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18 votes
2 answers
1k views

Proving algebraicity of compact Riemann surfaces without Chow's theorem

I am trying to write a report for a complex analysis class where I prove Riemann-Roch and apply it to prove algebraicity of compact Riemann surfaces. While writing this, I found that Riemann-Roch ...
29 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 ...
16 votes
0 answers
519 views

Gabriel's theorem for complex analytic spaces

Let $X,Y$ be noetherian schemes over $\mathbb{C}$. Then, it is known that $$ \text{Coh}(X) \simeq \text{Coh}(Y) \Rightarrow X \simeq Y, $$ by P. Gabriel(1962). Are there some results in the case of ...
1 vote
0 answers
156 views

Top cohomology of the canonical class of a compact non-Kähler manifold

Let $X$ be a complex compact manifold of complex dimension $n$. Let $K_X$ denote its canonical class. Is it true that the cohomology group $$H^n(X,K_X)$$ is one dimensional? Remark. If $X$ is Kähler ...
7 votes
1 answer
529 views

Do non-projective K3 surfaces have rational curves?

Define a compact Kähler surface $X$ to be a K3 surface if $X$ is simply connected, $K_X \simeq \mathcal{O}_X$, and $h^{0,1}=0$. If $X$ is projective, then a theorem typically attributed to Bogomolov ...
2 votes
1 answer
721 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 ...
4 votes
0 answers
197 views

Approximation of a holomorphic function vanishing at a submanifold by polynomials

Suppose $M$ is a complex affine algebraic manifold in ${\mathbb C}^n$ (I mean, a set of common zeroes of a system of polynomials on ${\mathbb C}^n$, which is at the same time a smooth manifold). ...
1 vote
0 answers
113 views

Quasi-plurisubharmonic function with polynomial decay

Let $(M, \omega)$ be a compact Kähler manifold. An $\omega$-quasi-plurisubharmonic function on $M$ is an upper semi-continuous function $\varphi : M \to \mathbb{R} \cup \{ - \infty \}$ such that $\...
3 votes
1 answer
292 views

Equivalent definitions of normality for complex algebraic varieties

In Kollár's article The structure of algebraic threefolds: an introduction to Mori's program he gives the following definition of a normal variety: Definition 5.4. Let $V \subset \mathbb{C}^n$ be an ...
5 votes
0 answers
189 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 ...
11 votes
1 answer
860 views

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 $f$...
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 ...
2 votes
0 answers
253 views

cohomology classes of complex submanifolds

I was wondering if there were restrictions in what the cohomology classes corresponding to complex submanifolds of a complex manifold could be. For example, say $T^4$ is regarded as a complex ...
8 votes
1 answer
431 views

Holomorphic deformation of complex structure on the real plane

It is known that each complex structures on $\mathbb{R}^2$ is biholomorphic to either $\mathbb{C}$ or the open unit disk $\Delta$. One can continuously deform one complex structure to the other as is ...
6 votes
1 answer
317 views

Cohomology of neighborhood of $\mathbb{C}\mathbb{P}^1$ in $\mathbb{C}\mathbb{P}^n$

Let $\mathbb{C}\mathbb{P}^1$ be embedded linearly to $\mathbb{C}\mathbb{P}^n$ with $n>1$. (Such an embedding is given in coordinates by $[x:y]\mapsto [x:y:0:\dots: 0]$.) Is it true that for any ...
3 votes
0 answers
169 views

Cohomology of a tubular neiborhood of submanifold vs cohomology of the formal neighborhood

Let $X$ be a smooth complex analytic manifold. Let $Z\subset X$ be a smooth compact analytic submanifold. Let $\hat Z$ be the formal neighborhood of $Z$. Let $U$ be an open neighborhood of $Z$. Is ...
8 votes
0 answers
277 views

Cohomology of complex manifold vs cohomology of its complex submanifold

Let $X$ be a smooth complex analytic manifold. Let $Z\subset X$ be a smooth compact analytic submanifold. Let $A$ be a holomorphic vector bundle over $X$. Assume that $$H^i(Z, A|_Z)=0 \mbox{ for any } ...
1 vote
0 answers
217 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 ...
2 votes
1 answer
314 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. ...
5 votes
1 answer
366 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
1 answer
273 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 ...
8 votes
1 answer
431 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
0 answers
156 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
1 answer
304 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
1 answer
252 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 ...
4 votes
2 answers
700 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,...
2 votes
1 answer
518 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 ...
4 votes
1 answer
229 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
0 answers
436 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
1k 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 ...
3 votes
1 answer
700 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 $\...
4 votes
2 answers
464 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 $\...
29 votes
2 answers
1k 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 necessarily ...
3 votes
1 answer
831 views

Are Lefschetz thimbles holomorphic manifolds?

I have a Lefschetz thimble defined by the stable flow of the gradient a holomorphic function toward a critical point (as defined e.g. in Witten arXiv:1001.2933 and F.Pham "Vanishing homologies and the ...