All Questions
34 questions
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 ...
- 66.3k
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 ...
user100272
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 ...
- 7,328
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 ...
- 283
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 ...
- 889
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 $f$...
- 337
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 ...
- 1,409
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 ...
- 21.8k
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 } ...
- 21.8k
7
votes
1
answer
527
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 ...
- 1,379
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 ...
- 21.8k
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 ...
- 21.8k
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 ...
- 1,409
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 $\...
- 21.8k
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 $\...
- 2,221
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,...
- 21.8k
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). ...
- 7,426
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 $\...
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 ...
- 229
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 ...
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 ...
- 21.8k
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 ...
- 21.8k
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.
...
- 21.8k
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 ...
- 1,111
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 ...
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 ...
- 21.8k
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 ...
- 21.8k
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 ...
- 21.8k
1
vote
0
answers
155
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 ...
- 21.8k
1
vote
0
answers
112
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 $\...
- 651
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 ...
- 21.8k
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 ...
- 21.8k
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 $...
- 21.8k
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 }...
- 21.8k