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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.
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
20
votes
Accepted
Which almost complex manifolds admit a complex structure?
In complex dimension 3 or more it is still an open conjecture
(which was re-stated Yau a couple of years ago in his UCLA lectures).
There is not a single known example of an almost complex manifold
of …
- 9,237
19
votes
Accepted
Three-dimensional compact Kähler manifolds
The main obstruction to existence of Kahler metric (in addition to Lefschetz
SL(2)-action and Riemann-Hodge relations in cohomology)
is homotopy formality: the cohomology ring of a Kahler manifold is …
- 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 …
- 9,237
12
votes
Accepted
Is the deformation limit of Ricci-flat Kahler manifolds Kahler?
There are counterexamples: a Moishezon manifold, which has trivial canonical class and is birationally equivalent to a hyperkahler manifold,
is also deformationally equivalent to a hyperkaehler manif …
- 9,237
11
votes
Weitzenböck Identities
The most general version of Weitzenbock identities (with coefficients
in appropriate universal enveloping algebras) is due to Uwe Semmelmann and Gregor Weingart: http://arxiv.org/abs/math/0702031
"The …
- 9,237
10
votes
How restrictive is having zero Chern numbers for a compact complex manifold ? Same for negat...
When a compact Kahler manifold satisfies $c_1=0$, it admits a Ricci-flat Kahler metric by Calabi-Yau, hence its tangent bundle is polystable (direct sum of stable bundles of the same slope). Then its …
- 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 …
- 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
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
8
votes
Coincide between Chern-connection and Levi-Civita connection
It is easier to prove this result for 1-forms, instead of vector fields. On (1,0)-forms, $\nabla^{0,1}=\bar\partial$ because the Levi-Civita connection is torsion-free, hence $\bigwedge(\nabla(\eta))= …
- 9,237
8
votes
Accepted
Proper family deformation retracts onto special fiber
Here is the reference:
Persson, Ulf,
On degenerations of algebraic surfaces,
Mem. Amer. Math. Soc. 11 (1977), no. 189.
Clemens, C. H. Degeneration of Kähler manifolds. Duke Math. J. 44 (1977), no. …
- 9,237
7
votes
Accepted
Finiteness of De Rham cohomology of smooth quasi-projective varieties
I think there are proofs which are much easier.
For example, you can try to compute the cohomology using the Morse theory. For that you need existence of Morse functions having finitely many critical …
- 9,237
7
votes
Accepted
"Simple" Kahler manifolds
A generic deformation of a Hilbert scheme of K3 and a generic torus have no
subvarieties, hence they are "simple" in the above sense. For a torus it's
well known, for a Hilbert scheme of K3 it's in my …
- 9,237
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