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Results tagged with complex-geometry
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user 10076
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.
4
votes
Accepted
Inclusion of (pulling back of) dualizing sheaves under normalization
$X$ is Gorenstein, so $\omega_X$ is a line bundle and hence so is $\nu^*\omega_X$. In particular, it is torsion-free and hence $\mu(\mathcal T)=0$, so $\mu$ indeed factors through $\omega_{\widetilde …
- 42.9k
4
votes
When is a holomorphic submersion with isomorphic fibers locally trivial?
Taking your question to the realm of schemes I think that assuming something like that ${\rm Aut} F$ has a natural scheme structure gives you something that could be considered the algebraic equivalen …
- 42.9k
5
votes
Structure of Kähler cone
The cone of curves of K3 surfaces is described in this and this papers.
- 4,713
2
votes
Question on Kähler/ample cone, cone of curves....
This is an answer to the two more questions asked in the comments. I started it out as a comment, but got tired of the space restriction....
I am not certain, but you are right, $NE(X)$ for this $X$ …
- 35.5k
3
votes
Topology of the preimage of a point for degree one holomorphic maps
This is probably not the most optimal way to do it, but this was what came to mind right away. Also, this is in some sense more general than what you ask and in some sense it is less. Finally, it is n …
- 2,821
2
votes
On minimal resolution of singularities and the type of singularities
You probably meant $\pi: X\to Y$ and not $\pi:Y\to X$. That way, any singularity that appears on a scheme of finite type over a field can be mapped to a smooth variety in a finite way. (This claim is …
- 2,821
3
votes
Accepted
Two morphisms possess the same Viehweg's variation
Since the definition only depends on the general fiber, and $\beta$ is birational, one may assume that $\beta$ is actually and isomorphism.
So, then $L$ is defined as a subfield with minimal transcend …
- 42.9k
3
votes
Gorenstein varieties: why the two definitions are equivalent?
As Donu mentioned, Gorenstein can be defined as Cohen-Macaulay and such that the canonical=dualizing sheaf is a line bundle. The point is that the dualizing complex is quasi-isomorphic to a sheaf if a …
- 42.9k
8
votes
Accepted
Basepoints in the canonical system of algebraic surfaces
In the general case when $n=\dim X$ is arbitrary
Kollár proves an analogue for minimal varieties of general type, although the bound is much worse than in the surface case.
In the situation at hand it …
- 42.9k
7
votes
Lefschetz hyper-plane theorem for singular projective varieties?
There are actually several versions of the Lefschetz hyperplane theorem for singular varieties. The main point is that this is ultimately a Hodge theoretic statement, one proof is using the Kodaira-Ak …
- 2,821
1
vote
Interesting examples of direct image bundles
If the fibers are compact Kähler manifolds, then the Hodge numbers of the fibers are constant and hence by Grauert's theorem, the direct images of the relative Hodge bundles are locally free (i.e., ve …
- 42.9k
5
votes
Accepted
Projective manifold whose anticanonical section is composed of two components
If you look at the proof of the theorem referenced in the answer to that linked question, near the bottom of p.801 and top of p.802 it is established that if you run an MMP (the chosen boundary diviso …
- 42.9k
6
votes
Question about Hodge number
Proposition. For compact complex manifolds of dimension $1$ (a.k.a. complex Riemann surfaces) the Hodge numbers are topological invariants.
Proof. The Hodge numbers are determined by the genus, which …
- 19.6k
5
votes
Accepted
Finite unramified analytic coverings vs finite etale coverings
(Using the notation from the question) $\mathcal F$ is a coherent sheaf of $\mathcal O_{\overline X}$-algebras. Then $Z={\rm Spec}_{\overline X}\, \mathcal F\to \overline{X}$ is a finite morphism betw …
- 42.9k
14
votes
Accepted
Generalisations of Riemann-Roch for surfaces
If $X$ is proper with rational singularities (and quotient and A-D-E (=Du Val) singularities are rational), then you can do most cohomology computations on a resolution.
Let $\pi:Y\to X$ be a resoluti …
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