The complex-manifolds tag has no usage guidance.

**27**

votes

**2**answers

966 views

### Can the holomorphic image of $(\mathbb{C}^*)^n$ be open but not dense

Let $M$ be a compact complex connected [but not necessarily kähler] $n$-manifold, and suppose we have a holomorphic map $$(\mathbb{C}^*)^n \to M$$ such that the image is open. Is the image necessarily ...

**5**

votes

**1**answer

455 views

### complex manifold with corner

I was reading Dominic Joycee article on Manifold with corner. He talk about manifold with corner modeled over $[0,\infty)^k\times \mathbb R^{n-k}$ for some $k\leq n$. From here i moved to Melrose ...

**3**

votes

**1**answer

912 views

### Algebraic De Rham cup product versus Betti cup product

Let $X$ be a smooth projective variety over $\mathbf{C}$ of complex dimension $2n$. Let
$C_1,C_2\subseteq X$ be two closed subvarieties of complex dimension $n$.
Then we get two Betti homology ...

**1**

vote

**2**answers

220 views

### Analytic isomorphisms above two etale maps

Le $X_1$, $X_2$ and $Z$ be smooth quasi-projective connected varieties defined over $\mathbf{C}$. Let $p_1:X_1\rightarrow Z$ and $p_2:X_2\rightarrow Z$ be finite etale maps. Assume that $f:X_1\...

**4**

votes

**0**answers

360 views

### Etymology of the O-notation for algebras of holomorphic functions

The notation $O(X)$ seems to be a quite standard notation for the algebra of all holomorphic functions on some connected domain in $\mathbb{C}^n$ (or a complex manifold). I would like to know where ...

**23**

votes

**2**answers

2k views

### Non-compact complex surfaces which are not Kähler

Not every complex manifold is a Kähler manifold (i.e. a manifold which can be equipped with a Kähler metric). All Riemann surfaces are Kähler, but in dimension two and above, at least for compact ...

**2**

votes

**1**answer

204 views

### What does non-levi flat point mean geometrically

Hello,
$CR$ manifold for example $S^1\times C^{n-1}$ is every where levi flat. Can I have example of $CR$ manifold which has at least one non levi flat point.
I can't see what the happening in Non-...

**13**

votes

**1**answer

843 views

### Splitting principle for holomorphic vector bundles

Let $E \to X$ be a vector bundle over a decent space $X$. Then there is a space $Z$ together with a map $p: Z \to X$ which induces a split injection on cohomology and such that $p^* E$ splits as a ...

**1**

vote

**2**answers

2k views

### Line bundles with complex connection

Suppose that we have a complex manifold $X$, and a line bundle $L$ over $X$. It is known that the line bundles over $X$ are parametrized by their Chern class, the Chern class being the cohomology ...

**7**

votes

**2**answers

370 views

### Non-bimeromorphic compactifications

Let $X$ be a (smooth, non-compact) complex space. By a compactification of $X$ we mean a compact complex space $\overline X$ which contains a dense open subset biholomorphic to $X$ (we shall identify ...

**3**

votes

**2**answers

392 views

### volume of complex hyperbolic manifolds

I would like to know if there are in the literature explicit computations of the volume of complex hyperbolic manifolds.
More precisely, let $\mathcal O$ be an imaginary quadratic number field, and ...

**0**

votes

**1**answer

228 views

### local kählerforms on complex manifold

hallo,
Let $M$ be a complex manifold. Assume we have a covering of $M$ by complex charts $\{U_{i}\}$. Furthermore assume that we have on each $U_{i}$ a Kählerform $\omega_{i}$ (i.e. $d\omega_{i} = 0$)...

**7**

votes

**1**answer

763 views

### Geometric interpretation of Simpson's correspondence

What is the exact geometric meaning of the Simpson's correspondence between Higgs bundles and local systems ? I know that it should have a rich geometric content but don't know an explicit geometric ...

**3**

votes

**3**answers

459 views

### Does isomorphisms of sheaf of holomorphic sections implies isomorphisms of two holomorphic vector bundles over the same complex space ?

My knowledge is very limited for complex geometry. I have the following question:
If we have two complex vector bundles $E\to X$ and $F\to X$ such that we have an isomorphism $\mathcal O\left(E\right)...

**1**

vote

**1**answer

179 views

### How to compute Kobayashi distance of compact Kaehler manifolds with postive Ricci curvature?

Recently I just learned the Kobayashi distance on complex manifolds and wants to get some feeling of how it looks like on exmaples of manifolds with positive Ricci curvature. I have a feeling that the ...

**10**

votes

**2**answers

532 views

### Classification of holomorphic disc bundles

I've had difficulty finding sources which treat the classification of holomorphic disc bundles over (compact and noncompact) Riemann surfaces. Note that by "bundle", I mean a holomorphic fiber bundle,...

**5**

votes

**1**answer

553 views

### On sufficient conditions on an analytic map to be algebraic(=regular)

Let $X$ and $Y$ be smooth quasi-projective varieties defined over $\mathbf{C}$ and let
$$
f:X(\mathbf{C})\rightarrow Y(\mathbf{C})
$$
be a holomorphic map (not necessarily regular=algebraic). Then it ...

**1**

vote

**0**answers

220 views

### glue together a sequence of holomorphic forms

hallo,
my problem is the following: i have a finite sequence of holomorphic $k-$forms $\alpha_{k}$, each defined on open subsets $U_{k} \subset M$, where $M$ is a complex $n$-dimensional manifold, ...

**4**

votes

**1**answer

708 views

### Are there any symplectic but not holomorphic Calabi-Yau manifolds in real dimensions 4 and 6?

Are there any symplectic but not complex Calabi-
Yau manifolds in real dimensions 4 and 6?

**5**

votes

**1**answer

296 views

### Rosenlicht theorem about uniruledeness and zeroes of holomorphic vector field on complex projective manifold

I heard that there is a theorem due to Rosenlicht which says the following:
Theorem. Let $X$ be a complex projective manifold and $V$ a non-trivial holomorphic vector field on $X$. Then $X$ is ...

**2**

votes

**1**answer

225 views

### biholomorphism complex manifold induced structure

Let $X$ be a $n$ dimensional complex manifold with complex structure $I$ and assume one has a diffeomorphism $f : \mathbb{C} \rightarrow X$ of some open set $U$ in $\mathbb{C}$ into its image $f(U)$. ...

**1**

vote

**3**answers

929 views

### Examples of Lie Algebroids

The concept of a Lie Algebroid is given an important geometric meaning in the framework of Generalized Complex Geometry. For reference, the (barebones) definition of a Lie Algebroid is a vector bundle ...

**7**

votes

**1**answer

458 views

### Complex analytic space with no (positive dim.) subscheme ?

Is there an example of a complex analytic space $X$ that doesn't have any (not necessarily open or closed) positive dimensional subspace $Y$ which is analytically isomorphic to (the complex analytic ...

**18**

votes

**1**answer

731 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

2k 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

337 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

428 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)-...

**1**

vote

**1**answer

467 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

596 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

899 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 ...

**4**

votes

**0**answers

534 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

577 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 ...

**6**

votes

**2**answers

724 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})=
H^1(\mathbb{P}^n(\mathbb{C}),\...

**6**

votes

**2**answers

935 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 ...

**16**

votes

**2**answers

1k 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

1k 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

618 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

632 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 ...

**19**

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 ...

**24**

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 ...