# 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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### Moduli space of almost complex structures as an algebro-geometric object

Let $M$ be a closed real-analytic manifold of dimension $2n$. Is it possible to make sense of the moduli space of real-analytic almost complex structures on $M$ as an algebro-geometric object (...

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### Affine algebraic variety as a set of common zeroes of holomorphic functions on ${\mathbb C}^n$

Let $V$ be an affine algebraic variety in ${\mathbb C}^n$, i.e. a set of common zeroes of some set $S$ of polynomials on ${\mathbb C}^n$:
$$
V=\{z\in {\mathbb C}^n:\ \forall p\in S\quad p(z)=0\}.
$$
...

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### Do analytic functionals form a cosheaf?

Let $X$ be a complex-analytic manifold.
Consider the sheaf of holomorphic functions $\mathcal{O}_X$ as a sheaf with values in the category of locally convex vector spaces. For $U\subseteq X$ open, we ...

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### Volume form under holomorphic automorphisms

$(M,\omega)$ is a compact Kaehler manifold and $f_{t,s}$ are 1-parameter group generated by holomorphic vector fields $V_s$. My question is whether the function $\frac{f_{t,s}^* \omega^n}{\omega^n}$ ...

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### Hypothetical uniqueness of an embedding of a Riemannian manifold to a compact Kähler one

Inspired by this question (Isometric embedding of a real-analytic Riemannian manifold in a compact Kähler manifold) I ask the following:
Suppose $X$ is a real analytic Riemannian manifold with a ...

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### Coordinate-free B.Feix's construction of a hyperkähler metric

In the 2001's paper 'Hyperk?hler metrics on cotangent bundles' B.Feix gives a construction of a hyperk?hler metric on a neighbourhood of zero section in $T^*X$ where $X$ is a real analytic K?hler ...

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### Symmetries for Julia sets of perturbations of polynomial maps

This is a naive question. Consider the
Julia sets
of the map
$$ z \mapsto z^n + \lambda / z^k $$
with $z,\lambda \in \mathbb{C}$, and the exponents $n,k \in \mathbb{N}$.
For example, for $n=k=3$, ...

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### 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 ...

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### 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.

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### 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 $M$ ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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, ...