Complex geometry is the study of complex manifolds and complex algebraic varieties. It is a part of both differential geometry and algebraic geometry.
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votes
1answer
157 views
Meaning of holomorphic Euler characteristics?
I wonder what holomorphic Euler characteristic $\chi(\mathcal{O}_X)$ of a variety represents. For example, I have seen someone fix $\chi(\mathcal{O}_C)=n$ for a complex curve $C$. What does this mean ...
8
votes
2answers
249 views
If $0$, $z_1$, $z_2$ and $z_3$ are concyclic, then $\frac{1}{z_1}$,$\frac{1}{z_2}$,$\frac{1}{z_3}$ are collinear
If the complex numbers $0$, $z_1$, $z_2$ and $z_3$ are concyclic, prove that $\frac{1}{z_1}$,$\frac{1}{z_2}$,$\frac{1}{z_3}$ are collinear.
I really can't seem to get anywhere on this problem, ...
4
votes
1answer
157 views
Proof of Hartogs's theorem
I'd be very grateful if someone could help me understand the proof of Hartogs's theorem appearing in Huybrechts' "Complex Geometry." The statement is:
Let $\mathbb{P}^n \subset \mathbb{C}^n$ be the ...
4
votes
1answer
99 views
Holomorphic Poincaré conjecture
Is there a notion of n-sphere for complex manifolds so that one could formulate a Holomorphic Poincaré conjecture ?
1
vote
1answer
61 views
Checking flat- and smoothness: enough to check on closed points?
I am currently studying varieties over $\mathbb{C}$, i know some scheme theory.
Let $f: X \rightarrow Y$ be a morphism of varieties. If we want to show flatness, is it enough to check the condition ...
8
votes
1answer
132 views
Are vector bundles on $\mathbb{P}_{\mathbb{C}}^n$ of any rank completely classified? (main interest $n=3$)
It is very well known that the group of line bundles on $\mathbb{P}_{\mathbb{C}}^n$ is exactly $\mathbb{Z}$. Are bundles of higher rank classified as well? If so, could anyone provide a nice ...
6
votes
2answers
97 views
Conditions such that taking global sections of line bundles commutes with tensor product?
Let us work with projective algebraic varieties over $k= \mathbb{C}$. If necessary we can also assume smoothness of the varieties.
Of course it is not in general true that given two line bundles $L, ...
4
votes
1answer
73 views
When are two tori biholomorphic? [duplicate]
If $\Lambda \subset \mathbb{C}$ is a lattice, let $T_{\Lambda}$ be the torus $\mathbb{C}/\Lambda$. My question is:
If $\Lambda_1, \Lambda_2 \subset \mathbb{C}$ are two lattices, when are ...
3
votes
2answers
216 views
Some references for potential theory and complex differential geometry
I am looking for references on two distinct (though related) topics.
Potential theory :
I read some time ago the book of Ransford (Potential Theory in the complex plane). It was great (intuitive ...
2
votes
2answers
66 views
The Roots of Unity and the diagonals of the n-gon inscribed in the unit circle
I want to prove that the sum of the squares of the diagonals of a regular $n$-gon inscribed in the unit circle is equal to $2n$. So what I've done is I considered the $n$th roots of unity and said ...
2
votes
1answer
130 views
(Continued:) finiteness of étale morphisms
I was writing a question, it became too long, and i decided to split it into two parts. I hope posting two questions at the same time is not a problem.
First question: Checking flat- and smoothness: ...
2
votes
0answers
201 views
Differential forms and a chain rule
Let $U$ be a Riemann surface and let $z:U\longrightarrow B(0,1)$ be a diffeomorphism, where $B(0,1)$ is the open unit disc in $\mathbf{C}$. So $z$ is a coordinate around $P=z^{-1}(0)$.
Let $Q\in U$ ...
1
vote
1answer
31 views
Holomorphic action of $C_{2}\times C_{2}$ on $\mathbb{P}^1$
I am looking for a holomorphic action of $C_{2}\times C_{2}$ on $\mathbb{P}^1$. Is it true that there is a unique effective action given by
$$
a:z\rightarrow -z, \ \ \ \ b:z\mapsto 1/z
$$
up to ...
1
vote
1answer
187 views
Weitzenböck Identities
The Wikipedia page for Weitzenböck identities is explicitly example based. I am looking for a reference which takes a more rigorous approach (as well as a discussion of the Bochner technique). In ...
