2
votes
1answer
57 views

Fundamental group of the following disc

What is the fundamental group of the following space in $\mathbf C^n$? This is the topological space given by $$\{(x_1,\ldots,x_n)\in \mathbf C^n-\{0\} : \vert x_1\vert < 1, \ldots, \vert x_n\vert ...
12
votes
2answers
759 views

What exactly is a Kähler Manifold?

Please scroll down to the bold subheaded section called Exact questions if you are too bored to read through the whole thing. I am a physics undergrad, trying to carry out some research work on ...
3
votes
1answer
91 views

Some questions about $S^n$

I have some questions about the $n$-sphere: I know that for $n=0,1,3$, $S^n$ forms a Lie group and I also know why it's true, but why is it not the case for other $n$? I have the same question for ...
4
votes
1answer
113 views

Holomorphic Poincaré conjecture

Is there a notion of n-sphere for complex manifolds so that one could formulate a Holomorphic Poincaré conjecture ?
2
votes
1answer
109 views

A question about complex manifolds

Let $(M,J_{M})$ be a almost complex manifold and $(N,J_{N})$ be a complex manifold. I want to prove that $F^{*}(\mathcal{O}_{N})\subset\mathcal{O}_{M}$ implies that $F:M\rightarrow N$ is almost ...
9
votes
4answers
153 views

holomorphic exoticness

A topological manifold is an exotic copy of another smooth manifold if it is homeomorphic to it, but not diffeomorphic (and when you switch diffeomorphic by homotopic, you get a fake copy, following ...