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Questions tagged [several-complex-variables]

For questions related to the study of functions of several variables, in particular the study of holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.

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Prove or disprove an integral identity

We define $$\mathbb{C}^{+n}:=\{ (z_1,\cdots,z_n)\in \mathbb{C}^n:\text{Im}[z_j]\geq 0 \text{ for } 1\leq j\leq n \}.$$ Let $h:\mathbb{C}^{+n}\to \mathbb{C}$ be a rational analytic function such that $...
Jonas Müller's user avatar
2 votes
1 answer
93 views

Confusion about (closed) submanifolds of Stein manifolds

I'm starting to study Stein manifolds and I think I'm confused by something that is probably very elementary (which possibly highlights how little I truly understand about basic manifold theory in ...
Maths Matador's user avatar
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19 views

Singular locus of analytic variety

Suppose $X$ is a complex manifold and $V \subseteq X$ is an analytic subvariety of $X$, i.e., $V$ is locally the zero locus of a finite collection of holomorphic functions. A point $x \in V$ is a ...
Frank's user avatar
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1 vote
1 answer
57 views

Zero set of analytic function $f$, $g$ on $\mathbb{C}^n$ concides implies that $f$ divides $g$.

If I have two analytic functions $f$ and $g$ on $\mathbb{C}^n$ such that $\{g=0\}\subset\{f=0\}$, what condition does it need to imply that $\frac{f}{g}$ is analytic? I think here we may need some ...
Holden Lyu's user avatar
1 vote
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17 views

Does the sequence of bounded symmetric square integrable holomorphic functions have a convergent subsequence?

Let $f$ be a bounded holomorphic function on $\mathbb D^2$ and $s : \mathbb C^2 \longrightarrow \mathbb C^2$ be the symmetrization map given by $s(z) = (z_1 + z_2, z_1 z_2),$ for $z = (z_1, z_2) \in \...
Anacardium's user avatar
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2 votes
0 answers
36 views

Pseudoconvex domains in one complex variables

I am trying to prove that every domain in $\mathbb{C}$ with $C^2$ boundary is (Levi) pseudoconvex. For that, suppose $\Omega$ is defined as $\rho(z)<0$, where $\rho$ is $C^2$ defining function in a ...
Soumya Ganguly's user avatar
1 vote
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15 views

Semi-continuity of Lelong number

Demailly gives the following some-continuity result for Lelong numbers: Proposition. Let $T_k$ be a sequence of closed positive currents of bidimension $(p, p)$ converging weakly to a limit $T$. ...
eulershi's user avatar
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8 views

Questions about the definition of generalized Lelong number by Demailly

According to Demailly, the definition of Lelong number is local, so assume $X$ is a Stein manifold, $\varphi$ is a continuous psh function (which means $e^{\varphi}$ is continuous. Let $T$ be a closed ...
eulershi's user avatar
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2 votes
1 answer
84 views

Showing that a particular function is holomorphic

Here is the problem: I have a diffeomorphism $F: \mathbb{C}^n \times \mathbb{C}^m \to \mathbb{C}^n \times \mathbb{C}^m$ over the projection $\mathbb{C}^n \times \mathbb{C}^m \to \mathbb{C}^m$, so in ...
Emory Sun's user avatar
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Explicitly showing examples of plurisubharmonicity

In light of the result that $-\log \delta_G$ is plurisubharmonic if and only if $G \subset \mathbb{C}^n$ is a pseudoconvex domain where is $\delta_G = d(z,\mathbb{C}^n-G)$ is the boundary distance ...
user262378's user avatar
1 vote
1 answer
27 views

Show that $\langle(f\circ\varphi_{\lambda})k_{\lambda}, (g\circ\varphi_{\lambda})k_{\lambda}\rangle=k_{\lambda}(\lambda)\langle f,g\rangle.$

Let $\Omega = \mathbb B_n,$ the unit ball in $\mathbb C^n$ and $L^2_a(\Omega)$ be the Bergman space endowed with the normalized volume measure on $\Omega.$ Let $k_{\lambda}$ be the associated Bergman ...
Anacardium's user avatar
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2 votes
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Does meromorphy in each variable imply joint meromorphy?

