All Questions
Tagged with several-complex-variables complex-geometry
87 questions
5
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1
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395
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Holomorphic Sard's theorem 2
My previous question on this topic had a negative answer, but Tom Goodwillie in the comments suggested a statement, which may be true, and even a strategy of how to prove it. I haven't been able to ...
- 5,529
6
votes
1
answer
640
views
Practically calculating the domain of a power series for function of several complex variables
For simplicity, let us consider a function $f$ holomorphic on a domain $D \subseteq \mathbb{C}^2$. We may therefore write $f$ as a sum of power series $$f(z) = \sum_{\nu_1 \nu_2 =0}^{\infty} c_{\nu_1 \...
- 1,379
3
votes
1
answer
177
views
Real solution of a complex equation with complex solution
Assume that $(M, [\lambda, \mu])$ defines an embeddable 3 dimensional CR structure where $\lambda$ is a real form and $\mu$ is a complex 1-form.
Because $M$ is embeddable, $\mu=dz$ for some ...
- 99
4
votes
1
answer
771
views
Understanding Remmert-Stein extension theorem
I'm trying to study the Remmert-Stein theorem in analytic geometry. This is an important result which can be used to prove the Proper Mapping theorem.
A preliminary result is stated in various books (...
- 675
3
votes
0
answers
637
views
English reference for Fischer-Grauert theorem and its generalization by Schuster
From this MSE question and its answer, and from this MO question I have learned of the following remarkable theorem of Wolfgang Fischer and Hans Grauert.
Theorem. A proper holomorphic submersion with ...
- 10.5k
9
votes
1
answer
662
views
Holomorphic Sard's theorem?
I have originally posted this question on math.SE, but it received little attention, so I repost it here.
Let $U\subset \mathbb{C}^{n}$ and $V\subset \mathbb{C}^{m}$ be open and connected. Let $\Phi:...
- 5,529
3
votes
0
answers
84
views
Discrete set of critical points of a holomorphic map
I have originally posted this question on math.SE, but it received no attention, so I repost it here.
Let $U$ be an open domain in $\mathbb{C}^{n}$. Let $m\ge n$ and let $F:U\to C^{m}$ be a ...
- 5,529
4
votes
1
answer
371
views
Kähler metric on compact complex manifolds with simple normal crossing divisor
Let $X$ be a reduced compact complex analytic space of $\dim_{\mathbb{C}}X\ge2$; by [KJ] definition 3.29, remark 3.44 and theorem 3.45, it admits a strong resolution $R(X)$ which is smooth, $E=\pi_X^{-...
- 828
4
votes
0
answers
292
views
Is there an Ax-Grothendieck result for entire functions?
The Ax-Grothendieck result states that any polynomial injective function from $\mathbb{C}^n$ to itself is surjective. Is there such a statement for entire functions ?
- 189
6
votes
1
answer
567
views
How to tell if it's a Moishezon morphism
Suppose that $f \colon X\rightarrow S$ is a proper morphism of reduced and irreducible complex spaces and $f$ is a smooth deformation in the sense of Kodaira and Spencer. If we know each fiber $X_s$, ...
- 1,121
-1
votes
1
answer
97
views
Zariski open set in orthogonal grassmanian [closed]
I am confused about the following question.
Consider $\mathbb C^4$ endowed with nondegenerate symmetric bilinear form $J:=\left(\begin{matrix}0&0&0&1\\0&0&1&0\\0&1&0&...
- 1,121
1
vote
0
answers
142
views
Construction of homogeneous Siegel domain from j-algebra
I am reading bounded homogeneous domain from Piatetski-Shapiro's
book ``Automorphic functions and the geometry of classical domains''
and have questions on how to construct homogeneous Siegel domain
...
- 637
3
votes
0
answers
130
views
cayley transformation of bounded symmetric domain
Can anyone write down the biholomorphic map between classical bounded symmetric domains(defiend by matrixs) with their siegel upperhalf plane models. I know if it's type 2, i.e $I-Z\bar{Z}^{t}>0$ ...
- 1,121
1
vote
1
answer
334
views
extension of holomorphic mappings
In 1971 Phillip.A.Griffith proved that
Let $B_n^\ast=\{z\in\mathbb{C}^n:0<\|z\|\le 1\}$ be the punctured ball in $\mathbb{C}^n$,and $f:B_n^\ast\rightarrow M$ a holomorphic mapping into a compact K\...
- 11
3
votes
0
answers
124
views
How to study Kähler metrics singular along a submanifold of codim 2?
Let $M$ be a compact complex manifold, $S\subset M$ a submanifold of codimension $2$, let $\omega$ be a k\"ahler metric on $M\setminus S$. Then we know by Reese Harvey's paper "Removable singularities ...
- 31
10
votes
0
answers
769
views
de Rham cohomology group and Dolbeault cohomology group on compact complex analytic spaces
My question is:
On a compact complex analytic space,since Hodge Theorem becomes invalid,is it true that the de Rham cohomology group $H^p_{DR}(M)$ and the Dolbeault cohomology group $H^{p,q}_{\bar{\...
- 101
19
votes
2
answers
1k
views
Classification of complex structures on $\mathbb{R}^{2n}$
Is there anything known about classification of complex structures on $\mathbb{R}^{2n}$ up to isomorphism for $n>1$? Say, are there finitely or infinitely many isomorphism classes? If there is a ...
- 21.8k
2
votes
1
answer
3k
views
generalization of fundamental theorem of algebra for several complex algebra [closed]
I am looking for a generalization to fundamental theorem of algebra for several complex variables functions or systems. If such theorem exists, it should concisely relates the number of zeros of ...
