The complex-manifolds tag has no wiki summary.

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### Basic questions on the Hilbert scheme/ Douady space

Let $X$ be a complex projective scheme (resp. complex analytic space). The Hilbert scheme (resp. Douady space) parameterizes closed subschemes (resp. complex analytic subspaces) of $X$. More ...

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### Flat family with special fiber $\mathbb{C}\mathbb{P}^1$

Let $C=Spec \mathbb{C}[t]/(t^{n+1})$. Let $X$ be an algebraic (or complex analytic) scheme, flat over $C$ with the structure morphism $f\colon X\to C$. Assume that the special fiber is isomorphic to ...

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### When flatness of a morphism implies smoothness?

EDIT: Let $f\colon X\to C$ be a flat proper morphism of complex algebraic (or analytic) varieties. Assume the special fiber over a point $p\in C$ is smooth.
Is it true that there exists a ...

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### A question related to the Grauert semi-continuity theorem

Let $f\colon X\to Y$ be a proper holomorphic map of complex analytic manifolds. Assume $f$ to be submersive for simplicity, but probably it is not important. Let $\mathcal{F}$ be a coherent sheaf on ...

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### A generalization of the Grauert direct image theorem

EDIT: Let $f\colon X\to Y$ be proper holomorphic submersive map of complex analytic manifolds. Let $\mathcal{F}$ be the sheaf of holomorphic sections of a holomorphic vector bundle over $X$. Assume ...

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### Kodaira-Spencer theory of deformation done right

I thought in asking this question on Math StackExchange, but by my experience I don' t think anyone will notice me. Recently, I started studying deformation of complex manifolds in the sense of ...

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### Why Cech cohomology does not compute sheaf cohomology on an open annulus

Let $A=\{z\in\mathbf{C}:1/2<|z|<1\}$ be an open annulus. Let us cover $A$ by 3 open sets:
$U_0,U_1$ and $U_2$ which we assume to be all homeomorphic to a 2 dimensional open disc. Moreover, we ...

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### What are the known examples of compact complex $n$-folds that do not have a strongly Gauduchon metric for $n >2$?

Gauduchon showed that every conformal hermitian structure on a compact complex $n$-fold contains an hermitian metric such that the associated 1,1-form $\omega$ satisfies $\partial ...

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### Hirzebruch's ICM talk

Is there an English translation of Hirzebruch's 1958 ICM lecture note:
KOMPLEXE MANNIGFALTIGKEITEN
Thank you very much!

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### Complex analytic set in infinite dimension

Let $E$ and $F$ be (infinite dimensional) complex Banach spaces and $f : E \rightarrow F$ a holomorphic map. Set $M = f^{-1}(\{0\})$. I have found the following results in the literature :
1) if no ...

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### Definition of a complex space

In the definition of a complex space (in the sense of Grauert), one defines a model space (one to which we require a complex space to be locally isomorphic) to be the support of the quotient sheaf ...

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### Solutions of the $\overline{\partial}$ equation in the upper half-plane

I am looking for references on solutions of the $\overline{\partial}$ equation in the upper half-plane (with conditions on the boundary). But this subject is completely outside my area of expertise ...

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### Existence of a map between automorphism group of universal covers

Let $f:X\to Y$ be a holomorphic map of holomorphic manifolds. You can assume that $dimY=1$. Let $\tilde X$ and $\tilde Y$ be universal covers of $X$ and $Y$ with group of holomorphic automorphisms ...

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### Off-diagonal holomorphic extension of real analytic functions on $\mathbb{C}^n \times\mathbb{C}^n$

I am struggling trying to understand an statement in a paper I am reading:
Let $M$ be a complex manifold of dimension $2n$. Let's consider a function $\xi$: $M$ $\rightarrow$ $\mathbb{C}$ whose ...

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### Do versions of the Nakai-Moishezon and Kleiman criteria hold for Moishezon manifolds, or other 'nice' spaces?

As I understand it, the Nakai-Moishezon criterion gives conditions for the existence of an ample divisor class on an arbitrary proper scheme, and Kleiman's criterion does the same for arbitrary ...

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### When is $\mathrm{Aut}(X)\times \mathrm{Aut}(Y)$ of finite index in $\mathrm{Aut}(X\times Y)$?

