Questions tagged [complex-manifolds]

For questions about or involving complex manifolds.

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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 \...
Michael Albanese's user avatar
17 votes
3 answers
6k 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}_ {\mathbb{...
anonymous's user avatar
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28 votes
7 answers
7k views

Elementary Proof of Riemann-Roch for Compact Riemann Surfaces

I am supposed to give a talk about the Riemann-Roch theorem to a seminar of first and second year graduate students. I want to do Riemann-Roch for compact Riemann surfaces, but I am open to perhaps ...
user avatar
4 votes
2 answers
675 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 precisely,...
asv's user avatar
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4 votes
1 answer
445 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 ...
Coffee's user avatar
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32 votes
1 answer
1k views

About a claim by Gromov on proper holomorphic maps

At p. 223 of his paper [G03], Mikhail Gromov makes the following claim: Let $X$, $Y$ be two complex manifolds (not necessarily compact or Kähler) of the same dimension and having the same even Betti ...
Francesco Polizzi's user avatar
29 votes
1 answer
4k 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 ...
Michael Albanese's user avatar
23 votes
2 answers
4k 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 ...
Michael Albanese's user avatar
18 votes
2 answers
1k 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}} ...
Michael Albanese's user avatar
18 votes
3 answers
1k 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 projective) ...
YangMills's user avatar
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18 votes
1 answer
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Is there an integrable complex structure on $\mathrm{SU}(3)$?

Is there a complex manifold diffeomorphic to $\mathrm{SU}(3)$? This question arises in a StackExchange discussion by HK Lee, Ted Shifrin and Jason DeVito: https://math.stackexchange.com/questions/...
Andrew McHugh's user avatar
18 votes
4 answers
3k 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 ...
algori's user avatar
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10 votes
2 answers
569 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 ...
diverietti's user avatar
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10 votes
1 answer
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Is every surjective holomorphic self-map on a compact complex manifold finite-to-one?

I have already asked this question on stack exchange, but I didn’t get any answer. Let $X$ be a compact connected complex manifold. Let $f:X \to X$ be a surjective holomorphic map. Is it true that $f$...
Mayuresh L's user avatar
10 votes
2 answers
3k 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 decomposition,...
LMN's user avatar
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9 votes
1 answer
1k 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 ...
LMN's user avatar
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8 votes
1 answer
425 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 ...
asv's user avatar
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8 votes
1 answer
1k 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 ...
LMN's user avatar
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7 votes
1 answer
533 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 of $\...
Renato G. Bettiol's user avatar
7 votes
3 answers
4k views

Hodge Index theorem for Complex Manifolds

The Riemann-Roch theorem is a result about Riemann Surfaces that was extended to the Hirzebruch–Riemann–Roch theorem, a result about compact complex manifolds. The Hodge Index theorem is a result ...
Aston Smythe's user avatar
7 votes
1 answer
562 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, 1)-...
user3566's user avatar
  • 169
6 votes
2 answers
1k 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})= H^1(\mathbb{P}^n(\mathbb{C}),\...
Hugo Chapdelaine's user avatar
6 votes
2 answers
3k views

When a Riemannian manifold is of Hessian Typ

When a Riemannian manifold is of Hessian Type (i.e., a Riemannian manifold which its metric is Hessian)
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5 votes
0 answers
174 views

Extension of holomorphic maps to smooth family of holomorphic maps

Let $\pi:X \to D^2$ be a family of diffeomorphic (but not isomorphic) complex manifolds. Each fiber is allowed to have boundary but is compact (maybe not Stein) and $D^2 \subset \mathbb{C}$ is a ...
Paul's user avatar
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4 votes
2 answers
2k 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 ...
Martin Brandenburg's user avatar
3 votes
2 answers
4k 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 ...
LMN's user avatar
  • 3,525
3 votes
0 answers
188 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 "...
Mikhail Bondarko's user avatar
3 votes
1 answer
363 views

(1/2) K3 surface or half-K3 surface: Ways to think about it?

I heard from string theorists thinking of the so-called "(1/2) K3 surface" or "half-K3 surface" as follows: Let $T^2 \times S^1$ be a 3-torus with spin structure periodic in all directions. $T^2 \...
wonderich's user avatar
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2 votes
1 answer
718 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 ...
Rhys Davies's user avatar
2 votes
1 answer
111 views

Quasiconformal map from a subset of $\mathbb{C}$ to a polytope

Question. Does a quasiconformal map exist between a subset of $\mathbb{C}$ (such as a unit disc or rectangle) and a polytope? Here, I take a polytope to be a two-dimensional surface that could be ...
Talmsmen's user avatar
  • 577
2 votes
0 answers
130 views

Symmetric group-cocycle descends to symmetric product

Let $C$ be a complex curve with universal covering $\tilde{C}$ (which in my case is the upper half plane). Any group-cocylce $e \in H^1(\pi_1(C^n),H^0(\tilde{C}{}^n,\mathcal{O}^{\times}))$ defines a ...
KuSi's user avatar
  • 53
0 votes
0 answers
142 views

Limit of a sequence of smooth varieties in Hilbert scheme

Let $\{Z_i\}_{i=1}^\infty$ be a sequence of smooth irreducible $k$-dimensional submanifolds of $\mathbb{C}\mathbb{P}^n$ which converges to a closed subscheme $Z$ in the sense of the Hilbert scheme of $...
asv's user avatar
  • 21.1k
0 votes
2 answers
3k views

Line bundles with complex connection

Suppose that we have a complex manifold $X$, and a line bundle $L$ over $X$. It is known that the line bundles over $X$ are parametrized by their Chern class, the Chern class being the cohomology ...
Blake's user avatar
  • 1,025
0 votes
1 answer
211 views

What is a sufficient condition for summability of formel power series? [closed]

There are several kind of summability , i accrossed differents conditions for applying for example Borel summation or laplace transform which let me mixed and confused , really i don't know if i have ...
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