8
votes
1answer
184 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
105 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
110 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
113 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
249 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
141 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
284 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
70 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
223 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 ...
5
votes
1answer
487 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 ...
2
votes
1answer
128 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
232 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
297 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
260 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
117 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
119 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 ...
4
votes
2answers
410 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 ...
2
votes
0answers
216 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 ...
0
votes
1answer
175 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 ...
1
vote
1answer
312 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 ...
2
votes
2answers
383 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}_ ...
9
votes
1answer
321 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 ...
5
votes
3answers
439 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 ...
1
vote
0answers
81 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 ...
1
vote
1answer
199 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 ...
5
votes
0answers
166 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 ...
1
vote
2answers
187 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. ...
4
votes
2answers
302 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 ...
2
votes
1answer
246 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 ...
16
votes
2answers
580 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 ...
14
votes
2answers
564 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}} ...
15
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 ...
8
votes
1answer
687 views

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 ...
1
vote
2answers
692 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 ...
1
vote
1answer
389 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 ...
26
votes
2answers
856 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 ...
3
votes
1answer
721 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 ...
1
vote
2answers
202 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 ...
4
votes
0answers
263 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 ...
18
votes
2answers
1k views

Non-compact complex surfaces which are not Kähler

Not every complex manifold is a Kähler manifold (i.e. a manifold which can be equipped with a Kähler metric). All Riemann surfaces are Kähler, but in dimension two and above, at least for compact ...
2
votes
1answer
191 views

What does non-levi flat point mean geometrically

Hello, $CR$ manifold for example $S^1\times C^{n-1}$ is every where levi flat. Can I have example of $CR$ manifold which has at least one non levi flat point. I can't see what the happening in ...
7
votes
2answers
324 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 ...
0
votes
1answer
224 views

local kählerforms on complex manifold

hallo, Let $M$ be a complex manifold. Assume we have a covering of $M$ by complex charts $\{U_{i}\}$. Furthermore assume that we have on each $U_{i}$ a Kählerform $\omega_{i}$ (i.e. $d\omega_{i} = ...
3
votes
3answers
428 views

Does isomorphisms of sheaf of holomorphic sections implies isomorphisms of two holomorphic vector bundles over the same complex space ?

My knowledge is very limited for complex geometry. I have the following question: If we have two complex vector bundles $E\to X$ and $F\to X$ such that we have an isomorphism $\mathcal ...
9
votes
2answers
448 views

Classification of holomorphic disc bundles

I've had difficulty finding sources which treat the classification of holomorphic disc bundles over (compact and noncompact) Riemann surfaces. Note that by "bundle", I mean a holomorphic fiber ...
5
votes
1answer
525 views

On sufficient conditions on an analytic map to be algebraic(=regular)

Let $X$ and $Y$ be smooth quasi-projective varieties defined over $\mathbf{C}$ and let $$ f:X(\mathbf{C})\rightarrow Y(\mathbf{C}) $$ be a holomorphic map (not necessarily regular=algebraic). Then it ...
1
vote
0answers
216 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, ...
2
votes
1answer
207 views

biholomorphism complex manifold induced structure

Let $X$ be a $n$ dimensional complex manifold with complex structure $I$ and assume one has a diffeomorphism $f : \mathbb{C} \rightarrow X$ of some open set $U$ in $\mathbb{C}$ into its image $f(U)$. ...
1
vote
3answers
737 views

Examples of Lie Algebroids

The concept of a Lie Algebroid is given an important geometric meaning in the framework of Generalized Complex Geometry. For reference, the (barebones) definition of a Lie Algebroid is a vector bundle ...
7
votes
1answer
419 views

Complex analytic space with no (positive dim.) subscheme ?

Is there an example of a complex analytic space $X$ that doesn't have any (not necessarily open or closed) positive dimensional subspace $Y$ which is analytically isomorphic to (the complex ...