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2
votes
0answers
51 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 ...
1
vote
2answers
112 views

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 ...
2
votes
1answer
195 views

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 ...
0
votes
0answers
67 views

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 ...
5
votes
1answer
181 views

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 ...
3
votes
1answer
83 views

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 ...
1
vote
0answers
117 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 ...
57
votes
1answer
2k views

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 ...
13
votes
1answer
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: ...
3
votes
1answer
69 views

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 ...
2
votes
0answers
90 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 ...
7
votes
1answer
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 $X$ ...
18
votes
1answer
712 views

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, ...
0
votes
0answers
70 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 ...
14
votes
2answers
333 views

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 ...
16
votes
2answers
739 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}} ...
3
votes
0answers
117 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 ...
2
votes
0answers
68 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 ...
1
vote
0answers
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 ...
19
votes
1answer
1k 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 ...
7
votes
1answer
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 ...
2
votes
1answer
157 views

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. ...
3
votes
1answer
170 views

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 ...
1
vote
1answer
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 ...
0
votes
0answers
130 views

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 $t$)?
17
votes
3answers
755 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 ...
3
votes
0answers
157 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 $X \subset \bar{X}$ ...
0
votes
1answer
227 views

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, ...
0
votes
1answer
155 views

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 ...
9
votes
1answer
335 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 ...
3
votes
1answer
199 views

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 ...
8
votes
1answer
255 views

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 ...
1
vote
0answers
127 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 ...
0
votes
1answer
142 views

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 ...
1
vote
1answer
139 views

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 ...
7
votes
2answers
360 views

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 ...
2
votes
2answers
315 views

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 ...
6
votes
1answer
674 views

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 ...
1
vote
1answer
313 views

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 ...
3
votes
1answer
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 ...
0
votes
0answers
117 views

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 ...
1
vote
1answer
303 views

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 ...
10
votes
6answers
643 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 ...
4
votes
2answers
698 views

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 ...
2
votes
1answer
176 views

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 ...
2
votes
3answers
322 views

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 ...
4
votes
0answers
251 views

Hirzebruch's ICM talk

Is there an English translation of Hirzebruch's 1958 ICM lecture note: KOMPLEXE MANNIGFALTIGKEITEN Thank you very much!
2
votes
2answers
855 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}_ ...
2
votes
1answer
340 views

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 ...
7
votes
3answers
1k views

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 ...