# Tagged Questions

**0**

votes

**2**answers

137 views

### Cohomology of Complements by an analytic subset?

Good moring,
Let $\Omega$ be a domain in $\mathbb{C}^n$ and $S\subset\Omega$ an analytic subset of codimension 1. What can we say about the cohomology group $H^1(\Omega\backslash S, \mathbb{Z})$? ...

**12**

votes

**1**answer

454 views

### Essential uniqueness of the real-analytic structure on $\mathbb R$

It is well-known that any $C^k$-smooth $1$-manifold homeomorphic to $\mathbb R$ is $C^k$-diffeomorphic to $\mathbb R$. The cases of $k\in{\mathbb N}\cup$ {$\infty$} may all be handled similarly by ...

**2**

votes

**1**answer

532 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

**0**answers

730 views

### Dolbeault cohomology of complex tori.

Let $T=\mathbb C^n/\Lambda$ a complex torus. It is completely elementary to prove that the de Rham cohomology of $T$ in degree $q$ is isomorphic to the $q$-th exterior power of the dual of $\mathbb ...

**16**

votes

**3**answers

1k views

### Holomorphic vector fields acting on Dolbeault cohomology

The question.
Let $(X, J)$ be a complex manifold and $u$ a holomorphic vector field, i.e. $L_uJ = 0$. The holomorphicity of $u$ implies that the Lie derivative $L_u$ on forms preserves the (p,q) ...

**3**

votes

**6**answers

917 views

### Dolbeault cohomology

Hello
I am trying to get a good book that explains the Dolbeault Cohomology, does anyone know of a good one?

**2**

votes

**4**answers

1k views

### Riemann Surfaces

In a complex analysis course I have been given the following definition:
Let X be a Riemann surface, denote by H(1,0) the space of all (1,0)-holomorphic forms on X and consider the quotient vector ...