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**18**

votes

**1**answer

677 views

### holomorphic K-theory

Topological K-theory is usually defined by setting $K(X)$ to be the groupification of the monoid $Vect_\mathbb{C}(X)$ of complex vector bundles over $X$ (with addition given by Whitney sum). However, ...

**7**

votes

**3**answers

1k views

### Is a non-compact Riemann surface an open subset of a compact one ?

Let $X$ be a non-compact holomorphic manifold of dimension $1$. Is there a compact Riemann surface $\bar{X}$ suc that $X$ is biholomorphic to an open subset of $\bar{X}$ ?
Edit: To rule out the case ...

**6**

votes

**2**answers

334 views

### Calabi-Yau manifolds and polygonal linkage configuration spaces: related?

I was reading about Calabi-Yau manifolds, about which I know little, and was wondering
if these (or related complex manifolds, perhaps K3 surfaces) can be viewed as configuration
spaces (or moduli ...

**7**

votes

**1**answer

417 views

### Is it always possible to extend a closed (1, 1)-form on a divisor to a closed (1, 1)-form on a tubular neighbourhood?

Let $X$ be a compact Kahler manifold, let $D$ be a smooth divisor in $X$, and let $U$ be a tubular neighbourhood of $D$ in $X$. Suppose that $D$ is Fano. Is it possible to extend every closed (1, ...

**1**

vote

**1**answer

446 views

### Almost Complex Integrability and Algebraic Varieties

Let $J$ be an almost complex structure on an algebraic variety $V$. As we all know, $J$ comes from a complex structure if the Nijenhuis tensor of $J$ vanishes. What I would like to know is if there ...

**2**

votes

**1**answer

549 views

### Plurisubharmonic exhaustion functions without critical points at infinity

A complex manifold $X$ is said to be weakly pseudoconvex if there exists on $X$ a smooth plurisubharmonic exhaustion function $\psi$.
For example, Stein manifolds are weakly pseudoconvex (in this ...

**2**

votes

**4**answers

886 views

### Noncompact Kahler manifolds with nonzero Ricci tensor but vanishing scalar curvature

Let us consider a noncompact K\"{a}hler manifold with vanishing scalar curvature but nonzero Ricci tensor. I'm wondering what can it tell us about the manifold. The example (coming from physics) has ...

**3**

votes

**0**answers

480 views

### What is the nature of the zero locus of a section of a coherent sheaf?

Suppose that $X$ is a reduced rigid space and $\scr{F}$ is a coherent sheaf on $X$. For a section $f\in {\scr F}(X)$, the zero locus of $f$ is the set of points $x\in X$ at which $f$ vanishes in the ...

**3**

votes

**2**answers

550 views

### Torsion in the Betti cohomology of complex surfaces

Q1. So what is the "simplest example" of a compact complex manifold of dimenension $2$, say $X$, for which $H_B^2(X,\mathbb{Z})$ has non-trivial torsion.
Q2. How do we think about these torsion ...

**5**

votes

**2**answers

598 views

### On the fundamental group of hypersurfaces

Let $H$ be a smooth projective hypersurface in $\mathbb{P}^n(\mathbb{C})$ where $n\geq 3$. Then by the Lefschetz hyperplane theorem we have that $H^1(H,\mathbb{C})=
...

**6**

votes

**2**answers

902 views

### Finite unramified analytic coverings vs finite etale coverings

Let $X$ be a smooth quasi-projective variety (so irreducible) over $\mathbf{C}$. We may think of $X$ as a complex manifold which we denote by $X^{an}$. Of course the topology on $X^{an}$ is finer ...

**15**

votes

**2**answers

920 views

### When can you reverse the orientation of a complex manifold and still get a complex manifold?

I'm told that $\overline{\mathbb{C}P^2}$, i.e. $\mathbb{C}P^2$ with reverse orientation, is not a complex manifold. But for example, $\overline{\mathbb{C}}$ is still a complex manifold and ...

**3**

votes

**2**answers

945 views

### holomorphy of inverse map

Let $M,N$ be complex manifolds and $f : M \to N$ be a bijective holomorphic map. Is then $f^{-1}$ also holomorphic?
The open mapping theorem implies that $f^{-1}$ is continuous. In order to apply the ...

**0**

votes

**2**answers

598 views

### Diffeomorphism group of the unit sphere of complex n-space

What is the largest subgroup of the diffeomorphism group of a (2n-1)-sphere that only contains members that are holomorphic in the coordinates assigned to the sphere by taking it to be the unit sphere ...

**12**

votes

**2**answers

600 views

### Highly connected, compact complex manifolds

Here are four remarks about the homology and homotopy type of a compact, complex manifold $M$:
If $M$ is Kähler, then it is symplectic and thus $H^2(M,\mathbb{R}) \ne 0$. (Also, as explained in a ...

**17**

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) ...

**12**

votes

**3**answers

1k views

### A topological consequence of Riemann-Roch in the almost complex case

This question originated from a conversation with Dmitry that took place here
Is there a complex structure on the 6-sphere?
The Hirzebruch-Riemann-Roch formula expresses the Euler characteristic of ...

**23**

votes

**3**answers

2k views

### Topologically distinct Calabi-Yau threefolds

In dimensions 1 and 2 there is only one, respectively 2, compact Kaehler manifolds with zero first Chern class, up to diffeomorphism. However, it is an open problem whether or not the number of ...