Let $f:\mathbb{C}\times\mathbb{C}\mapsto\mathbb{C}$ be a complex valued function which is meromorphic in each of the two variables separately. Is it meromorphic on $\mathbb{C}\times\mathbb{C}$? It is ...
Akash Yadav's user avatar
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2 votes
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Problem in computing a sum.

For a multi-index $\alpha = \left (\alpha_1, \cdots, \alpha_n \right )$ we write $\left \lvert \alpha \right \rvert = \alpha_1 + \cdots + \alpha_n$ and $\alpha! = \alpha_1! \cdots \alpha_n!.$ For $z = ...
Anacardium's user avatar
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1 vote
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Open mapping theorem for spaces of holomorphic functions

In studying several complex variables, I came across an exercise in what is effectively an application of the open mapping theorem for Frechet spaces without actually using said theorem. The situation ...
Maths Matador's user avatar
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Show that a domain with $C^2$ boundary is strictly Levi psuedoconvex at a special boundary point.

We let $D \subset\subset \mathbb{C}^n$ be a domain with $C^2$ boundary and let $P \in bD$. We also suppose that there is some neighborhood $U$ of $P$, $v \in C^2(U)$ strictly plurisubharmonic, with $v(...
LukeJ's user avatar
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1 answer
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Showing that the set of strictly Levi pseudoconvex boundary points of $C^2$ domain is not empty.

This question comes from R. Michael Range's Holomorphic Functions and Integral Representations in Several Complex Variables, exercise 2.2.14.a: Let $D$ be a relatively compact domain in $\mathbb{C}^n$ ...
LukeJ's user avatar
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4 votes
1 answer
159 views

A manifold with an almost-complex structure that admits a holomorphic frame is integrable

Let $(M,I)$ be an almost-complex manifold such that every point $m\in M$ has a neighborhood $U\ni m$ and holomorphic functions $f_1, \dots, f_n:U\to\mathbb C$ such that $\text{span}(df_1(m), \dots, ...
Derivative's user avatar
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Definition of local parametrisation of complex submanifold

I am currently reading Range's Holomorphic functions and integral representations in several complex variables and would like some clarification on the following definition. Let $M$ be a complex ...
Maths Matador's user avatar
2 votes
0 answers
83 views

Understanding GAGA correspondence from some basic questions

On page 78 of Hartshorne's "Algebraic Geometry," we encounter the following correspondence: Proposition 2.6: Let $k$ be an algebraically closed field. There exists a natural, fully faithful ...
Lelong  Wang's user avatar
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22 views

Is an irreducible analytic subset locally complete intersects at general point?

Let $X$ be an reduced and irreducible complex analytic space of dimension $m$ and $Z$ an $n$-dimensional irreducible analytic subset of $X$ with reduced structure. Then we say that $Z$ locally ...
Lelong  Wang's user avatar
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65 views

Boundary of unit ball with $1$-norm is a Levi-flat hypersurface

Let $\mathbb{D}$ denote the unit disc in the complex plane centered at origin. Let us define the set $M=\{(z_1, z_2) \in \mathbb{D}^2 : |z_1|+|z_2|=1\}$. Also, let $\Omega=\mathbb{D}^2\setminus \{(0,0)...
Curious's user avatar
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Relation between local defining functions for the boundary.

Definition $:$ Let $G \subseteq \mathbb C^n$ be a domain. The boundary $\partial G$ of $G$ is said to be smooth at $z_0 \in \partial G$ if there is an open neighbourhood $U = U(z_0) \subseteq \mathbb ...
Akiro Kurosawa's user avatar
1 vote
0 answers
30 views

Local structure of a germ of analytic set

I'm reading Complex Analytic and Differential Geometry by J-P. Demailly. In section 2.4.2, the author gives the following proposition: Suppose $A=V(\mathcal{J})$, where $J$ is a prime ideal of $\...
eulershi's user avatar
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Question on region of convergence

We know that the unit circle can be a boundary of complete convergence or complete nonconvergence or of conditional convergence for power series in one complex variable...If the subset of the boundary ...
Richard Peterson's user avatar
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0 answers
87 views

commutativity of $\iota^*$ and ${\bar\partial}^*$ / $\mathcal G$ on smooth differential froms

Let $(X,\omega_X)$ be a compact Kaehler manifold. Denote by $\iota:Y\rightarrow X$ the natural embedding, where $Y$ is a submanifold in $X$. Recall that we have the De Rham-Kodaira Hodge ...
Invariance's user avatar
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3 votes
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Terminology for Complex Algebraic Geometry with Complex Conjugation

Semialgebraic geometry is essentially real algebraic geometry but with the defining polynomial relations allowed to be inequalities rather than just equalities. This doesn't make sense over $\mathbb{C}...
Harry Wilson's user avatar
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2 votes
0 answers
43 views

Analytically continuing a function of two complex variables.

