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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.
3
votes
Subset of a complex manifold whose intersection with every holomorphic curve is analytic
I think that the answer is no.
Take a complex torus $X$ without any holomorphic curve. If the result you are asking for were true, it would imply (in the empty sense) that every subset $A \subset X$ i …
- 66.3k
6
votes
Accepted
Fundamental group of a smoothing of a complex surface
Non surprisingly, this usually involves Seifert-Van Kampen theorem, but the actual computation can be a tricky one. However, since you have just one singularity, life will be probably easier.
For an e …
- 66.3k
3
votes
Accepted
How do we define the type of a singularity on a cubic surface?
All the singularities involved in this classification are Rational Double Points. These singularities are taut, in other words, their analytic type is uniquely determined by the configuration of curve …
- 66.3k
14
votes
Accepted
Non-Kähler pseudo-Kähler manifolds
I think that the easiest example of compact pseudo-Kähler manifold which does not admit any Kähler metric is the Kodaira-Thurston manifold. See for instance the introduction of
Yamada, Takumi, Ricci f …
- 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
2
votes
Examples of smooth compact Kähler manifolds with semipositive canonical class
As explained by abx in his comment, any variety of general type whose canonical class $K = \det \Omega^1$ is not ample provides an example of what you are looking for.
If you need an explicit series o …
- 66.3k
1
vote
Surjectivity of $H^2(X,\mathbb C)\to H^2(X,\mathcal O)$
For compact complex surfaces (i.e., when $\dim X =2$) the map is always surjective. See Theorem 2.10, p.141 in
Barth, Wolf P.; Hulek, Klaus; Peters, Chris A. M.; Van de Ven, Antonius, Compact complex …
- 66.3k
4
votes
Fixed-point free holomorphic involutions
The general recipe for constructing such varieties is the following.
Start with your favorite smooth variety $Y$ such that $\operatorname{Pic}^0(Y) \neq 0$ (for instance, this condition is automatical …
- 66.3k
5
votes
Varieties with few trisecant lines
You can have a look at Ingrid Bauer's paper
Bauer, I., The classification of surfaces in $\mathbb{P}^5$ having few trisecants, Rend. Semin. Mat., Torino 56, No. 1, 1-20 (1998). ZBL0965.14029.
It turns …
- 66.3k
1
vote
Direct image of a sheaf with nowhere vanishing sections
Let me give a short answer generalizing abx's comment.
If we have a (branched or unbranched) cover of degree $n \geq 1$, say $f \colon X \to Y$, with $X$ connected, then $$f_* \mathcal{O}_X= \mathcal{ …
- 66.3k
5
votes
Accepted
Milnor hypersurface
Since the question was not answered on MSE, I will provide a short answer here.
The expression $H_{ij}$ is a bi-homogeneous polynomial of bi-degree $(1, \, 1)$, hence it defines an effective divisor i …
- 66.3k
0
votes
How would a topologist explain "every Riemann surface of genus $g$ is hyperelliptic if and o...
The fact that every curve of genus $2$ is hyperelliptic comes from the fact that the canonical $g_2^1$ induces a hyperelliptic involution.
If $g \geq 3$, the general curve is not hyperelliptic. In fac …
- 66.3k
3
votes
Quotients of complex manifolds by symmetric group
For a series of examples in dimension $n=2$, take a degree $2k$ hypersurface $H_{2k} \subset \mathbb{P}^2$. Correspondingly, there is a double cover $X_{2k} \to \mathbb{P}^2$, branched over $k$, so $X …
- 66.3k
9
votes
Accepted
Does any projective bundle on a compact complex manifold have an associated holomorphic vect...
It is classically known that this is true when $\dim X=1$ ("ruled surfaces" = "geometrically ruled surfaces").
It is also true when $\dim X=2$, provided that $H^2(X, \, \mathcal{O}_X)=H^3(X, \, \mathb …
- 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