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Results tagged with complex-geometry
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user 123448
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.
6
votes
1
answer
569
views
Equivalent definitions of Kodaira dimension
The Kodaira($-$Iitaka) dimension of a line bundle $L$ on a complex manifold $X$ can be defined either in three ways:
The maximal dimension of the image of the rational maps $φ_{|mL|} : X \dashrighta …
- 1,141
5
votes
0
answers
138
views
Are open subsets of a $\sigma$-compact LCH space $\mathcal{K}$-analytic?
I'm reading Guedj and Zeriahi's Degenerate Complex Monge-Ampère Equations Chapter 4 which talks about capacities. Specifically Corollary 4.13 claims that when $X$ is a locally compact Hausdorff $\sigm …
- 1,141
2
votes
1
answer
377
views
Are "transverse" hyperplane sections of nondegenerate irreducible projectice varieties alway...
Let $X \subseteq \mathbb{P}^n$ be a irreducible complex projective variety. It is called nondegenerate if it is not contained in a hyperplane in $\mathbb{P}^n$.
Assuming $X$ is nondegenerate and irred …
- 1,141
10
votes
0
answers
267
views
Looking for counterexamples: Are maximal tori in the automorphism groups of smooth complex q...
Let $X$ be a smooth quasiprojective variety over $\mathbb{C}$. It has a group of (algebraic) automorphisms $
\DeclareMathOperator{\Aut}{Aut}
\Aut(X)$.
Define a torus in $\Aut(X)$ to be a faithful alge …
- 1,141
5
votes
1
answer
439
views
Is the determinant line bundle of a coherent sheaf functorial (between sheaves of the same r...
The determinant line bundle of a coherent sheaf $\mathcal{F}$ on an $n$-dimensional (smooth) analytic space is defined as
\begin{equation}
\det \mathcal{F} := \bigotimes_i^n (\det \mathcal{E}_i)^{ …
- 1,141
4
votes
0
answers
101
views
Serre vanishing on one-point blow-ups
This is basically the last step of problem 5.3.7 in Huybrechts' Complex Geometry.
Let $X$ be a complex manifold, $x \in X$, $E$ a holomorphic vector bundle on $X$ and $L$ a positive line bundle. Denot …
- 1,141
7
votes
2
answers
606
views
Does Peetre's theorem hold in complex analysis?
Let $E, F$ be two smooth vector bundles over a smooth manifold $M$. Peetre's theorem states that any $\mathbb{R}$-linear morphism $D: \mathcal{E} \to \mathcal{F}$ of the sheaves of sections of $E$ and …
- 1,141