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# 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.

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### Structure of the module of derivations on the space of Holomorphic functions

Maybe this is well-known, maybe not. Let $\Omega\subset \mathbb{C}$ be connected open and non-empty. It can be shown that if $d\in\mathfrak{Der}(\mathcal{H}(\Omega))$ (i.e. $d$ is a derivation of ...
0answers
26 views

### Constructing certain Global section with prescribed zero locus over Stein manifold

Let $X^n$ be a Stein manifold (complex submanifold in $\mathbb{C}^N$ for some large $N$). Let $D = \{(z,z)\in X\times X: z\in X\}$ be the diagonal in $X\times X$. I'm looking for some holomorphic ...
0answers
52 views

### Understanding the universal sheaf locally

Suppose I have a projective, flat morphism $\pi : X \to S$ between smooth projective varieties over $\mathbb{C}$. By the work of Simpson, there exists a moduli space $M = M(X/S, v)$ of torsion-free (...
0answers
32 views

### Field of definition from deformation rigidity

It is known that a smooth complex quasi-projective variety which is deformation-rigid (e.g. any holomorphic deformation inside an ambient space is trivial) can be defined over a number field. Can one ...
0answers
75 views

### Existence of finite etale covering and cohomology of the profinite completion of the fundamental group

Let $X$ be a connected complex-analytic space. , $G = \pi_1(X)$ the fundamental group of $X$ , $\hat{G}=\varprojlim G/N$ its profinite completion. Let $\beta\in H^2(X,\mathcal{O}_X^\times).$, say ...
0answers
61 views

### Where can I learn about the discrete symmetries of the complex projective plane (or space)?

I understand that $CP^1$ is the Riemann Sphere. I guess all its discrete symmetries were known for a long time and well-classified. (But suggestions or good references where this is worked out in a ...
0answers
133 views

### Holomorphic functions to complex torus

Let $X$ be a complex algebraic variety and $T$ a complex torus (not necessarily algebraic). Assume that $X$ is a proper subset of its completion $\bar{X}$. Let $f:X \to T$ be a holomorphic map. Are ...
0answers
96 views

### Bridgeland stability for restricted Kahler moduli?

Let $X$ be a simply-connected, smooth, projective Calabi-Yau threefold. To my understanding, Bridgeland introduced stability conditions on triangulated categories to give a proper mathematical ...
1answer
171 views

### Locally affine varieties and du Val singularities

Let me start with an apologetic disclaimer: I am very far from an algebraic geometer, so this question might be crudely formulated. I have a specific question about du Val singularities, but while ...
1answer
296 views

### Does the $\overline{\partial}$ operator have closed image?

Let $X$ be a complex-analytic manifold, not necessarily compact. Does $\overline{\partial} : C^\infty(X) \rightarrow \Omega^{0,1}(X)$ have closed image with respect to the Fréchet topology given by ...
1answer
60 views

### Disk with punctures and convex geodesical hull of the punctures isomorphic?

Consider a unit disk with marked points $z_i$, $i=1, \dots , n$ on its boundary. Let us call this surface $X$. As it is well known, the disk can be equipped with an hyperbolic metric and is then ...
0answers
78 views

### Spectral gaps for spin manifold Laplace spectrum

For a (compact) spin manifold, we know that the eigenvalues $\lambda_n$ of the Dirac operator are countable, with finite multiplicity, and satisfy  |\lambda_n| \to \infty, ~~~ \text{ as } n \to \...
0answers
78 views

### Condition for Integrability of an Almost Complex Structure

The following question concerns a remark made in the paper: Lebrun, C., Complete Ricci-flat K?hler metrics on $\mathbb{C}^n$ need not be flat, Proceedings of Symposia in Pure Mathematics, Volume 52 ...
0answers
187 views

### Is $h^1(X,O_X)$ always equal to the dimension of the Albanese?

Let $X$ be a projective integral scheme over $\mathbb{C}$. If $X$ is smooth, then $\mathrm{h}^1(X,\mathcal{O}_X)$ is the dimension of the Albanese variety of $X$. Probably, even if $X$ is normal, ...
0answers
95 views

### Toric Fan for the Du Val's singularities D_n and E_n

Let us consider the Du Val's singularities. i.e. https://en.wikipedia.org/wiki/Du_Val_singularity. It is well known that they are classified by ADE, because the exceptional divisors arising in the ...

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