1
vote
1answer
86 views

A question on Aut$(N)$ and Aut$(N/G)$

Let $N$ be a complex manifold and $G$ is a finite group freely acting on $N$. Define another complex manifold $M$ as $M=N/G$. I would like to study Aut$(M)$, the (holomorphic) automorphism group of ...
-2
votes
1answer
118 views

If a subspace is simply connected, then the space itself is simply connected

Let $X$ be a "nice" connected topological space. Let $U\subset X$ be a non-empty subspace. Suppose that $U$ is simply connected. Is $X$ simply connected? In my application, I'm actually thinking ...
6
votes
1answer
284 views

Projective closure in the Zariski and Euclidean topologies

In Smith's An Invitation to Algebraic Geometry, following the definition of the projective closure of an affine variety, it was remarked that "the closure may be computed in either the Zariski ...
2
votes
1answer
59 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 ...
6
votes
2answers
227 views

Why is $x^2$ +$ y^2$ = 1, where $x$ and $ y$ are complex numbers, a sphere?

I've heard $x^2 + y^2$ = 1, where $x$, $y$ are complex numbers, is supposed to be a sphere with two points removed, or also a cylinder. The problem is I've been trying to wrap my head around this for ...
0
votes
1answer
208 views

A consequence of Runge's theorem

I'd like to have a reference for the proof of the following fact of complex analysis. I think it follows from Runge's theorem, but I don't know how to prove it. Fact. Let $U \subseteq V \subseteq ...
1
vote
1answer
83 views

The $0$-section of sheaf

I ran into a problem with a defintion in Complex Analysis as follows: A sheaf $\mathscr S$ over a paracompact Hausdorff space $X$ with a map $f: \mathscr S \to X$ such that (1) $f$ is surjective and ...
3
votes
0answers
145 views

Definition of a complex analytic space

A complex analytic space is a topological space (say, Hausdorff and second countable) such that each point has an open neighborhood homeomorphic to some zero set $V(f_1,\ldots,f_k)$ of finitely many ...