# 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.

1,979 questions
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### Question on PDEs which are related to certain geometric problems (e.g. Calabi conjecture, Gauduchon conjecture)

There are interesting symmetric functions $P_k$ arising from differential geometry and PDEs, where $P_k$ is given by \begin{aligned} P_k(\lambda) = \prod_{1\leq i_{1}<\cdots < ...
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### Hodge decomposition and degeneration of the spectral sequence

I am teaching a course on Hodge theory and I realised that I don't understand something basic. Let first $X$ be a compact Kahler manifold. Let $H^{p,q}(X)=H^q(X,\Omega^p_X)$ where $\Omega^p_X$ is the ...
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### Conformal mappings that preserve angles and areas but not perimeters?

Conformal mappings from $U$ to $V$, both subsets of $\mathbb{C}$, locally preserve angles. But, in general, such mappings neither preserve areas nor preserve perimeters. Q. Are there examples of ...
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### Hodge decomposition of the symmetric product of a curve

Let X be a smooth projective connected curve over $\mathbb{C}$ and let $n>1$ be an integer. Let $Y= Sym^n_X$ be the $n$-th symmetric product of $X$. Is there, for every $i$, a nice formula ...
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### What are hyperkähler metrics used for?

It seems that a lot of effort has been devoted to endow holomorphic-symplectic manifolds with hyperkähler metrics. It started with Calabi [4] with $T^*\mathbb{CP}^n$. Other examples include coadjoint ...
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### Sections of tangent bundle on hypersurface

If $X\subset \mathbb{P}^n$ is a smooth hypersurface (more generally a complete intersection) of dimension at least 2, and if $K_X+\mathscr{O}_X(-1)\geq 0$, why is it true that $H^0(T_X(1))=0$? (...
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### Proper modifications of $\mathbb{C}^{n}$

Let $f:X\to Y$ be a proper surjective holomorphic map between two $n$-dimensional connected complex manifolds $X$ and $Y$. $X$ is called a proper modification of $Y$ if there are nowhere dense compact ...
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### Can we extend a logarithmic form to some appropriate compactification?

Given some simple normal crossings divisor $D$ on a complex manifold $X$, which is not assumed to be compact. Given a form $\omega\in H^0(X,\Omega_X^1(\log D))$, when is it possible to find a ...
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### Complex Structure on Manifold of Maps

Suppose $M$ is a compact smooth manifold and $V$ is a compact complex manifold. I want to show that the spaces $C^{k,\alpha}(M,V)$ and $W^{k,p}(M,V)$ (the latter for $kp>\dim M$) are complex ...
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### Does the holomorphic curvature determine the connection?

Let $M$ be a simply connected complex manifold (of dimension greater than one), $L$ a line bundle, and $\nabla$ a connection on $L$ with possibly singularities along a divisor $D$. We define the ...
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### non-existence of global coordinates

Assume we have a smooth manifold, $M$, of dimension $n$. (An example of interest is the case when $M$ is a compact and orientable Riemann surface of genus $g$, but the question is intended to be broad....
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### Chow theorem in $\mathbb{C}^2$

I have the following question the answer to which I cannot find in the literature (but it must have been studied): Suppose that $M\subset\mathbb{C}^2$ is a real surface which may locally be written ...
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### Differential operator of globally unbounded order on connected complex manifold?

Let $X$ be a connected complex manifold. Consider the module $\mathcal{D}_X$. I recall hearing somewhere that one has to be careful with regards to differential operators that are not globally of ...
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### Universal cover of ladderly puntured complex plane

The twice punctured complex plane $\mathbb{C}-\{0,1\}$ has as its universal cover the upper half plane via elliptic modular function. I am looking for the constructions of the covering map from the ...
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### Is Serre duality related to Pontryagin duality?

I am wondering if there is some relationship between Serre duality and Pontryagin duality for compact complex manifolds. In this case Serre duality reduces to the commutativity of Hodge-star operator ...
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### Is singular Cauchy operator bounded in Morrey spaces?

The singular Cauchy operator is defined by $$S_\Gamma :f \to \int_\Gamma \frac{f(\xi)}{\xi-z} d\xi , z\in \Gamma.$$ Is this operator bounded in Morrey spaces and weighted Morrey spaces? i.e. is there ...
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### Can the base of an elliptically fibered Calabi-Yau threefold be an Enriques surface?

For this question, a Calabi-Yau manifold or variety of dimension $n$ is defined as a non-singular projective variety with trivial canonical bundle and $h^{i,0} = 0$ unless $i = 0$ or $i = n$. If ...
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### Good reduction of abelian varieties over valuation rings via coverings

Let $\mathcal{O}_K$ be a valuation ring with fraction field $K$, and let $A$ be an abelian variety over $K$. Suppose that there is a smooth proper scheme $\mathcal{X}$ over $\mathcal{O}_K$ whose ...
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### Do abelian varieties have Neron models over arbitrary valuation rings?

Let $\mathcal{O}_K$ be a valuation ring with fraction field $K$. Let $A$ be an abelian variety over $K$. Does $A$ have a Neron model? If $\mathcal{O}_K$ is a discrete valuation ring, then this is ...
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### Explicit descriptions of a flop

I want to know how to describe explicitly the flop of the following flop contraction. Because the construction is so natural and simple, I was wondering such descriptions should already exist in the ...
### What is the unitary $1$-parameter group generated by a vector field on a manifold?
If $M$ is a (compact, Riemannian) manifold, and one complexifies its tangent bundle, is it possible to give a meaning to the "unitary group"-like expression $t \mapsto \exp (\mathrm i t X)$ when $X$ ...