All Questions
Tagged with several-complex-variables reference-request
23 questions
14
votes
2
answers
2k
views
What´s essential to learn about complex spaces and several complex variables for an algebraic geometer?
Hi, I don´t know if this question is suitable for this site. The field of several complex variables is too broad, so I would like to know what´s essential to learn about complex spaces and several ...
- 163
7
votes
1
answer
248
views
Are there such things as non-trivial entire semigroups?
I'll state the theorem I am posing up front, and then explain why I think this theorem appears to be true. I am asking if anyone can prove it, or knows references to where it is proved. Please, ...
user78249
7
votes
1
answer
351
views
Oka-Grauert principle, up to the boundary
Let $Z\subset \mathbb{C}^n$ a domain of holomorphy with smooth boundary $\partial Z$ and closure $\bar Z$. There is a natural notion of holomorphic vector bundle over $\bar Z$, given in terms of ...
- 779
7
votes
1
answer
723
views
Complex manifolds with corner?
I was reading Dominic Joycee article on Manifold with corner. He talk about manifold with corner modeled over $[0,\infty)^k\times \mathbb R^{n-k}$ for some $k\leq n$. From here I moved to Melrose ...
- 227
6
votes
1
answer
240
views
The approximation property for some spaces of holomorphic functions
I am reading a circle of papers which use arguments based on Fredholm determinants of nuclear operators to compute numerical quantities associated to real-analytic and holomorphic dynamical systems. ...
- 6,206
4
votes
1
answer
396
views
Reference for the converse of Cartan's Theorem B
Cartan's theorem B states that every coherent analytic sheaf on a Stein manifold is acyclic. Hartshorne claims that the converse to this theorem also holds, but doesn't supply a reference. Does ...
4
votes
2
answers
355
views
On a variation of Hartogs' separate analyticity theorem
Let $f(z_1,z_2,\ldots,z_n)$ be a function on $\mathbf{C}^n$ such that for all $i$, the restriction
$$
[z_i\mapsto f(z_1,z_2,\ldots,z_n)]
$$
is a "rational function".
(added: to be precise ...
- 7,609
3
votes
1
answer
386
views
A question about Lelong number
If $f$ is plurisubharmonic (not identically $-\infty$) on a neighbourhood of $0$ then the Lelong number of $f$ at $0$ is defined by $$\nu_{f}(0) = \liminf_{|z|\rightarrow 0}\dfrac{f(z)}{\log|z|}.$$
My ...
- 33
3
votes
0
answers
219
views
Schwartz's theorem without English language reference
I'm reading the paper "Spectral Synthesis And The Pompeiu Problem" by Leon Brown, Bertram M. Schreiber and B. Alan Taylor,
Annales de l’Institut Fourier 23, No. 3, 125-154 (1973), MR352492, ...
- 421
3
votes
0
answers
102
views
What domains are there other than unit ball and polydisc, on which Caratheodory metric is known?
What are few (bounded)domains in $\mathbb{C}^n$ on which the explicit expression of Caratheodory metric is known. For example, unit ball and unit polydisc.
- 95
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
3
votes
0
answers
89
views
Trace of a weighted composition operator on Bergman space
I am reading a series of papers by Pollicott, Jenkinson and coauthors which make use of the following type of result:
Theorem: Let $\mathbb{D} \subset \mathbb{C}^d$ be a bounded, connected open set. ...
- 6,206
2
votes
1
answer
103
views
Fatou-Bieberbach domain in $\Bbb C^*\times\Bbb C^*$
According to Forstneric's book, pag 123, it is a long standing problem whether it exists a Fatou-Bieberbach domain in $\Bbb C^*\times\Bbb C^*$.
The idea is to search for an $F:\Bbb C^2\to\Bbb C^2$ ...
- 779
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
2
votes
0
answers
62
views
Geometric meaning of Catlin multi types
Can someone working in the area of several complex variables explain the geometric idea behind the Catlin multitype. I have seen the technical definition, but unable to grasp the idea behind this.
...
- 95
2
votes
0
answers
70
views
Regular exposable points on the boundary of compacts in Stein manifolds
Given a Stein manifold $Y$, there exists $\rho$, a $\mathscr C^2$-smooth strictly plurisubharmonic exhausting function for $Y$, such that the set of critical points $C=\{z\in Y\;:\;d\rho(z)=0\}$ is ...
- 779
2
votes
0
answers
75
views
Notation and geometry facts in a paper on the Diederich-Fornæss index
I am reading this article by Bingyuan Liu on the Diederich-Fornæss index.
I am having some problems with both the notation and the geometrical side.
1)I don't know what kind of objects $N,L$ are ...
- 779
1
vote
0
answers
55
views
References for the generalized Dirichlet problem for plurisubharmonic functions on the bidisc
In the paper "On a Generalized Dirichlet Problem for Plurisubharmonic Functions and Pseudo-Convex Domains. Characterization of Silov Boundaries" Theorem 5.3, the following result is obtained ...
- 53
1
vote
0
answers
36
views
Precise definition of locally closed complex curve
In Stein Manifold and Holomorphic Mappings, by Forstnerič, I refer to Definition 8.9.9:
An exposed point is a point belonging to a certain subset $\Sigma$ of $\Bbb C^2$, enjoying certain properties.
...
- 779
1
vote
0
answers
61
views
Holomorphic dynamical systems defined on a contractible bounded open subset of $\Bbb{C}^n$
Let $U$ be a contractible bounded open subset of $\Bbb{C}$. There is a standard classification of possible dynamical behaviors of holomorphic maps $f:U\rightarrow U$:
Attracting Case: There is an ...
- 3,599
1
vote
0
answers
294
views
Can an entire function have every root function?
My question is an amalgamation of two previous questions. The first question I'd like to draw attention to is here. It asks whether there can exist a non trivial semigroup defined on $\mathbb{C}$
$$\...
user78249
0
votes
1
answer
104
views
About maxima of injective holomorphic maps on $\mathbb{C}^n$
I am hoping the following is true. Mention of related ideas/topics are appreciated.
Suppose $F:\mathbb{C}^n \to \mathbb{C}^n$ is a injective holomorphic mapping such that $F(0)=0$ and $dF(0) = I_n$ ...
- 239
0
votes
0
answers
122
views
How to solve $\sqrt{-1}\partial\bar{\partial}u=\omega$
I'm looking for references on the study of the equation $\sqrt{-1}\partial\bar{\partial}u=\omega$,especially when $\omega$ is a k\"ahler metric on $\Omega\setminus S$,where $\Omega\subset \mathbb{C}^n$...
- 11