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26
votes
2answers
881 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 ...
22
votes
3answers
2k views

Topologically distinct Calabi-Yau threefolds

In dimensions 1 and 2 there is only one, respectively 2, compact Kaehler manifolds with zero first Chern class, up to diffeomorphism. However, it is an open problem whether or not the number of ...
18
votes
2answers
2k 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 ...
16
votes
3answers
1k views

Holomorphic vector fields acting on Dolbeault cohomology

The question. Let $(X, J)$ be a complex manifold and $u$ a holomorphic vector field, i.e. $L_uJ = 0$. The holomorphicity of $u$ implies that the Lie derivative $L_u$ on forms preserves the (p,q) ...
16
votes
2answers
599 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 ...
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 ...
15
votes
1answer
551 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, ...
15
votes
2answers
845 views

When can you reverse the orientation of a complex manifold and still get a complex manifold?

I'm told that $\overline{\mathbb{C}P^2}$, i.e. $\mathbb{C}P^2$ with reverse orientation, is not a complex manifold. But for example, $\overline{\mathbb{C}}$ is still a complex manifold and ...
14
votes
2answers
600 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}} ...
14
votes
0answers
611 views

Horrible sets and blowups in Hubbard's Teichmuller theory

Edit: I can rephrase this question this way: When blowing up every point in the $x$-axis in $\mathbb{C}^2$ by means of an inverse limit of finite blowups, how can anything be 'left over'? The horrible ...
12
votes
2answers
584 views

Highly connected, compact complex manifolds

Here are four remarks about the homology and homotopy type of a compact, complex manifold $M$: If $M$ is Kähler, then it is symplectic and thus $H^2(M,\mathbb{R}) \ne 0$. (Also, as explained in a ...
12
votes
3answers
1k views

A topological consequence of Riemann-Roch in the almost complex case

This question originated from a conversation with Dmitry that took place here Is there a complex structure on the 6-sphere? The Hirzebruch-Riemann-Roch formula expresses the Euler characteristic of ...
12
votes
1answer
635 views

Splitting principle for holomorphic vector bundles

Let $E \to X$ be a vector bundle over a decent space $X$. Then there is a space $Z$ together with a map $p: Z \to X$ which induces a split injection on cohomology and such that $p^* E$ splits as a ...
10
votes
6answers
605 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 ...
10
votes
3answers
493 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 ...
9
votes
3answers
885 views

Primitive Cohomology Useful?

In her book, after proving the hodge decomposition, Voisin spends time discussing primitive cohomology $H^r(X, \mathbb{C})_{prim} = \ker L^{n-r+1} \subset H^r(X, \mathbb{C})$ (where $L$ is the ...
9
votes
1answer
342 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 ...
9
votes
2answers
461 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 ...
9
votes
1answer
301 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 ...
8
votes
1answer
406 views

necessary and sufficient condition for existence of $SU(3)$-structure on 6-manifolds

Is there any necessary and sufficient condition for existence of $SU(3)$-structure on 6-manifolds $M$?
8
votes
2answers
841 views

Torsion in cohomology of smooth manifolds

I've been interested in the possible (singular) cohomology groups of complex projective algebraic varieties, and there are lots of theorems that give various restrictions on these (Hodge ...
8
votes
1answer
703 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 ...
8
votes
1answer
215 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 ...
8
votes
0answers
271 views

Complexification of a complex manifold

Hi all, Let $M$ be a real-analytic manifold and let $N$ be a complexification of $M$ (in other words, $M$ sits in $N$ as a totally real submanifold). Suppose $M$ has an (integrable) complex ...
7
votes
2answers
331 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 ...
7
votes
1answer
607 views

Geometric interpretation of Simpson's correspondence

What is the exact geometric meaning of the Simpson's correspondence between Higgs bundles and local systems ? I know that it should have a rich geometric content but don't know an explicit geometric ...
7
votes
1answer
409 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, ...
7
votes
1answer
423 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 ...
6
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 ...
6
votes
2answers
313 views

Calabi-Yau manifolds and polygonal linkage configuration spaces: related?

I was reading about Calabi-Yau manifolds, about which I know little, and was wondering if these (or related complex manifolds, perhaps K3 surfaces) can be viewed as configuration spaces (or moduli ...
6
votes
1answer
1k views

Self-intersection and the normal bundle

Let $X/k$ be a surface nonsingular and proper over an algebraically closed field $k$. Let $C \subset X$ be a nonsingular curve. Then it is clear that the self-intersection $(C \cdot C)_X$ is ...
6
votes
2answers
880 views

Finite unramified analytic coverings vs finite etale coverings

Let $X$ be a smooth quasi-projective variety (so irreducible) over $\mathbf{C}$. We may think of $X$ as a complex manifold which we denote by $X^{an}$. Of course the topology on $X^{an}$ is finer ...
6
votes
1answer
531 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 ...
5
votes
2answers
522 views

On the fundamental group of hypersurfaces

Let $H$ be a smooth projective hypersurface in $\mathbb{P}^n(\mathbb{C})$ where $n\geq 3$. Then by the Lefschetz hyperplane theorem we have that $H^1(H,\mathbb{C})= ...
5
votes
3answers
448 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 ...
5
votes
1answer
304 views

Minimal distance spheres in complex projective spaces

My question has to do with distance spheres in $\mathbb CP^{n+1}$. I am interested in knowing what is the radius $r$ of a distance sphere $S(r)$ around a point that makes it a minimal submanifold ...
5
votes
1answer
527 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 ...
5
votes
1answer
544 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 ...
5
votes
0answers
169 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 ...
4
votes
2answers
582 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 ...
4
votes
1answer
436 views

Are complex varieties Kahler? - Algebraic, non-projective complex manifolds

Let $X/\mathbb{C}$ a nonsingular proper variety and $X_{an}$ it's associated analytic space. Is $X_{an}$ necessarily Kahler? Certainly we know this if $X$ is projective. A complex torus is algebraic ...
4
votes
2answers
313 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 ...
4
votes
2answers
412 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 ...
4
votes
2answers
262 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 ...
4
votes
1answer
280 views

Rosenlicht theorem about uniruledeness and zeroes of holomorphic vector field on complex projective manifold

I heard that there is a theorem due to Rosenlicht which says the following: Theorem. Let $X$ be a complex projective manifold and $V$ a non-trivial holomorphic vector field on $X$. Then $X$ is ...
4
votes
1answer
399 views

complex manifold with corner

I was reading Dominic Joycee article on Manifold with corner. He talk about manifold with corner modeled over $[0,\infty)^k\times \mathbb R^{n-k}$ for some $k\leq n$. From here i moved to Melrose ...
4
votes
0answers
237 views

Hirzebruch's ICM talk

Is there an English translation of Hirzebruch's 1958 ICM lecture note: KOMPLEXE MANNIGFALTIGKEITEN Thank you very much!
4
votes
0answers
270 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 ...
3
votes
2answers
811 views

holomorphy of inverse map

Let $M,N$ be complex manifolds and $f : M \to N$ be a bijective holomorphic map. Is then $f^{-1}$ also holomorphic? The open mapping theorem implies that $f^{-1}$ is continuous. In order to apply the ...
3
votes
2answers
733 views

Tangent bundle and normal bundle in self-product

$\newcommand{\I}{\mathcal{I}}$ Let $X$ a variety smooth over the complex numbers. Then we know that $\Omega_{X/\mathbb{C}}$ is the (usual) pullback of the conormal sheaf $\I/\I^2$ where $\I$ the sheaf ...