Let $\mathrm{Aut}(X)$ denote the group of biholomorphic autmorphisms of the (non-compact) complex manifold $X$. If $X$ and $Y$ are two (non-compact) complex manifolds, then $\mathrm{Aut}(X)$ and ...

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### Foliations by holomorphic curves on complex surfaces

On a complex surface, does there exist a non-singular foliation by holomorphic curves that is NOT a holomorphic foliation, i.e. transversally holomorphic foliation?
The surface should be compact and ...

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### Stokes theorem for Grassmanians

This question is extensively rewritten by David Speyer; the original version is below.
The Grassmannian $G(k,n)$ is the quotient $SU(n)/S[U(k) \times U(n-k)]$. Let's write $\pi$ for the map $SU(n) ...

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### Cohomology of submanifold complements

Let $X$ be a finite-dimensional complex manifold (possibly non-compact). Let $\mathcal{H}$ be a union of codimension-$1$ submanifolds such that the local picture is that of intersecting hyperplanes. I ...

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### Kodaira Spencer map and versal deformation

First I want to clarify what I mean by the Kodaira-Spencer map.
Let's have a family of deformations $\pi:\mathcal{X}\rightarrow B$ of a complex manifold $X=\mathcal{X}_0:=f^{-1}(0)$ (by that I mean ...

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### Least area minimal hypersurface of $\mathbb C P^{n+1}$

After a few lectures on min-max for minimal hypersurfaces and isoperimetric problems, and seeing in several instances that the least area minimal hypersurface of the round sphere is an equator, I was ...

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### Holomorphic separation and the existence of strictly plurisubharmonic functions

Recall that a complex manifold is Stein if it is holomorphically convex and separable. If we assume holomorphically convex alone, then there is Cartan-Remmert reduction to say how far it is from being ...

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### complex Morse function on a four-manifold

If we have a complex Morse function on a complex four-manifold, $f: X\to \mathbb{C} $, can we tell from the function how the genus of inverse images $f^{-1}(z)$ (for regular values) may change? under ...

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### Holomorphic objects associated with a compact complex manifold?

Good morning,
I'm just curious about the following. With a compact Kahler manifold, we can associate an Albanese torus. This helps us a lot study the manifold.
My question: Are there other ...

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### Another fibration with a given singular fiber class.

Let $f:X\rightarrow C$ be a fibration of complex manifold over a smooth curve $C$. Let $D$ be an irreducible component of singular fibers. How can one prove that $X$ does not admit any fibration with ...

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### Toroidal embedding

Its known ( see " The birational geometry of degenerations") that there exist a smooth one parameter family (i.e. total space is smooth) of two dimensional complex toris over unit disk whose central ...

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### On a Hirzebruch surface.

I am trying to solve exercise in Huybrechts's book 'Complex geometry'
While solving problems, one problem kept me from going forward.
That is,
The surface $\Sigma_n=\mathbb{P}$ $(\mathcal{O}_ ...

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### Betti numbers of Proper nonprojective varieties

This is a question about pathologies.
Let $X/\mathbb{C}$ be an irreducible projective variety smooth over $\mathbb{C}$. Then, the singular cohomology groups $H^i(X, \mathbb{C})$ have a hodge ...

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### Two questions on complex geometry

I have two questions on complex geometry.
First one is that why the existence of almost complex structure on tangent bundle on real 2n-dimensional manifold is a topological question?
Wikipedia ...

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### When a Riemannian manifold is of Hessian Typ

When a Riemannian manifold is of Hessian Type (i.e., a Riemannian manifold which its metric is Hessian)

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### ${\bar{\partial}}$-geometrically formal ?

A compact Riemannian manifold is called geometrically formal if the wedge product of two $d$-harmonic forms is $d$-harmonic. Are there any known results for when a non-Kahler compact complex ...

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### When does $Aut(X)=Bir(X)$ hold?

Let $X$ be a projective complex manifold. Under what condition do we have the equality $Aut(X)=Bir(X)$? Here $Aut(X)$ denotes the group of holomorphic automorphisms of $X$ and $Bir(X)$ the group of ...

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### necessary and sufficient condition for existence of $SU(3)$-structure on 6-manifolds

Is there any necessary and sufficient condition for existence of $SU(3)$-structure on 6-manifolds $M$?

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### Primitive Cohomology Useful?

In her book, after proving the hodge decomposition, Voisin spends time discussing primitive cohomology $H^r(X, \mathbb{C})_{prim} = \ker L^{n-r+1} \subset H^r(X, \mathbb{C})$ (where $L$ is the ...

