All Questions
74 questions
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
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
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
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