All Questions
Tagged with complex-geometry smooth-manifolds
47
questions
0
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66
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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
2
votes
0
answers
233
views
Smooth compactification of complex varieties and uniqueness
Since I'm working in differential geometry, for the following I'm strictly interested in the smooth setting over $\mathbb{C}$ and its relation to the setting over $\mathbb{R}$.
Here are a few useful ...
- 1,755
3
votes
0
answers
192
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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
5
votes
1
answer
445
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Threefolds with the same Betti numbers and the same Chern numbers
By a threefold, I mean a compact complex manifold of dimension three.
My question is a simple one:
Are there known INFINITELY many non-homeomorphic threefolds that have the same Betti numbers and the ...
- 1,831
1
vote
0
answers
575
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Proof of Ehresmann's theorem
In Huybrechts' book Complex geometry: An introduction p.269, Proposition 6.2.2, the author gives a proof of the following theorem
(Ehresmann)
Let $\pi:\mathcal X\to B$ be a proper family of ...
- 449
-3
votes
1
answer
976
views
Pull back a vector field [closed]
In Voisin's book Hodge theory and complex algebraic geometry, I Section 9.1.2, p.223, the author writes:
Let $\phi:\mathcal X\to B$ be a family fo complex manifolds. The differential $\phi_*$ is a ...
- 449
11
votes
1
answer
575
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Examples of 6-manifolds without an almost complex structure
Question: I am searching for examples for closed (hence orientable ), smooth $6$-manifolds without an almost complex structure.
Finding such an example is equivelant to finding a manifold where the ...
- 6,975
1
vote
0
answers
82
views
Submersion function from a product space
Let $\Phi(x,y) \colon U_N \times U_M \to \mathbb{C}^n$ be a submersion, where $U_N \subset \mathbb{C}^N$ and $U_M \subset \mathbb{C}^M$.
Under which condition on $\Phi$ can I find some $s \in \...
10
votes
1
answer
639
views
Algebraic atlas on smooth manifolds
A real/complex rational atlas on a smooth closed manifold $M$ is an atlas with charts homeomorphic to Euclidean open sets in $\Bbb{R}^n$/$\Bbb{C}^n$ covering $M$ and real/complex rational transition ...
- 1,481
2
votes
0
answers
201
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Intuition behind Nakano positivity
I am learning about Nakano positivity of hermitian vector bundles, which is the strongest notion of positivity we can ask. I don't understand what is the geometric meaning of it. Let me briefly ...
- 1,237
7
votes
0
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96
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Uniqueness of Fano varieties
It is a theorem of Kollár–Miyaoka–Mori that there is a finite number of deformation families of smooth, complex Fano $n$-folds for each $n$ (hence also a finite number of diffeomorphism types).
My ...
- 6,975
0
votes
1
answer
197
views
Same fiber of induced covering map [closed]
Consider a holomorphic map $h: X \to E$ between compact, connected, complex analytic manifolds Let $p: \tilde{E}\to E$ be the universal cover, and denote by $\tilde{h}: \tilde{X}\to\tilde{E}$ the pull-...
- 19
3
votes
0
answers
417
views
Integration over a Surface without using Partition of Unity
Suppose we are given a compact Riemann surface $M$, an open cover $\mathscr{U}=\{U_1,U_2,\dots\}$ of $M$, charts $\{(U_1,\phi_1),(U_2,\phi_2),\dots\}$, holomorphic coordinates, $\phi_m:p\in U_m\mapsto ...
- 277
1
vote
1
answer
137
views
Equalizer of local analytic isomorphisms
Let $a,b : V\to W$ be two morphisms of smooth complex analytic spaces.
Assume $a$ and $b$ are local analytic isomorphisms.
Does the equalizer $U$ of $a,b$ exist as a smooth complex analytic ...
- 180
14
votes
1
answer
810
views
An almost complex structure on $S^2\times ...\times S^2 / \mathbb{Z_2}$
Consider the product of $2n$ two-spheres $X_n=(S^2)^{2n}$. This manifold admits an orientation preserving involution that preserves the product structure and acts as the (orientation reversing) ...
- 14.1k