I am aware of the identity theorem and how it allows us to extend the definition of a complex analytic function from $A \subset \mathbb{C}$ to a larger domain $D$ that is an open subset of the complex ...
mathphy24's user avatar
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0 answers
61 views

Estimates for Montel's theorem in higher dimensions

As I'm beginning to study the theory of several complex variables, a lot of the early proofs that I've seen from the textbooks I'm using (Range's Holomorphic Functions and Integral Representations in ...
Maths Matador's user avatar
1 vote
0 answers
62 views

Show that $\widehat {f}$ is holomorphic in each variable.

I am going through Theorem $1.1$ from the book Holomorphic Functions and Integral Representations in Several Complex Variables by Michael Range (Page no. $43$) which states the following $:$ Theorem $...
Anacardium's user avatar
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Domain of convergence of a power series is a complete Reinhardt domain.

I am studying the book From Holomorphic Functions to Complex Manifolds by Grauert and Fritzsche. In proposition $2.7$ the authors showed that domain of convergence of a power series in $\mathbb C^n$ ...
Anacardium's user avatar
  • 2,612
1 vote
0 answers
70 views

On the definition of exhaustion functions

I'll first express my confusion:why can the exhaustion function of a domain in $\mathbb{C}^n$ be bounded? More precisely: In p. 45 of the book Partial differential equations in several complex ...
msecauchy's user avatar
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0 answers
83 views

About the ring of formal power series convergent on a neighborhood of $0$

Remember that $R_{n}=\mathbb{C}\{x_{1}, \cdots, x_{n}\}$ is the ring of formal power series convergent on a neighborhood of $0$. Note that $R_n$ is a local ring with maximal ideal $m_{n}:=\langle x_{1}...
user 987's user avatar
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1 vote
0 answers
37 views

What does it mean to "take perturbations", e.g. $\sigma \rightarrow \sigma + \delta \sigma$, and how can I extend this to complex numbers??

In this paper the author has an equation (30) which reads $\int_\Omega \sigma |\nabla u|^2 dx + \sum_\ell \int_{e_\ell} z_\ell \left(\sigma \nabla u \cdot \hat{n} \right)^2 dS = \sum_\ell V_\ell I_\...
George Gatling's user avatar
4 votes
1 answer
125 views

Is non-constant holomorphic map $f:\mathbb P^n\to \mathbb P^n$ surjective?

Let $\mathbb P^n$ denote the $n$-dimensional complex projective space. From Forster's book Lectures on Riemann surfaces p.11, Theorem 2.7, we know any non-constant holomorphic map $f:\mathbb P^1\to \...
Tom's user avatar
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0 votes
1 answer
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Show that this domain is Levi pseudoconvex and find all strictly Levi pseudoconvex points on boundary.

I'm troubling with an Exercise that comes from the book $\mathit{Holomorphic~Functions~in~Several~Complex~Variables~}$by R.Michael Range. Exercise 2.6 Define$~r(z)$ for$~z\in \mathbb{C}^2~$by$$r(z)=Re(...
xmu_student1776's user avatar
0 votes
1 answer
110 views

$-\log\delta_{\Omega}$ is plurisubharmonic.

I was reading the section of Pseudoconvexity and plurisubharmonicity of Hörmander's book (Introduction to complex analysis in several variables), and I have a doubt regarding the proof of the ...
user 987's user avatar
  • 645
0 votes
1 answer
78 views

Subharmonicity is a local property.

