# Questions tagged [complex-geometry]

Complex geometry is the study of complex manifolds and complex algebraic varieties, and, by extension, of almost complex structures. It is a part of both differential geometry and algebraic geometry.

1,991 questions

**4**

votes

**0**answers

46 views

### Arnold's theorem on small denominators and holomorphic tubular neighborhoods

By a theorem of Grauert, along a curve with negative self-intersection a complex surface is locally biholomorphic to a neighborhood of the zero section of that curve inside its normal bundle. For ...

**0**

votes

**0**answers

77 views

### A coherent sheaf is a vector bundle over subvariety?

Over a complex manifold, can every coherent sheaf be seen as a holomorphic vector bundle over an analytic subset?
Thanks in advance.

**3**

votes

**0**answers

49 views

### Metric with singularities on Riemann Surfaces and the associated Laplacians

I have asked this question on Math Stack Exchange
Metric with singularities and associated Laplacian
but I have not got any answers/comments, therefore I post this question on the MO.
Suppose $M$ ...

**-1**

votes

**0**answers

204 views

### A Riemann Surface can be characterized by its Lie Algebra? [on hold]

I was studying the geometric properties of Lie algebras.
Let $X$, $Y$ be two Riemann surfaces. Suppose that this surfaces meet a finite set of ordinary differential equations. Let $L_X$ and $L_Y$ ...

**3**

votes

**1**answer

92 views

### Hyperkähler ALE $4$-manifolds

It is well known that Kronheimer classified all hyperk?hler ALE $4$-manifolds. In particular, any such manifold must be diffeomorphic to the minimal resolution of $\mathbb C^2/\Gamma$ for a finite ...

**6**

votes

**1**answer

185 views

+50

### The Construction of a Basis of Holomorphic Differential 1-forms for a given Planar Curve

Working over the complex numbers, consider a function $F\left(x,y\right)$ and a curve $C$ defined by $F\left(x,y\right)=0$.
I know that to construct the Jacobian variety associated to $C$, one ...

**2**

votes

**0**answers

234 views

### Integration over a Surface without using Partition of Unity

Suppose we are given a compact Riemann surface $M$, an open cover $\mathscr{U}=\{U_1,U_2,\dots\}$ of $M$, charts $\{(U_1,\phi_1),(U_2,\phi_2),\dots\}$, holomorphic coordinates, $\phi_m:p\in U_m\mapsto ...

**3**

votes

**0**answers

71 views

### Large isometry groups of Kaehler manifolds

Let $M$ be a closed simply-connected Kaehler manifold that is not isomorphic to a product of lower-dimensional Kaehler manifolds. Pick an orientation for $\mathbb{C}$; this endows $M$ with an ...

**2**

votes

**0**answers

77 views

### Connectedness of isometry group of closed Kaehler manifolds

Let $(M, g, J)$ be a closed Kaehler manifold. Is there some more-or-less non-tautological condition ensuring that its group of orientation-preserving isometries is connected? Also the same question, ...

**6**

votes

**2**answers

257 views

### Is the symmetric product of an abelian variety a CY variety?

Let $n>1$ be a positive integer and let $A$ be an abelian variety over $\mathbb{C}$. Then the symmetric product $S^n(A)$ is a normal projective variety over $\mathbb{C}$ with Kodaira dimension zero ...

**12**

votes

**0**answers

238 views

### Donaldson-Thomas Theory and “Quantum Foam” for Mathematicians

Let $X$ be a smooth, projective Calabi-Yau threefold. From an algebro-geometric perspective, the Donaldson-Thomas invariants $\text{DT}_{\beta, n}(X)$ are virtual counts of ideal sheaves on $X$ with ...

**3**

votes

**0**answers

95 views

### Symplectic Chern class of holomorphic symplectic manifold

I've posted this question already on math.stackexchage. Unfortunately, it did not receive any answers even though there was a bounty on it. So maybe somebody of you could help me. I apologize if this ...

**1**

vote

**0**answers

72 views

### Can every non-compact Kähler manifold be realized as the analytification of a smooth variety?

I was just thinking about how we have nice theorems relating compact K?hler manifolds to the algebraic setting, but I was wondering if anything interesting holds in the non-compact case?
i.e. Given ...

**1**

vote

**0**answers

28 views

### Complex differentials and measured singular foliations

I'm trying to understand the technical basis of singular foliations for pseudo-Anosov diffeomorphisms, and I've hit a bit of a strange calculation I'm having a hard time verifying/unpacking. In Fathi, ...

**4**

votes

**0**answers

129 views

### Constructing new complex manifolds out of old

It is not difficult to build new manifolds out of old in the smooth category, for example
taking the direct product or constructing a fiber bundle,
taking the level set of a regular value of a smooth ...