The complex-manifolds tag has no usage guidance.

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### How much of modern algebraic geometry is there in modern complex(algebraic, analytic, differential) geometry?

Good day to you, people of mathoverflow. I'll get to the point. I wonder how much of modern abstract algebraic geometry is there in modern complex geometry?
What do I mean by complex geometry? ...

**16**

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**0**answers

683 views

### 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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366 views

### 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 structure....

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194 views

### Toroidal compactifications

Does anyone know if there is an intrinsic definition of a toroidal compactification (over $\mathbb{C}$)?
Something like: Let $X$ be an algebraic variety over $\mathbb{C}$. Then $X \subset \bar{X}$ ...

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197 views

### 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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**0**answers

260 views

### Hirzebruch's ICM talk

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

**4**

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**0**answers

360 views

### Etymology of the O-notation for algebras of holomorphic functions

The notation $O(X)$ seems to be a quite standard notation for the algebra of all holomorphic functions on some connected domain in $\mathbb{C}^n$ (or a complex manifold). I would like to know where ...

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542 views

### What is the nature of the zero locus of a section of a coherent sheaf?

Suppose that $X$ is a reduced rigid space and $\scr{F}$ is a coherent sheaf on $X$. For a section $f\in {\scr F}(X)$, the zero locus of $f$ is the set of points $x\in X$ at which $f$ vanishes in the ...

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99 views

### How many compact complex 3-folds with $b^1 = b^2=h^{1,2}=0$?

Are there any compact complex 3-folds with Betti numbers, $b^1 = b^2 = 0 $ and Hodge number, $h^{1,2}=0$? If yes, then how plentiful are they?

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61 views

### What is the relation between holomorphic blow-up and symplectic blow-up?

McDuff has shown us exactly how the symplectic blow-up procedure along a symplectic submanifold affects the symplectic structure in the ambient space, i.e., if $\omega$ is the original symplectic form,...

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122 views

### Do Kähler realizations of symplectic manifolds exist?

If $(S, \omega)$ is a smooth (not necessarily analytic!) symplectic manifold, does there exist a (almost-)Kähler manifold $K$ and a surjective Poisson submersion $\pi : K \to S$?
Think of $\Bbb R ^{...

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121 views

### K-theory of coherent sheaves on complex manifolds: references and gamma-filtration?

For a complex manifold $X$ one has an exact category of locally free coherent sheaves; so it seems to be no problem to define certain $K$-theory (I do not know whether the $K$-groups given by the "...

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110 views

### Generalized Isotropic almost complex structures

Let $(M,g)$ be a Riemannian manifold, $TM$ it's tangent bundle, $\mathcal{H}TM$ be the horizontal sub-space of $TTM$ with respect to $g$, $\mathcal{V}TM$ be the vertical sub-space of $TTM$ and $K$ be ...

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96 views

### Darboux-like coordinates on a Kähler manifold

If $(M, g, J, \Omega)$ is a Kähler manifold, do there exist local coordinates in which 2 out of the 3 geometrical structures look nice? I have Darboux coordinates in which $\Omega$ looks nice, but ...

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96 views

### Can a class be represented by both a $(p,q)$ form and a $(p',q')$ form?

Suppose $X$ is a complex manifold.
If $X$ is Kahler, the cohomology groups decompose into subgroups represented by $(p,q)$ forms.
If $X$ is not Kahler, I think the decomposition may not hold?
Is ...

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73 views

### Sections of inverse image sheaf of sheaf of sections of vector bundle

Let $\eta \colon Y \to Z$ be holomorphic map between complex manifolds, $E$ holomorphic vector bundle over $Z$, $\eta ^\ast E$ its pullback over $Y$, ${\cal O}_E$ sheaf of sections of $E$, and $\eta^{-...

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146 views

### Coherence of $\mathcal O_X[T]$

Let $X$ a complex manifold, and $\mathcal O_X$ the sheaf of holomorphic functions. Oka Coherence Theorem states that $\mathcal O_X$ is coherent (as $\mathcal O_X$-module).
How to prove that also the ...

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**0**answers

326 views

### 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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164 views

### 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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165 views

### 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 $p,...

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357 views

### 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 ...

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83 views

### Riemann surface defined by a Beltrami differential

Let $R$ be a Riemann surface, $\omega$ and $\tau$ respectively a holomorphic and an antiholomorphic 1-form on $R$. Locally $\omega=fdz$ and $\tau=gd\overline{z}$ with $\partial_{\overline{z}}f=0$ and $...

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122 views

### Homeomorphism of fibers of holomorphic maps

EDIT (after the comment by Jason Starr): Let $X$ be a complex algebraic (or, more generally, analytic) variety, possibly singular and non-compact. Let $f\colon X\to D^*$ be a proper algebraic morphism ...

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130 views

### Exactness of the relative de Rham complex restricted to subschemes

I think that the statement below about relative de Rham complex is true (am I wrong?) If it is the case, a reference would be very helpful. (I admit that the statement sounds somewhat technical and ...

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91 views

### 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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220 views

### glue together a sequence of holomorphic forms

hallo,
my problem is the following: i have a finite sequence of holomorphic $k-$forms $\alpha_{k}$, each defined on open subsets $U_{k} \subset M$, where $M$ is a complex $n$-dimensional manifold, ...

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79 views

### How to investigate the harmonocity of holomorphic vector fields?

Let $(M,g,J)$ be a Kahler manifold and $\nabla$ be its Levi-Civita connection. We know that $\Delta _gX=||\nabla X||^2X$ is the characterizing equation for harmonic unit vector fields.
I dont know ...

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76 views

### Hurwitz's theorem for a system of functions

First, let me define a notation of $H(G_1\times G_2 \times \ldots \times G_m)$.
We say that $f\in H(G_1\times G_2 \times \ldots \times G_m)$ if $$f:G_1\times G_2 \times \ldots \times G_m \rightarrow \...

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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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70 views

### 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 ...