The complex-manifolds tag has no usage guidance.

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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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**1**answer

257 views

### 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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418 views

### 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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596 views

### 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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**1**answer

322 views

### 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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302 views

### 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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**1**answer

839 views

### 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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311 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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**1**answer

167 views

### 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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161 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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185 views

### 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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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 ...

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**1**answer

592 views

### 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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956 views

### 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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**1**answer

421 views

### 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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**3**answers

480 views

### 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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659 views

### 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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339 views

### ${\bar{\partial}}$-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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661 views

### 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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**1**answer

473 views

### 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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89 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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### 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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280 views

### 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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202 views

### 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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673 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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**1**answer

2k views

### 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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328 views

### 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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1k views

### 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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**1**answer

599 views

### 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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**1**answer

296 views

### 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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379 views

### 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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**1**answer

272 views

### 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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351 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 ...

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356 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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773 views

### Moishezon manifolds with vanishing first Chern class

Suppose $M$ is a Moishezon manifold with $c_1(M)=0$ in $H^2(M,\mathbb{R})$. Does it follow that $K_M$ is torsion in $\mathrm{Pic}(M)$?
This is true whenever $M$ is Kähler (and therefore ...

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**1**answer

277 views

### recurrence formula for *i*-th Chern class of $CP^n$

one can show that the relation between first Chern class and second Chern class of $CP^n$ is
$\frac{2(n+1)}{n} c_2 (M)=c_1 (M)^2$
here $c_1 (M)^2=c_1 (M)∧c_1 (M)$.
So is there any recurrence ...

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750 views

### Does equality of Laplacians imply Kähler?

This question follows on from this one.
Let $(X, \omega)$ be a Hermitian manifold and define the Laplacians $\Delta_{\partial} = \partial\partial^* + \partial^*\partial$ and $\Delta_{\bar{\partial}} ...

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2k views

### Almost Complex Structure approach to Deformation of Compact Complex Manifolds

I don't know much about the deformation of compact complex manifolds, I've only read chapter 6 of Huybrechts' book Complex Geometry: An Introduction. There are two parts to this chapter. The second ...

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**1**answer

804 views

### recognizing Kahler manifolds of complex dimension n

Is there new classification of Kahler manifolds of complex dimension n and new results for necessary and sufficient conditions for a manifold being Kahler? I know if redactivity of Lie algebra on ...

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### Question about an estimate in Hörmander's proof of Cartan's Theorem B

I have been working through the proof of Cartan's Theorem B that Hörmander gives in his book 'Introduction to Complex Analysis in Several Variables'. When I began, I skipped over some of the initial ...

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763 views

### On the algberaicity of the universal elliptic curve associated to a torsion free subgroup

So let $\Gamma\subseteq SL_2(\mathbf{Z})$ be a finite index subgroup (not necessarily a congruence subgroup). Recall that we have an action of $SL_2(\mathbf{R})$ on
...

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471 views

### How to study the nonregular part of a finite branched holomorphic covering?

A finite branched holomorphic covering is a holomorphic map $f : V \to W$
between holomorphic varieties $V$ and $W$ such that
$f$ is a finite branched covering (in the topological sense)
There is a ...

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**1**answer

594 views

### Are Lefschetz thimbles holomorphic manifolds?

I have a Lefschetz thimble defined by the stable flow of the gradient a holomorphic function
toward a critical point (as defined e.g. in Witten arXiv:1001.2933 and F.Pham "Vanishing homologies and the ...

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958 views

### Can the holomorphic image of $(\mathbb{C}^*)^n$ be open but not dense

Let $M$ be a compact complex connected [but not necessarily kähler] $n$-manifold, and suppose we have a holomorphic map $$(\mathbb{C}^*)^n \to M$$ such that the image is open. Is the image ...

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451 views

### complex manifold with corner

I was reading Dominic Joycee article on Manifold with corner. He talk about manifold with corner modeled over $[0,\infty)^k\times \mathbb R^{n-k}$ for some $k\leq n$. From here i moved to Melrose ...

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890 views

### Algebraic De Rham cup product versus Betti cup product

Let $X$ be a smooth projective variety over $\mathbf{C}$ of complex dimension $2n$. Let
$C_1,C_2\subseteq X$ be two closed subvarieties of complex dimension $n$.
Then we get two Betti homology ...

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218 views

### Analytic isomorphisms above two etale maps

Le $X_1$, $X_2$ and $Z$ be smooth quasi-projective connected varieties defined over $\mathbf{C}$. Let $p_1:X_1\rightarrow Z$ and $p_2:X_2\rightarrow Z$ be finite etale maps. Assume that ...