All Questions
Tagged with complex-manifolds cv.complex-variables
93 questions
4
votes
0
answers
148
views
Universal cover of Kodaira surface
From an earlier question, the universal cover of a Kodaira fibered surface is a bounded domain in $\mathbb{C}^2$. It is also not the polydisk or the ball. Can we say more about the structure of the ...
- 1,159
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
2
votes
2
answers
354
views
Holomorphic maps into a symmetric product of Riemann surface
Let $X$ and $Y$ be compact Riemann surfaces that are both hyperbolic (i.e. genus > 1). A classical result of de Franchis implies that the space of non-constant holomorphic maps from $X$ into $Y$ is a ...
- 1,159
3
votes
0
answers
90
views
Two questions on homogeneous domains
Let $G$ be a domain in $\mathbb{C}^{n}$ and let $Aut(G)$ be the group of biholomorphic selfmaps of $G$. $G$ is called:
(1) homogeneous if $Aut(G)$ acts transitively on $G$, i.e. for any $z,w\in G$ ...
- 5,529
2
votes
1
answer
321
views
Are induced morphisms on cohomology strict with respect to the hodge filtration in the non Kähler case?
For a complex manifold $X$ there is the Hodge filtration on cohomology, induced by the filtration on the complex of holomorphic forms given by:
$$ F^r\Omega_X^p:=\begin{cases}\{0\}\qquad\text{if }r&...
- 359
0
votes
1
answer
284
views
Lelong number of curvature of Kawamata's hermitian metric
Let $X,Y$ are two projective varieties and $f:X\to Y$ is an Iitaka
fibration. Consider the following singular hermitian metric
$$h(\sigma,\sigma)=\left(\int_{X_y}|\sigma|^{\frac{2}{m!}}\right)^{m!}...
user21574
4
votes
3
answers
687
views
Finite covers of punctured Riemann surfaces
Let $X$ be a compact Riemann surface, i.e. compact smooth complex analytic (hence automatically algebraic) curve. Let $A\subset X$ be a finite subset, and $X_0:=X\backslash A$.
Let $Y_0$ be a smooth ...
- 21.8k
1
vote
2
answers
654
views
Examples of pluripolar sets
I have a very basic question on pluripolar sets. First remind their definition.
Let $\Omega\subset \mathbb{C}^n$ be a domain. A subset $E\subset \Omega$ is called pluripolar if there exists a ...
- 21.8k
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
131
views
Hurwitz's theorem for a system of functions
First, let me define a notation of $H(G_1\times G_2 \times \ldots \times G_m)$.
We say that $f\in H(G_1\times G_2 \times \ldots \times G_m)$ if $$f:G_1\times G_2 \times \ldots \times G_m \rightarrow \...
- 11
18
votes
2
answers
1k
views
Vanishing of Dolbeault cohomologies and Steinness
That Stein manifolds have all $(p,q), p \geq 0, q \geq 1$ vanishing Dolbeault cohomology groups is more or less standard. I am a little bit confused about the reverse implication: whether the ...
- 243
2
votes
1
answer
314
views
Flatness of a morphism of complex analytic spaces
Let $f\colon X\to D$ be a morphism of a complex analytic space $X$ into the 1-dimensional disk $D$. Assume for simplicity that $X$ has a single irreducible component which may not be reduced.
...
- 21.8k
5
votes
1
answer
366
views
A weak analytic version of the valuative criterion of properness
EDIT: Let $f\colon X\to Y$ be a morphism of complex analytic spaces (not necessarily smooth or reduced). Assume that
(a) $f$ is injective on points;
(b) $f$ is local imbedding near each point $x\in ...
- 21.8k
1
vote
1
answer
273
views
When a proper morphism of schemes is a closed imbedding?
Let $X$ and $Y$ be finitely presented schemes over $\mathbb{C}$. Let $f\colon X\to Y$ be a proper morphism. Let us assume that for any finitely presented scheme $S$ the induced map
$$Mor_{Sch}(S,X)\to ...
- 21.8k
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
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
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
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
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
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
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
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
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
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 $\...
2
votes
3
answers
732
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 ...
- 955
4
votes
1
answer
476
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 ...
- 601
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
1
vote
2
answers
237
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. ...
- 287
5
votes
2
answers
471
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 ...
- 53
9
votes
1
answer
931
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 ...
- 19.3k
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
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
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
8
votes
0
answers
964
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 ...
- 1,111
2
votes
1
answer
259
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 Non-...
- 541
1
vote
1
answer
220
views
How to compute Kobayashi distance of compact Kaehler manifolds with postive Ricci curvature?
Recently I just learned the Kobayashi distance on complex manifolds and wants to get some feeling of how it looks like on exmaples of manifolds with positive Ricci curvature. I have a feeling that the ...
- 637
10
votes
2
answers
811
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 bundle,...
- 1,221
1
vote
0
answers
234
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, ...
- 19
2
votes
1
answer
357
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 $f(U)$. ...
- 23
11
votes
3
answers
3k
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 ...
- 23.3k
3
votes
1
answer
994
views
Plurisubharmonic exhaustion functions without critical points at infinity
A complex manifold $X$ is said to be weakly pseudoconvex if there exists on $X$ a smooth plurisubharmonic exhaustion function $\psi$.
For example, Stein manifolds are weakly pseudoconvex (in this ...
- 7,902
14
votes
2
answers
780
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 ...
- 56.6k
21
votes
4
answers
2k
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) ...
- 6,247