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2
votes
0answers
65 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
197 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
237 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 $X$ ...
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 ...
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 ...
14
votes
2answers
334 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 ...
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 ...
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
158 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
205 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 $t$)?
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 ...
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 ...
2
votes
2answers
316 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 ...
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 ...
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 ...
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
305 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 ...
6
votes
1answer
676 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 ...
4
votes
2answers
700 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 ...
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
1answer
341 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 ...
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 ...
3
votes
1answer
126 views

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 ...
2
votes
1answer
150 views

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 ...
1
vote
1answer
250 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 ...
4
votes
2answers
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 ...
10
votes
3answers
583 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 ...
1
vote
1answer
321 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) ...
4
votes
2answers
287 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 ...
5
votes
1answer
787 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 ...
2
votes
0answers
293 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 ...