Questions tagged [several-complex-variables]

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Putnam 2020 inequality for complex numbers in the unit circle

The following simple-looking inequality for complex numbers in the unit disk generalizes Problem B5 on the Putnam contest 2020: Theorem 1. Let $z_1, z_2, \ldots, z_n$ be $n$ complex numbers such that ...
darij grinberg's user avatar
23 votes
3 answers
8k views

Functions of several complex variables: book recommendations?

Can anyone recommend a good comprehensive introduction to functions of several complex variables that a) is fairly up to date, b) isn't a geometry or an algebra book only, but takes multiple ...
19 votes
2 answers
2k views

Laurent series in several complex variables

Is there a good generalisation of Laurent series for several complex variables? I am interested in generalised power series that have some terms with negative powers, but not too many. In single ...
rimu's user avatar
  • 749
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 ...
asv's user avatar
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18 votes
7 answers
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Fundamental motivation for several complex variables [closed]

I have 3 general abstract reasons to care about complex analysis in a single variable: The laplacian is, up to a constant multiple, the only isometry invariant PDO in the plane, and so it is ...
18 votes
2 answers
2k views

motivation for multiplier ideal sheaves

What is the origin of multiplier ideal sheaves?It was introduced ny Nadel.Yum Tong Siu,his advisor in his plenary lecture in 2002 icm mentions some thing that it arose in pde.Can anyone kindly ...
Koushik's user avatar
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17 votes
3 answers
728 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(\...
Steven Gubkin's user avatar
16 votes
2 answers
2k views

Why only $\bar\partial$ but not $\partial$ in Dolbeault cohomology

While I learn about $\partial$ and $\bar{\partial}$ operators, I had some questions about the reason why people prefer $\bar\partial$ over $\partial$. Specifically, When defining Dolbeault ...
Ramanasa's user avatar
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14 votes
1 answer
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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 \...
Steven Gubkin's user avatar
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 ...
user17868's user avatar
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11 votes
1 answer
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Dual of the space of all bounded holomorphic functions

Let $\mathbb{B}$ be the open unit ball in $\mathbb{C}^n, n\geq 1$ and let $H^\infty (\mathbb{B})$ be the space of all bounded holomorphic functions on $\mathbb{B}$. It is well known that $H^\infty (\...
Anindya Biswas's user avatar
10 votes
3 answers
892 views

Complex manifold with boundary

My question is of local nature. Let $$f:\mathbb C^n\to\mathbb R$$ be a $C^\infty$ function that vanishes at $0\in \mathbb C^n$, with non-zero derivative. Then, around $0\in \mathbb C^n$, $$M:=f^{-1}(0)...
André Henriques's user avatar
10 votes
1 answer
852 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}^...
Steven Gubkin's user avatar
10 votes
0 answers
765 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{\...
whitacre's user avatar
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9 votes
1 answer
2k views

If $f$ is $C^{\infty}$ and $f^2$ is analytic, then $f$ is analytic

Assume that $f:\mathbb{C}^n\rightarrow \mathbb{C}$ is a $C^{\infty}$ function such that $f^2$ is (complex) analytic. Then one can show that $f$ is analytic. (Note: Liviu Nicolaescu and Alexandre ...
O.R.'s user avatar
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9 votes
1 answer
624 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:...
erz's user avatar
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7 votes
2 answers
946 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 ...
Steven Gubkin's user avatar
7 votes
1 answer
240 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, ...
user avatar
7 votes
1 answer
369 views

Analogue of Grauert's upper semi-continuity for Bott–Chern cohomology

In Coherent analytic sheaves, one has the following theorem due to Grauert: Let $f: X \rightarrow Y$ be a holomorphic family of compact complex manifolds with connected complex manifolds $X, Y$ and $V$...
Invariance's user avatar
7 votes
1 answer
303 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 ...
Jan Bohr's user avatar
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7 votes
1 answer
701 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 ...
Jonujohn's user avatar
  • 217
7 votes
0 answers
723 views

Snazzy applications of Several Complex Variables techniques

I am starting to dive into a study of Several Complex Variables. I would like to have a few guiding examples of "big payoffs". These should be very natural sounding theorems which depend on a lot of ...
6 votes
1 answer
183 views

Do we have the Oka coherence theorem for finite group actions?

We first consider the sheaf of holomorphic functions $\mathcal{O}(\mathbb{C}^n)$ on $\mathbb{C}^n$. By Oka coherence theorem, $\mathcal{O}(\mathbb{C}^n)$ is coherent over itself. Now we consider a ...
Zhaoting Wei's user avatar
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6 votes
1 answer
583 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 \...
AmorFati's user avatar
  • 1,349
6 votes
1 answer
264 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 ...
Mauro Porta's user avatar
6 votes
1 answer
207 views

$(-2)$-curves in complex $3$-folds

Let $X$ be a smooth complex $3$-fold, and let $C \subset X$ be an embedded smooth rational curve whose normal bundle $N_{C/X}$ is isomorphic to $\mathscr{O}(-1) \oplus \mathscr{O}(-1)$. Is it true ...
user691704's user avatar
6 votes
1 answer
229 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. ...
Ian Morris's user avatar
  • 6,176
6 votes
0 answers
234 views

Bezout theorem for germs of holomorphic functions

UPDATE. It was pointed out by @Dmitri that two smooth curves given by $f=y$ and $g=y+x^k$ in $\mathbb C^2$ provide a simple counterexample. Let $f_1, \ldots, f_p, g_1, \ldots, g_q$ be germs of ...
Dmitri Zaitsev's user avatar
5 votes
2 answers
523 views

$\frac{\partial f}{\partial \overline{z}}=0$ in distributional sense implies $f$ is holomorphic

Let $f=(u,v)\in \mathscr{D}'(U,\mathbb{C})$ be a distribution, where $U\subset\mathbb{C}=\mathbb{R}^2$ is an open set and $u$ and $v$ are the projection of $f$ onto the real and imaginary axis (ie $\...
No-one's user avatar
  • 1,037
5 votes
2 answers
1k views

zeros of holomorphic function in n variables

Conjecture: Let $f:{\mathbb C}^n\rightarrow{\mathbb C}$ be an entire function in $n$ complex variables. Assume that for every $x\in{\mathbb R}^n$ there exists a $y_x\in{\mathbb R}^n$ such that $f(x+...
M Mueger's user avatar
  • 605
5 votes
1 answer
247 views

How does pseudoconvexity restrict the topology?

