# Tagged Questions

**1**

vote

**1**answer

92 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

**2**answers

233 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

**1**answer

136 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

**1**answer

280 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

**0**answers

66 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

**1**answer

216 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

**1**answer

474 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

**1**answer

126 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

**0**answers

230 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

**1**answer

295 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

**3**answers

254 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

**1**answer

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

**1**answer

113 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

**2**answers

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

**0**answers

214 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

**1**answer

174 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

**1**answer

300 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

**2**answers

368 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

**1**answer

316 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

**3**answers

436 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

**0**answers

80 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

**1**answer

194 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

**0**answers

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

**2**answers

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

**2**answers

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

**1**answer

242 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

**2**answers

579 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

**2**answers

561 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

**1**answer

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

**1**answer

684 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

**2**answers

688 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

**1**answer

385 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

**2**answers

853 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

**1**answer

710 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

**2**answers

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

**0**answers

258 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

**2**answers

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

**1**answer

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

**2**answers

322 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

**1**answer

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

**3**answers

425 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

**2**answers

446 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

**1**answer

523 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

**0**answers

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

**1**answer

205 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

**3**answers

734 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

**1**answer

416 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 ...

**7**

votes

**1**answer

406 views

### Is it always possible to extend a closed (1, 1)-form on a divisor to a closed (1, 1)-form on a tubular neighbourhood?

Let $X$ be a compact Kahler manifold, let $D$ be a smooth divisor in $X$, and let $U$ be a tubular neighbourhood of $D$ in $X$. Suppose that $D$ is Fano. Is it possible to extend every closed (1, ...

**2**

votes

**4**answers

840 views

### Noncompact Kahler manifolds with nonzero Ricci tensor but vanishing scalar curvature

Let us consider a noncompact K\"{a}hler manifold with vanishing scalar curvature but nonzero Ricci tensor. I'm wondering what can it tell us about the manifold. The example (coming from physics) has ...

**3**

votes

**2**answers

517 views

### Torsion in the Betti cohomology of complex surfaces

Q1. So what is the "simplest example" of a compact complex manifold of dimenension $2$, say $X$, for which $H_B^2(X,\mathbb{Z})$ has non-trivial torsion.
Q2. How do we think about these torsion ...