5
votes
1answer
277 views

Universal covering map from $\mathcal{H}$ to $\mathbb{C}\setminus \mathbb{Z}\oplus i\mathbb{Z}$ (the countably punctured complex plane)

It's a consequence of the uniformization theorem for simply connected Riemann surfaces that the universal cover of $\mathbb{C}\setminus(\mathbb{Z}\oplus i\mathbb{Z})$ ($\mathbb{C}$ punctured at all ...
5
votes
1answer
306 views

Branched Regular Cover over 4-times punctured sphere

This is probably trivial but has been bothering me all day. Suppose $f:\Sigma_g\to \mathbb{S}^2$ is a $g+1$ fold branched conformal map with $\Sigma_g$ a connected genus $g$ surface and $f$ having ...
4
votes
1answer
447 views

Is a simply connected Ricci-flat Kaehler manifold a Calabi-Yau manifold?

Hi, I have the following question: Let $(M,\omega, J)$ be a simply connected Kaehler manifold with Ricci-flat Kaehler metric. How can one show that $M$ is a Calabi-Yau manifold. By Calabi-Yau ...
3
votes
3answers
363 views

Jordan curve theorem for cylinders

Hello, I would like to know if the following result is true: Let $A,B$ be two embedded circles in $S^2$ which do not intersect and let $C$ be the $\textit{closed}$ region bounded by $A$ and $B$ ...
1
vote
0answers
156 views

Plane Curve invariants via Contour Integrals

We learn in complex analysis class how to find the winding number of the contour $\Gamma$ around the origin. \[ n = \frac{1}{2\pi i} \oint \frac{dz}{z} = \frac{1}{2\pi i} \oint d(\log z) = ...
2
votes
1answer
336 views

Are Lefschetz thimbles holomorphic manifolds?

I have a Lefschetz thimble defined by the stable flow of the gradient a holomorphic function toward a critical point (as defined e.g. in Witten arXiv:1001.2933 and F.Pham "Vanishing homologies and the ...
2
votes
3answers
303 views

connectedness of the complement of the zero set of a polynomial $P: SL(N,\mathbb{C})^n \rightarrow \mathbb{C}$

I know that the complement of the zero set of a polynomial $P: \mathbb{C}^n \rightarrow \mathbb{C}$ is connected in $\mathbb{C}^n$ (by the way, can anybody suggest a reference?). Is it possible to ...
4
votes
2answers
764 views

Fundamental Theorem of Algebra, Theorems of Brouwer and Borsuk

Several months ago I was browsing through a question posted here ("Applications of Brouwer’s fixed point theorem"); amongst the comments attached it was mentioned that one could derive both Borsuk's ...
2
votes
1answer
498 views

Notions related to De Rham Cohomology

In R^2, we have the following abelian groups, some of which have R vector space structures, or even C vector space structures. Closed forms/exact forms real parts of analytic functions/harmonic ...
11
votes
9answers
3k views

Good book on Riemann surfaces and Galois theory?

I'm supervising an undergraduate project on Galois theory and covering spaces. I want to have him read about the fact that from a branched cover of a Riemann surface you get an extension of its field ...
2
votes
1answer
546 views

How to calculate Dr. Curt McMullen's expanding eigenvalues for totally degenerate groups?

What is required in order to derive the expanding eigenvalues of Dr. Curt McMullen's torus orbifold bundles over the circle and the corresponding totally degenerate groups, as presented in Section 3.7 ...
4
votes
1answer
297 views

An analytic subset as a singular homology class of a compact manifold

We know every differential manifold can be triangulable. Let $M$ be a compact complex manifold of dimension $m$ and V be an analytic subset of dimension $s$ of $M.$ If $V$ has no singularity then $V$ ...
10
votes
2answers
658 views

Fundamental Groups of compact Complex manifolds?

Hi, are limitations on the fundamental group for compact complex manifolds known? Can an arbitrary (finite represantable) group be the fundamental group of a compact complex manifold? Thanks
16
votes
1answer
749 views

Irrational Numbers and the Riemann Surface of a Multi-Valued Function

Suppose a meromorphic function $f(z)$ has two poles, with residues $1$ and $\gamma$, respectively. Then the topology of the Riemann surface of the anti-derivative of $f(z)$ depends on whether or not ...