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### Divisors, factorisations of matrix valued functions, and the Lorentz group

How to construct a complex projective variety with several classes of non-intersecting divisors? How to keep the answer concrete and simple, so that explicit calculations can be done? And the problem ...

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### Torsion in cohomology of smooth manifolds

I've been interested in the possible (singular) cohomology groups of complex projective algebraic varieties, and there are lots of theorems that give various restrictions on these (Hodge ...

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### On linear automorphism on positive definite matrices.

I saw a statement in [Murakami, On automorphisms on Siegel domains] that every linear automorphism $\phi$ on the set of positive definite matrices can be represented as conjugation: i.e. there is a ...

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### Basis for hodge decomposition of Elliptic K3 Surfaces

We know the hodge numbers of K3 Surfaces. To work out some ideas, I'd like to know an explicit basis for the hodge decomposition $H^{p,q}$ of a smooth Elliptic K3 Surface over $\mathbb{C}$ (for all ...

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### How do fibers of the functor Algebraic Varieties $\to$ Complex Analytic Spaces look like?

There's already a question (which got several interesting answers) asking about examples of the phenomenon of non (essential) injectivity of the functor $U:Alg\to AnEsp$, mapping each algebraic ...

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### k-Hyperbolic manifolds

A complex manifold $N$ is $k$-hyperbolic ($\dim N \geq k$) if any holomorphic map from $\mathbb C^k$ to $N$ has rank strictly less than k. Brody hyperbolic manifolds are $1$-hyperbolic for example. ...

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### Horrible sets and blowups in Hubbard's Teichmuller theory

Edit: I can rephrase this question this way: When blowing up every point in the $x$-axis in $\mathbb{C}^2$ by means of an inverse limit of finite blowups, how can anything be 'left over'? The horrible ...

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### Self-intersection and the normal bundle

Let $X/k$ be a surface nonsingular and proper over an algebraically closed field $k$. Let $C \subset X$ be a nonsingular curve. Then it is clear that the self-intersection $(C \cdot C)_X$ is ...

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### Complex structures on $R^{2N}$ with complex annulus

Let $M$ be a complex manifold of dimension $N\ge2$ such that
$\qquad$(1) $M$ is diffeomorphic to $R^{2N}$,
$\qquad$(2) There is a compact set $K\subseteq M$ such that $M\setminus K$ is biholomorphic ...

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### Tangent bundle and normal bundle in self-product

$\newcommand{\I}{\mathcal{I}}$ Let $X$ a variety smooth over the complex numbers. Then we know that $\Omega_{X/\mathbb{C}}$ is the (usual) pullback of the conormal sheaf $\I/\I^2$ where $\I$ the sheaf ...

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### Are complex varieties Kahler? - Algebraic, non-projective complex manifolds

Let $X/\mathbb{C}$ a nonsingular proper variety and $X_{an}$ it's associated analytic space. Is $X_{an}$ necessarily Kahler? Certainly we know this if $X$ is projective.
A complex torus is algebraic ...

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### Inclusion of logarithmic de-Rham complex into differentials

Let $X$ be a complex manifold and $D$ a normal crossing divisor. Let $U = X - D$ and $j: U \rightarrow X$ the natural map. Voisen observes that there is a natural inclusion $$\Omega^k_X(\log D) ...

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### Minimal distance spheres in complex projective spaces

My question has to do with distance spheres in $\mathbb CP^{n+1}$. I am interested in knowing what is the radius $r$ of a distance sphere $S(r)$ around a point that makes it a minimal submanifold ...

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### On the jacobian origin of CM abelian varieties

Let $K$ be a CM field of degree $2n$ over $\mathbf{Q}$ and let $\mathcal{O}_K$ be its ring of integers. Let $\Phi=(\phi_1,\ldots,\phi_n)$ be a CM type of K. Then it is known that complex torus ...

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### Complexification of a complex manifold

Hi all,
Let $M$ be a real-analytic manifold and let $N$ be a complexification of $M$ (in other words, $M$ sits in $N$ as a totally real submanifold). Suppose $M$ has an (integrable) complex ...

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### On $\pi_1$ of an algebraic surface

I have the following question. Any input would be appreciated. If you have a counterexample, I would be very interested to know.
Question. Let $(X, \omega, J)$ be a Kahler manifold of complex ...