A domain of holomorphy in $\mathbb{C}^n$ has vanishing de rham cohomology in real dimensions greater than $n$ - half of it's cohomology is missing. Are there any other restrictions? If I give you a ...
Steven Gubkin's user avatar
5 votes
1 answer
242 views

Non-constant holomorphic map onto a smooth curve

Let $\Gamma$ be a smooth projective curve in $\mathbb{P}^2$ and let $U$ be an open neighborhood of $\Gamma$. Denote by $\Gamma_1,\Gamma_2,\ldots,\Gamma_n$ a finite collection of smooth curves ...
Curiosity's user avatar
  • 293
5 votes
1 answer
630 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 ...
user42804's user avatar
  • 1,091
5 votes
1 answer
434 views

A sequence that tell us if a holomorphic function of several variables is identically zero

Is there any sequence $ \{ Z_{\nu} \}_{\nu \in \mathbb{N}}$ in $\mathbb{C}^{n}$, $Z_{\nu} \rightarrow 0$, such that any holomorphic function in $\mathbb{C}^{n}$ which vanishes in $Z_{\nu}$ for all $\...
theStudent's user avatar
5 votes
1 answer
382 views

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 ...
erz's user avatar
  • 5,385
5 votes
1 answer
380 views

Morrey & Grauert - real analytic vector bundles admits analytic Riemannian metric

In theorem 1.2 of Brian Conrad's handout Operations with Pseudo-Riemannian metrics, the author writes Theorem 1.2. Every $C^p$ vector bundle $E\to M$ over a $C^p$ manifold with corners $0\leq p\leq \...
Arrow's user avatar
  • 10.3k
5 votes
1 answer
589 views

Are polynomials dense in holomorphic $L^p(\mathrm{Gauss})$ for $p < 1$?

Let $\mu$ be standard Gaussian measure on $\mathbb{C}^n$, i.e. $d\mu = \frac{1}{(2 \pi)^n} e^{-|z|^2/2}\,dz$, and fix $0 < p < 1$ (note carefully). Suppose $g$ is holomorphic on $\mathbb{C}^n$...
Nate Eldredge's user avatar
5 votes
1 answer
543 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 $X_s$, ...
user42804's user avatar
  • 1,091
5 votes
0 answers
166 views

Example of an $H^1$ function on the bidisk that is not a product of two $H^2$ functions

Fix $n \in \mathbb{N}$ and consider the Hardy space $H^1 := H^1(\mathbb{D}^n)$, consisting of holomorphic functions $f$ on the unit polydisk $\mathbb{D}^n=\mathbb{D}\times\dots\times\mathbb{D}$ such ...
EG2023's user avatar
  • 51
5 votes
0 answers
133 views

Are open subsets of a $\sigma$-compact LCH space $\mathcal{K}$-analytic?

I'm reading Guedj and Zeriahi's Degenerate Complex Monge-Ampère Equations Chapter 4 which talks about capacities. Specifically Corollary 4.13 claims that when $X$ is a locally compact Hausdorff $\...
Carlos Esparza's user avatar
5 votes
0 answers
537 views

a question on Hodge and Atiyah's paper "integrals of the second kind on an algebraic variety"

I have a question on Hodge and Atiyah's paper "Integrals of the second kind on an algebraic variety". It is about the exact sequence below formula (14) and above formula (15) on page 71: $$H_{2n-q}(S)...
user42804's user avatar
  • 1,091
5 votes
0 answers
231 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 ...
Mauro Porta's user avatar
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 ...
user95633's user avatar
4 votes
1 answer
407 views

Is the determinant line bundle of a coherent sheaf functorial (between sheaves of the same rank)?

The determinant line bundle of a coherent sheaf $\mathcal{F}$ on an $n$-dimensional (smooth) analytic space is defined as \begin{equation} \det \mathcal{F} := \bigotimes_i^n (\det \mathcal{E}_i)^{⊗...
Carlos Esparza's user avatar
4 votes
1 answer
386 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
1 answer
271 views

complex dynamic system and affine algebraic variety

Let $M^n$ be a $n$-dimensional noncompact complex manifold. In "The density property for complex manifolds and geometric structures II", Dror Varolin showed that some open set of $M$ is ...
Xiaoyang Chen's user avatar
4 votes
1 answer
959 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 $...
user108005's user avatar
4 votes
1 answer
295 views

Existence of plurisubharmonic functions on complex manifolds

Let $X$ be a noncompact complex manifold which contains no positive dimensional compact analytic sets. Conjecture: There must be strictly plurisubharmonic functions on $X$ . Is it true?
jack lion's user avatar
  • 391
4 votes
1 answer
178 views

Space of holomorphic embeddings of open unit ball in ${\mathbb C}^n$

Let $B$ be the open unit ball in $\mathbb C^n$. Consider the space $\mathcal F$ of holomorphic embeddings of $B$ in $\mathbb C^n$ equipped with the compact-open topology. (A holomorphic embedding of $...
user102829's user avatar
4 votes
1 answer
685 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 (...
Chertopkhanov on Malek Adel's user avatar