All Questions
Tagged with complex-geometry complex-manifolds
220
questions
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66
views
Details of the proof of the inequality $ \int_{X}\left(2 r \mathrm{c}_{2}(E)-(r-1) \mathrm{c}_{1}^{2}(E)\right) \wedge \omega^{n-2} \geq 0.$
I'm trying to make sense of the following proof.
Let $E$ be a holomorphic vector bundle of rank $r$ on a compact hermitian manifold $(X, g)$. If $E$ admits an Hermite-Einstein structure then $$
\int_{...
- 103
0
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0
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126
views
Function of several complex variables with prescribed zeros
I accidentally stumbled upon a problem of complex analysis in several variables, and I have a hard time understanding what I read, it might be related to the Cousin II problem but I cannot say for ...
- 1,332
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votes
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33
views
Contraction of an inclusion with respect to Kobayshi hyperbolic metric
Suppose that $X = \mathbb{C}^n - \Delta_X$ and $Y = \mathbb{C}^n - \Delta_Y$, where $\Delta_X$ and $\Delta_Y$ are unions of hyperplanes in $\mathbb{C}^n$ such that $\Delta_Y \subset \Delta_X$, $\...
- 41
2
votes
1
answer
203
views
Prefactor $2\pi i$ for Tate-Hodge structure
A rather basic question. What was the original reason to consider the underlying $\mathbb{Z}$-module of the - as canonical object regarded - Tate-Hodge structure $\mathbb{Z}(1)$ to be given as $2 \pi ...
- 5,780
16
votes
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answers
502
views
Gabriel's theorem for complex analytic spaces
Let $X,Y$ be noetherian schemes over $\mathbb{C}$.
Then, it is known that
$$
\text{Coh}(X) \simeq \text{Coh}(Y) \Rightarrow X \simeq Y,
$$
by P. Gabriel(1962).
Are there some results in the case of ...
- 869
0
votes
0
answers
96
views
Principle of degeneration as precursor of Zariski's connectedness theorem (geometric intuition)
I have following question about so-called "principle of degeneration"
in algebraic geometry (which in modern terms is an immediate consequence
of Zariski's main theorem and goes in it's ...
- 5,780
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160
views
“Holomorphic” bump function
I was wondering in what sense can I construct a holomorphic “bump function”? Now, of course we cannot really construct a holomorphic bump function in the usual sense, but I have a much rougher idea in ...
- 371
3
votes
0
answers
160
views
Action of complex Lie group on Dolbeault cohomology
Let $M$ be a compact complex manifold acted holomorphically by a complex Lie group $G$. Let $F$ be a holomorphic $G$-equivariant vector bundle over $M$.
Consider the natural representation of $G$ in (...
- 21.3k
3
votes
0
answers
192
views
An almost complex structure on $\Bbb S^n$ induces a cross product on $\Bbb R^{n+1}$
It is known that the only spheres that admit an almost complex structures are $\Bbb S^2$ and $\Bbb S^6$ (Borel and Serre, 1953). In particular, $\Bbb S^4$ cannot be given an almost complex structure (...
- 1,097
4
votes
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answers
114
views
Non-Kähler complex structure on $S^2 \times T^4$
Consider $M = S^2 \times T^4$. Then we can construct a non-Kähler complex structure as follows. Let $L$ be a line bundle over $\mathbb{P}^1$ such that there are two sections $s_1, s_2 \in H^0 (L)$ ...
- 651
1
vote
1
answer
92
views
Common holomorphic forms for two distinct complex structures
Let $S$ be a closed real surface having two complex structures $c_1$ and $c_2$ which are not biholomorphic (so $S$ is a Riemann surface with genus at least 1). Consider $\omega$ a 1-form on $S$ which ...
- 331
2
votes
0
answers
132
views
Symmetric group-cocycle descends to symmetric product
Let $C$ be a complex curve with universal covering $\tilde{C}$ (which in my case is the upper half plane). Any group-cocylce $e \in H^1(\pi_1(C^n),H^0(\tilde{C}{}^n,\mathcal{O}^{\times}))$ defines a ...
- 53
0
votes
1
answer
267
views
Universal covering of symmetric product
Let $C$ be a 1-dimensional complex manifold whose universal covering is provided by the half-plane $\mathcal{H}=\{z \in \mathbb{C} \mid \operatorname{Im}z>0\}$. The symmetric product $C^{(n)} = C^n ...
- 53
1
vote
0
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148
views
Top cohomology of the canonical class of a compact non-Kähler manifold
Let $X$ be a complex compact manifold of complex dimension $n$. Let $K_X$ denote its canonical class. Is it true that the cohomology group
$$H^n(X,K_X)$$
is one dimensional?
Remark. If $X$ is Kähler ...
- 21.3k
7
votes
1
answer
302
views
When $H^{p,q}_{\bar\partial}(X)$ can be seen as a subspace of $H^k(X,\mathbb C)$?
It is known that for a $\partial\bar\partial$-manifold $X$ (a compact complex manifold satisfies the $\partial\bar\partial$-lemma), the Bott-Chern cohomology $H_{BC}^{\bullet,\bullet}:=\frac{\ker \...
- 449