The complex-manifolds tag has no usage guidance.

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### 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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### Examples of pluripolar sets

I have a very basic question on pluripolar sets. First remind their definition.
Let $\Omega\subset \mathbb{C}^n$ be a domain. A subset $E\subset \Omega$ is called pluripolar if there exists a ...

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### Moduli of stable bundles - analytic approach

Consider a compact Riemann surface $X$ of genus $\ge 2$, and consider the set $M$ of stable holomorphic vector bundles of rank $n$ and degree $d$ on $X$, up to isomorphism.
At that point, one states ...

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### Kahlerness of the projectivized cotangent bundle [duplicate]

Let $X$ be a smooth not necessarily compact complex manifold which admits a Kahler metric. Is it true that its projectivized cotangent bundle also admits a Kahler metric? If not, are there sufficient ...

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### Why should the Kaehler form be closed? [closed]

As the question says, why should the Kaehler form be closed? Like people start from a fundamental 2-form (say, a 2-from $\mathcal{K}$) and then set they set the condition that in order for the ...

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### Can the $(n-1)$ power of the hermitian form on a compact complex $n$-fold be $\partial {\bar{\partial}}$-exact?

Let $\omega$ be the hermitian form for an hermitian metric on a compact complex manifold. Can $\omega^{n-1}$ be $\partial {\bar{\partial}}$-exact? (We know that our hermitian metric must necessarily ...

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### 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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### Is there a complex surface into which every Riemann surface embeds?

This question was previously asked on Math SE.
Every Riemann surface can be embedded in some complex projective space. In fact, every Riemann surface $\Sigma$ admits an embedding $\varphi : \Sigma ...

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234 views

### Is there an integrable complex structure on SU(3)?

Is there a complex manifold diffeomorphic to SU(3)?
This question arises in a StackExchange discussion by HK Lee, Ted Shifrin and Jason DeVito:
...

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### compact almost complex submanifolds of complex Lie groups

I find the following Corollary 1.21:
Question: does there exist any complex Lie groups $G$ such that there are some compact almost complex submanifolds (for example, $\mathbb{C}P^m$) of $G$? I want ...

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### 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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254 views

### Does there exist an algebraic space with large fundamental group but no finite etale covers by schemes

Does there exits a smooth proper algebraic space $X$ over $\mathbb C$ with "large" fundamental group such that no finite etale cover of $X$ is a scheme?
By "large" fundamental group I mean that $X$ ...

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### holomorphic K-theory

Topological K-theory is usually defined by setting $K(X)$ to be the groupification of the monoid $Vect_\mathbb{C}(X)$ of complex vector bundles over $X$ (with addition given by Whitney sum). However, ...

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### 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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### Vanishing of Dolbeault cohomologies and Steinness

That Stein manifolds have all $(p,q), p \geq 0, q \geq 1$ vanishing Dolbeault cohomology groups is more or less standard. I am a little bit confused about the reverse implication: whether the ...

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### 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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### 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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### 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 ...

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137 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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### 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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205 views

### A question on the twistor space of a manifold

Let $M$ be either (a) self-dual conformal 4-manifold, or (b) hypercomplex $4n$-manifold.
In either case one can construct the twistor space $Z$ (in the case (b) $Z=\mathbb{C}\mathbb{P}^1\times M$ as a ...

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### Flatness of a morphism of complex analytic spaces

Let $f\colon X\to D$ be a morphism of a complex analytic space $X$ into the 1-dimensional disk $D$. Assume for simplicity that $X$ has a single irreducible component which may not be reduced.
...

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### A weak analytic version of the valuative criterion of properness

EDIT: Let $f\colon X\to Y$ be a morphism of complex analytic spaces (not necessarily smooth or reduced). Assume that
(a) $f$ is injective on points;
(b) $f$ is local imbedding near each point $x\in ...

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203 views

### Relation between Milnor fiber and its restriction via vanishing cycles

I am reading these notes on nearby and vanishing cycles, where an initial assumption is made: the author talks about a complex analytic function $f:X\to \mathbb C$, assuming $X$ is closed in an open ...

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### Ricci flow on non-compact manifold

Suppose $\omega$ defines a Kähler metric on a non-compact complex manifold. Does
the Kähler-Ricci flow equation always have a solution (for small $t$)?

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### 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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### 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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### Moment map coordinates in tours action

I am trying to understand the proof of lemma 3.1, in this paper
In proof, they say that $g(dz_i,d\tau_k)=dz_i(\nabla\tau_k)=0$ I don't understand first and second equality.In second they say, ...

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### When a proper morphism of schemes is a closed imbedding?

Let $X$ and $Y$ be finitely presented schemes over $\mathbb{C}$. Let $f\colon X\to Y$ be a proper morphism. Let us assume that for any finitely presented scheme $S$ the induced map
$$Mor_{Sch}(S,X)\to ...

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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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### Connection between Strebel differentials, ribbon graphs, and Belyi maps

In this paper, a nice story is woven regarding the connection between quadratic differentials on Riemann surfaces, so-called 'ribbon graphs' drawn on those surfaces, and Belyi maps. However, I am ...

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### Different notions of convergence of complex subvarieties

Let $X$ be a smooth complex algebraic variety (or, better, complex analytic manifold). Let $\{C_i\}$ be a sequence of compact algebraic subvarieties (resp. analytic reduced subspaces) which converges ...

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### 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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### Hilbert scheme of a closed subscheme

Let $X$ be a complex algebraic variety. Its Hilbert scheme represents the functor $G$ from schemes to sets given by $$G(S)=\{Z\subset X\times S|\, Z \mbox{ is a closed subscheme, flat and proper over ...

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### Hilbert scheme of an infinitesimal neighborhood of a subvariety

Let $X$ be a complex algebraic variety. Let $C\subset X$ be a compact (reduced) subvariety. Let $C^{(n)}$ denote the $n$th infinitesimal neighborhood of $C$ inside $X$. Let $Hilb(X)$ denote the ...

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### Non-bimeromorphic compactifications

Let $X$ be a (smooth, non-compact) complex space. By a compactification of $X$ we mean a compact complex space $\overline X$ which contains a dense open subset biholomorphic to $X$ (we shall identify ...

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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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### 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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### 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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162 views

### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### Is a non-compact Riemann surface an open subset of a compact one ?

Let $X$ be a non-compact holomorphic manifold of dimension $1$. Is there a compact Riemann surface $\bar{X}$ suc that $X$ is biholomorphic to an open subset of $\bar{X}$ ?
Edit: To rule out the case ...