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Results tagged with complex-geometry
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user 9871
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.
9
votes
1
answer
2k
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Deform a compact Kähler manifold to a non Kähler one
Could you give me an example of a compact Kähler manifold which analytically deforms to a non Kähler one?
For example, there is no hope to find a complex structure on a Hopf manifold in order to make …
- 7,902
4
votes
5
answers
941
views
Uniformity of ampleness
Let $X$ be a smooth projective variety over $\mathbb C$ and $A\to X$ an ample line bundle.
Is there an integer $k_0$ such that for all line bundle $L\to X$, the tensor product $A^{\otimes k}\otimes …
- 7,902
10
votes
1
answer
373
views
Non projective hyperbolic compact complex space
A famous conjecture by Kobayashi (perhaps slightly revisited subsequently) states that every compact hyperbolic Kähler manifold $X$ has ample canonical bundle.
This implies in particular that $X$ is …
- 7,902
13
votes
1
answer
993
views
Examples of Brody hyperbolic affine varieties which are not Kobayashi hyperbolic
Let $X$ be a complex space.
We say that $X$ is Brody hyperbolic if there is no non-constant holomorphic map $f\colon\mathbb C\to X$.
We say that $X$ is Kobayashi hyperbolic if the Kobayashi pseudo …
- 7,902
12
votes
2
answers
2k
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Negative holomorphic sectional curvature
Let X be a complex hermitian manifold with hermitian form $\omega$. How can you prove that if $\omega$ has negative holomorphic sectional curvature, then its scalar curvature is negative, too?
- 7,902
11
votes
0
answers
200
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Holomorphically convex manifolds and Bergman complete manifolds
Suppose $X$ is a complex manifold which admits the Bergman metric (for definitions, see for instance Kobayashi's book "Hyperbolic Complex Spaces"). Suppose moreover that the Bergman metric of $X$ is c …
- 7,902
2
votes
0
answers
224
views
Hermitian connections on real hypersurfaces of $\mathbb C^{n+1}$
I would like to find some non-trivial and "geometric" examples of hermitian connections (that is compatible with a give hermitian metric) on complex hermitian vector bundles over a smooth real manifol …
- 7,902
6
votes
1
answer
638
views
A line bundle not big but with good intersection numbers
Let $X$ be a complex projective manifold of complex dimension $n$ and $A\to X$ an ample line bundle. Let $L\to X$ be a line bundle such that
$$
c_1(L)^k\cdot c_1(A)^{n-k}>0,\quad k=1,\dots,n.
$$
Is it …
- 7,902
20
votes
1
answer
2k
views
Rational or elliptic curves on Calabi-Yau threefolds
Let $X$ be a Calabi-Yau threefold. From a complex analytic point of view, it is widely believed that it should not be Kobayashi hyperbolic, that is it should always admit some non-constant entire map …
- 7,902
3
votes
2
answers
226
views
Uniformity of injectivity for maps associated to linear systems
Let $X$ be a compact complex manifold and $L\to X$ a holomorphic line bundle (without any a priori assumption on its positivity).
Suppose that for each $x,y\in X$, with $x\ne y$, there exists a $k_0 …
- 7,902
10
votes
0
answers
515
views
About the Bloch conjecture on entire curves
The Bloch conjecture states the following:
Bloch's conjecture. Let $X$ be a compact complex Kähler variety such that the irregularity $q = h^0(X,\Omega^1_X)$ is larger than the dimension $n = \dim X$ …
- 7,902
10
votes
0
answers
304
views
Examples of quasi-negative but not negative holomorphic sectional curvature
Let $(X,\omega)$ be a compact Kähler manifold and call $\operatorname{HSC}_{\omega}(x,[v])$ the holomorphic sectional curvature of the Chern connection of $\omega$ at the point $x\in X$ in the directi …
- 7,902
12
votes
0
answers
2k
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Dolbeault cohomology of complex tori.
Let $T=\mathbb C^n/\Lambda$ a complex torus. It is completely elementary to prove that the de Rham cohomology of $T$ in degree $q$ is isomorphic to the $q$-th exterior power of the dual of $\mathbb C^ …
- 7,902
10
votes
2
answers
591
views
Non-bimeromorphic compactifications
Let $X$ be a (smooth, non-compact) complex space. By a compactification of $X$ we mean a compact complex space $\overline X$ which contains a dense open subset biholomorphic to $X$ (we shall identify …
- 7,902
6
votes
1
answer
1k
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Blowing-up an ordinary double point, then contracting the exceptional locus to a curve
Let $X\subset\mathbb P^4$ a projective hypersurface with an ordinary double point at $o\in X$.
Blow-up $\mathbb P^4$ at $o$ and let $E\simeq\mathbb P^3$ the exceptional divisor of this blow-up. Consi …
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