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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.
41
votes
Accepted
Which Kahler Manifolds are also Einstein Manifolds?
This question can be interpreted in two different ways.
1) Which Kahler manifolds admit a Kahler metric that is at the same time Einstein?
2) Which Kahler manifolds admit an Einstein metric?
If y …
- 28.9k
35
votes
Accepted
Complex vector bundles that are not holomorphic
Here is the answer to the question, kindly explained to me by Burt Totaro.
EDITED. This is an OPEN PROBLEM.
0) Apparently in the case of $\mathbb CP^n$ existence of a complex bundle without holomor …
- 28.9k
32
votes
Accepted
Is the wedge product of two harmonic forms harmonic?
It is easy to construct examples on Riemann surfaces of genus $>1$. Take any surface like this. Let $A$ and $B$ be two harmonic $1$-forms, that are not proportional. Then $A \wedge B$ is non-zero, but …
- 28.9k
20
votes
Accepted
Accumulation of algebraic subvarieties: Near one subvariety there are many others (?)
The answer is yes (as in the case of Sasha's answer we use ramified covers)
Proof. Let $X$ be any variety in $\mathbb CP^n$. Take a section $s_m$ of $O(m)$ on $X$ such that $s_m$ is not equal to $m$ …
- 28.9k
19
votes
Accepted
Unique almost complex structure up to diffeomorphism
You were lucky to find the only possible example. If you take any manifold of dimension $\ge 4$ you can pick an almost complex structure that is integrable in some closed ball and make it non-integrab …
- 28.9k
18
votes
Accepted
Can one bound the Todd class of a 3-dimensional variety polynomially in c_3
The case of complex manifolds of higher dimension is very different from the case of complex surfaces. So the answer to the question about complex $3$ folds is no, there exists a real 6-dimensional si …
- 28.9k
16
votes
Accepted
K3 surface of genus 8
To be able calculate the degree it is worth to read a bit of Griffiths-Harris about Grassmanians (chapter 1 section 5). To prove that $S$ is $K3$ one needs to caluclate the canonical bundle of $G$, us …
- 28.9k
15
votes
Accepted
Injective maps on cohomology and Kahler manifolds
In order to have this property it is sufficient to require that $\phi^{-1}(y)$ is a non-zero cycle in $H_*(X,\mathbb Q)$, where $y$ is a generic point in $Y$. This holds indeed when $X$ and $Y$ are Ka …
- 28.9k
14
votes
Accumulation of algebraic subvarieties: Near one subvariety there are many others (?), 2
One little counterexample (to both versions of the question). On a quintic $3$-fold in $\mathbb CP^4$ there are $2875$ lines. You can take any of these lines. An analytic neighbourhood of such a line …
- 28.9k
14
votes
Accepted
When can you reverse the orientation of a complex manifold and still get a complex manifold?
If you take an odd dimensional complex manifold $X$ with holomorphic structure $J$ then $-J$ defines on $X$ a holomorphic structure as well. And, of course, $J$ and $-J$ induce on $X$ opposite orienta …
- 28.9k
14
votes
Accepted
Kähler metrics for projective space that are not the Fubini-Study metric
Every complex manifold that admits one Kahler metric $w$ admits a lot of them, indeed $w+i\partial \bar \partial f$ is a Kahler metric if the second derivatives of $f$ are not too large. This is why, …
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
12
votes
Is there any holomorphic version of the tubular neighborhood theorem?
To start with something positive, it is indeed true that whenever you have a $\mathbb CP^1$ with negative self-intersection in a complex surface it has a standard holomorphic neighbourhood.
On the o …
- 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 …
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