- 383
1
vote
1
answer
164
views
Lifting quadratic forms on the cotangent bundle to higher level forms
Backround
In several complex variables, an essential tool is Hormander's machinery for solving the $\overline{\partial}$ problem with $L^2$ estimates.
If $\alpha$ is a $(p,q+1)$ form on a domain $\...
- 12.1k
10
votes
1
answer
989
views
Computing Dolbeault cohomology of some simple domains
I have seen computations of the Dolbeault cohomology groups on compact Kahler manifolds using Hodge theory.
I have never seen the computation of Dolbeault cohomology for simple domains in $\mathbb{C}^...
- 12.1k
2
votes
2
answers
539
views
Is there an Oka-Grauert principle for homogeneous spaces?
Suppose we have a fibration over the punctured disc (i.e., a deformation of complex manifolds) such that each fiber is a homogeneous space. Is the total space a product of a fiber with the punctured ...
- 1,121
17
votes
3
answers
764
views
Can all $L^2$ holomorphic functions on a domain vanish at a particular point?
Let $\Omega \subset \mathbb{C}^n$. Is it possible that there is a point $p \in \Omega$ such that every $f \in A^2(\Omega) = L^2(\Omega) \cap \mathcal{O}(\Omega)$ has a zero at $p$? The space $A^2(\...
- 12.1k
2
votes
1
answer
373
views
Does the "Ohsawa-Takegoshi theorem without bounds" have a name?
There are many theorems which now could be called "The Ohsawa-Takegoshi" theorem. Of these, the most basic is roughly the following:
Let $\Omega \subset \subset \mathbb{C}^n$ be a psuedoconvex ...
- 12.1k
6
votes
1
answer
270
views
Are open immersions in analytic geometry transverse?
lately I have been interested in functional analysis, especially with a view towards its applications in the world of (complex) analytic geometry. I have been using R. Taylor's book Several complex ...
- 513
5
votes
0
answers
241
views
Density of rational functions in open Stein
I repost here, after I tried here.
Lately I have been wondering on this problem: if $U \subset \mathbb C^n$ is an open Stein and I denote by $\mathcal R(U)$ the set of rational functions on $\mathbb ...
- 513
5
votes
1
answer
692
views
Can someone tell me properties of Douady space?
I want to know the parallel properties of Douady space with respect to Hilbert scheme. For example I want to know what is the irreducible component of Douady space, what if I consider a family of ...
- 1,121
14
votes
1
answer
1k
views
What is the "complex third derivative"?
Background
I am including this information about real higher order derivatives because it does not seem to be common knowledge. I also include a review of the complex Hessian.
If $f:\mathbb{R}^n \...
- 12.1k
2
votes
0
answers
100
views
Translation of "Über kompakte homogene Kählersche Mannigfaltigkeiten"
Has anyone translated Borel and Remmert's 1962 paper titled:
Über kompakte homogene Kählersche Mannigfaltigkeiten?
- 89
3
votes
1
answer
439
views
Singularities of the Remmert reduction of a holomorphic convex manifold
Let $X$ be a holomorphically convex manifold, namely, for any infinite discrete subset of $X$ there exists a holomorphic function on $X$ which is unbounded on this set, then a theorem of Remmert says ...
- 285
4
votes
1
answer
1k
views
the group of all biholomorphic group automorphisms of complex tori
My background is complex geometry, but when I confront complex tori, I feel it is easier to consider it as a compact connected complex Lie group although I just know the definition of Lie group.
Let $...
- 197
3
votes
1
answer
385
views
Explicit form for hermitian structure $h$ with respect to $\omega$
Let $(M,\omega)$ be a symplectic manifold. and $\pi:L\to M$ be a complex line bundle , we denote $h$ as hermitian structure,i.e. if $s,s'$ are smooth sections of $L$ and if $X$ is a vector field on $M$...
user21574
2
votes
0
answers
82
views
Is the Szego projection on a codim-$k$ CR manifold an integral operator?
The Szego projection on a CR manifold $M$ is defined to be the orthogonal projection from $L^2(M)$ to the closed subspace $H^2(M),$ where
$$H^2(M) = \{f \in L^2(M)\ |\ \bar{\partial}_{b}f = 0\ \...
- 363
4
votes
1
answer
1k
views
Any non-constant surjective holomorphic map between connected compact complex manifolds of equal dimension is a ramified finite-sheet covering
I'm reading this site:holomorphy of inverse map
There is a statement made by Colin Tan at the last answer made by himself.
Any non-constant surjective holomorphic map between connected compact ...
- 41
2
votes
0
answers
221
views
Automorphisms of the affine quadric $z_1^2+\dots+z_n^2=1$
Denote by $V$ the vanishing locus of the affine quadric $z_1^2+\dots+z_n^2-1$ in $\mathbb{C}^n$. Clearly $V$ is a complex manifold. Let $g:V\to V$ be an automorphism of $V$ (a biholomorphism, not ...
- 783
1
vote
0
answers
403
views
Weakened jacobian conjecture for entire functions
A rudin's theorem is the assertion that any polynomial injection between affine spaces of the same dimension has a polynomial inverse, and the inverse is also given by polynomials.
The jacobian ...
- 2,106
0
votes
2
answers
878
views
Extension of pluriharmonic functions
Suppose $M$ is a complex manifold and $\Omega$ a (edit: bounded) pseudoconvex domain in $M$. Let $u:M\setminus\Omega\to\mathbb{R}$ be a pluriharmonic function. Is it true that $u$ has a pluriharmonic ...
- 11
8
votes
2
answers
994
views
Another proof of the bidisc and the ball are biholomorphically inequivalent?
Does this outline of a proof work?
Consider the ball and the bidisc in $\mathbb{C}^2$. Give each space its Bergman metric. To show that the ball and the bidisc are not holomorphic, it is enough to ...
- 12.1k