Let $D\subseteq \mathbb{C}$ be a domain. Definition. A function $u:D\to\mathbb{R}\cup\{-\infty\}$ is called subharmonic if $u$ is upper semicontinuous and if for every compact set $K\subseteq D$ and ...
user 987's user avatar
  • 645
2 votes
0 answers
76 views

$n$-th root of meromorphic functions of several complex variables

Let $f:\Omega\rightarrow\mathbb{C}$ be a "nice" meromorphic function of several complex variables on some domain $\Omega$. I wonder if the following claim is true. Claim. $f$ admits a global ...
GTA's user avatar
  • 186
0 votes
0 answers
40 views

On Domains of Holomorphy

Let $D\subseteq \mathbb{C}^{n}$ be a domain. We say that $D$ is a domain of holomorphy if, for every domain $V\subseteq \mathbb{C}^{n}$ for which $V\cap \partial D\not=\emptyset$ and for every ...
user 987's user avatar
  • 645
3 votes
0 answers
56 views

Localizing domain near strongly pseudoconvex boundary point

Let $D \subset \mathbb{C}^n$ be a smoothly bounded pseudoconvex domain and let $p \in \partial D$ be a strongly pseudoconvex boundary point . Often it is useful to localize the boundary of $D$ near ...
Clyde's user avatar
  • 903
1 vote
0 answers
61 views

The ball take away a submanifold of codimension 1 is a domain of holomorphy

Let $\mathbb{B}^n \subset \mathbb{C}^n$ be the unit ball and $V \subset \mathbb{B}^n$ be a closed complex submanifold of codimension one. How do we show that $\mathbb{B}^n \setminus V$ is a domain of ...
Clyde's user avatar
  • 903
4 votes
1 answer
204 views

Intuition for Reinhardt domains

I'm taking a complex analysis course, and in a lecture on several complex variables my professor defined Reinhardt domains in $\mathbb{C}^n$ as domains invariant under the action of the n-dimensional ...
Scv's user avatar
  • 41
2 votes
1 answer
150 views

Maximum principle for several complex variables

Suppose that we have an analytic function on the polydisc {$ \{z \in \mathbb{C}^n: \left|z_i\right| < 1, \forall i=1,\dots,n \} $} and continuous on the boundary. Can a non-constant function take ...
sr chunchurria's user avatar
1 vote
0 answers
22 views

Do these two quantities behave the same?

Let $\Phi_z(w)$ denote an involutive automorphism of the unit ball, $w,z\in \mathbb{B}^n$ and $f:\mathbb{B}^n\to \mathbb{C}$ be a holomorphic function. My question is the following: Let's take the ...
Beslikas Thanos's user avatar
4 votes
1 answer
123 views

A possible criterion for a function on a complex analytic space to be holomorphic

Let $(X, \mathcal O)$ be a reduced complex analytic space, and $f:X \rightarrow \mathbb C$ be a continuous function. Assume that for all holomorphic mappings $F$ from unit disc $D \subset \mathbb C$ ...
Matsmir's user avatar
  • 2,555
5 votes
3 answers
420 views

Textbooks on Complex Analytic Spaces

I'm looking for textbooks or monographs on complex analytic spaces. I'm aware of Coherent Analytic Sheaves by Grauert/Remmert and Several Complex Variables by Gunning/Rossi, but more references would ...
Lukas Heger's user avatar
  • 21.8k
1 vote
1 answer
164 views

Analogue to Osgood's lemma for rational functions

Osgood's Lemma says that if a function of several complex variables is continuous on an open set $D$ and holomorphic on each variable then it is holomorphic. Is there a similar result but for rational ...
sr chunchurria's user avatar
0 votes
0 answers
31 views

Integration of total derivative for several complex variables functions

Let $f: \mathbb{C}^m \rightarrow \mathbb{P}^1$ be a function. The total derivative of $f$ denoted by $Df = \sum_{j=1}^{m}z_j \frac{\partial f}{\partial z_j}$. Moreover, $D^{k+1}f = D(D^kf)$. Let us ...
Sayantan Maity's user avatar
1 vote
0 answers
41 views

Complex Monge Ampere measure with finite mass

I was reading this paper by Bedford-Taylor. It was proven in Theorem 2.4 that the Monge Ampere measure $(dd^c u)^n$ has a finite mass on any polydisc when $u$ is PSH and continuous up to the boundary ...
Zack math's user avatar
2 votes
0 answers
31 views

Connected component of globally analytic subset

I am reading the book Introduction to complex analytic geometry by Stanislaw Lojasiewicz. In Chapter II, we have that a subset $Z$ of a complex manifold $M$ is said to be a principal analytic subset ...
Curious's user avatar
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