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Results tagged with complex-geometry
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user 7460
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.
10
votes
Accepted
Is polynomial convexity a topological invariant?
The answer is no.
In fact, Kallin has shown in [Kal64] that the union of three disjoint closed balls is polynomially convex, but the union of three disjoint closed polydisks needs not to be polynomi …
- 66.3k
3
votes
Accepted
Compact complex surface that admits a Kodaira fibration is Kahler
Let $f \colon S \longrightarrow B$ be a Kodaira fibration, and let $F$ be a general fibre. Then by [Kas68, Thm. 1.1] we have $g(B) \geq 2$ and $g(F) \geq 3$.
In particular, $S$ contains no rational …
- 66.3k
28
votes
Accepted
Must an algebraic variety with trivial tangent bundle be an abelian variety?
More generally, in the complex case the following result holds.
Theorem. Let $X$ be a compact Kähler manifold which is complex parallelisable, i.e. such that
$T_X$ is holomorphically trivial. …
- 66.3k
16
votes
Accepted
Is a holomorphic family whose fibers are all smooth locally trivial?
By Grauert-Fischer Theorem, a smooth family of compact complex manifold is locally trivial if and only if all the fibers are analytically isomorphic.
However, there exist smooth families $\pi \colon …
- 66.3k
7
votes
Deformations of Kähler manifolds where Hodge decomposition fails?
If any example exists, then the general fibre of the family cannot be projective.
In fact, Dan Popovici ["Limits of projective manifolds under holomorphic deformations", arXiv.09102032] recently pro …
- 66.3k
9
votes
The asymptotic growth of global sections of powers of a complex line bundle
Actually, it is possible to prove the following statement. Set
$$\mathbb{N}(L, X):=\{a > 0 \ | \ h^0(X, aL) \geq 1 \}$$
and let $d$ be the largest common divisor of the elements of $\mathbb{N}(L, X …
- 66.3k
3
votes
Existence of nodal curves in a linear system
When $S=\mathbb{P}^2$ and $\mathcal{L}= H= \mathcal{O}_{\mathbb{P}^2}(1)$,
this question is discussed in Sernesi's book Deformation of Algebraic schemes, Chapter 4. In particular Corollary 4.7.19 page …
- 66.3k
4
votes
Accepted
Lifting property for proper morphism
Regarding your general question, the answer is no.
Take any hyperbolic projective variety $Y$ (for instance, a ball quotient) of dimension $n$, and project it generically onto $\mathbb{P}^n$. Removing …
- 66.3k
2
votes
Restricting a non-constant map to an ample divisor
This is a partial answer, too long to be a comment.
Let me give a family of examples where $D$ can be found (every effective divisor $D$ works, actually).
Take any smooth, projective variety $X$ admit …
- 66.3k
6
votes
Lifting a birational map of $X/G$ to a birational map of $X$?
It seems to me that the answer is no even if the action of $G$ is free.
In fact, assume that $G$ acts freely and set $Y:= X/G$. Let $f \colon Y \to Y$ be an automorphism and $f_* \colon \pi_1(Y) \to …
- 66.3k
7
votes
Kähler metric on projectivised bundle
As pointed out by Georges Elencwajg, the answer is yes.
However, if one substitutes the assumption "holomorphic vector bundle" with the weaker "complex vector bundle", the answer is no.
In fact, th …
- 66.3k
3
votes
Projective embedding of a compact complex surface
Yes, this is precisely Theorem (6.2), p. 160 of
Barth, Wolf P.; Hulek, Klaus; Peters, Chris A. M.; Van de Ven, Antonius, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. …
- 66.3k
16
votes
Accepted
Kähler metric on a Zariski open subset of a non-Kähler manifold
One can consider the following example.
A Moishezon manifold $M$ is a compact connected complex manifold such that the field of meromorphic functions on $M$ has transcendence degree equal to the compl …
- 66.3k
12
votes
Accepted
Extending holomorphic functions
Yes, there are several generalizations of Hartogs Extension Theorem that hold on Stein spaces.
For a good survey you can look at the paper by Øvrelid and Vassiliadou Hartogs Extension Theorems on St …
- 66.3k
13
votes
Accepted
Hypersurface of complex projective space
As noticed by Sasha in his comment, the answer is no.
The following proof also shows that this result cannot be generalized for higher values of the degree.
For any smooth complex hypersurface of d …