Popovici has proved t …
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6
votes
Question about Hodge number
For a Kaehler surface, the Hodge numbers are topological invariants.
By Hodge Index Theorem, the signature of the Poincare pairing is equal
to 2h^{2,0} +2 - h^{1,1}, hence h^{2,0} is a topological inv …
- 9,237
0
votes
Examples of symplectic non-Kahler classes.
This is probably the simplest example.
Take the Fubini-Study form $\omega$ on $CP^2$.
Then $-\omega$ is symplectic, but never Kaehler,
because by Yau's theorem $CP^2$ admits a
unique (standard) comp …
- 9,237
3
votes
Accepted
Global Definition of the Almost Complex Structure of a Complex Manifold
the construction of the canonical $J$ for a complex manifold is what I'm interested in
Given a complex manifold, you have a bundle of (1,0)-forms within complexified 1-forms
which is generated (o …
- 9,237
9
votes
Accepted
Different occurences of the word 'period' in algebraic geometry
The second and the third are pretty much equivalent.
Indeed, "the period" in XIX century sense is essentially
the same as the discrepancy between the branches of a
multi-valued function, obtained as a …
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