# Questions tagged [several-complex-variables]

The several-complex-variables tag has no usage guidance.

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### Automatic plurisubharmonicity for a non-negative function

I feel confused about a point in this very short paper. On the top of page 3, it is claimed that:
If $S$ is a totally real submanifold in a compact almost complex manifold $(X,J)$, then any function ...

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### Equations needed to define a normal complex surface singularity

This questions is highly related with this other question of mine: Irreducible surface singularity that is not a local set-theoretical complete intersection I just thought that a different look at the ...

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### Irreducible surface singularity that is not a local set-theoretical complete intersection

I have been looking for a criterion for the germ of an irreducible complex surface singularity $(X,x)$ to be a set-theoretical complete intersection.
A germ $(X,x)$ of an isolated complex singularity ...

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110 views

### Computing the convex hull of a region of $\mathbb{C}^2$

Consider a function $f(z, w)$ of two complex variables. The function is symmetric with respect to $z$ and $w$. When $\Re(z)>0$ and $\Re(w)>0$, the function is analytic in its two variables. When ...

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### On Remmerts reduction

Let $(X,0)$ be a normal surface singularity. An let $\pi: \tilde{X} \to X$ be the minimal resolution. Now, we can apply a result of Oliveira (exploiting previous work by Laufer) and obtain a 1-...

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### Interpretation of deformation of complex structure

Let $X$ be a smooth complex analytic space and let $D$ be the unit disk in $\mathbb{C}$. Let $\omega:Y \to D$ be a deformation of complex structures of $X$ in the sense that (1) $\omega^{-1}(0) \simeq ...

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### Analytic continuation of a Dirichlet series with several complex variables

For $w_1,w_2,z_1,z_2\in\mathbb{C}$ with $\operatorname{Re}(w_1)>0$ and $\operatorname{Re}(w_2)>0$, define
\begin{equation*}
U(w_1,w_2;z_1,z_2):=\prod_{p}\left(1-\frac{e^{z_1}}{p^{1+w_1}}-\frac{e^...

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269 views

### Does Bergman metric induce the standard topology?

I am a physics student and am interested in the study of invariant metrics. I have searched several textbooks, including those fat books of Krantz, but the following concern seems not to be mentioned ...

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### 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 ...

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198 views

### Exponential Sequence of Sheaves

Let $(X, \mathcal{O}_X)$ be a complex analytic space in the sense of Grauert, i.e., a $\mathbb{C}$-analytic ringed space which is locally isomorphic to a local model. We may assume that $X$ is a ...

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452 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)...

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### A question on f.g. ideals of $\textrm{Hol}(\mathbb{C}^2,\mathbb{C})$

Suppose that $I$ and $J$ are finitely generated ideals of the ring $\textrm{Hol}(\mathbb{C}^2,\mathbb{C})$ of all entire functions in two complex variables.
Then is $I\cap J$ finitely generated too?

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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 (\...

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### 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 ...

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### Re-expressing $\operatorname{Hess}_{\rho}(L,N)\cdot(\nabla_N\overline L)\rho$

Let $\Omega\subseteq\Bbb C^n$ be a pseudoconvex domain.
Let $r,\rho$ two defining functions for $\Omega$. Then it is known that they are related by $\rho=re^{\psi}$ for a suitable real smooth ...

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### 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 ...

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65 views

### Can this sum be rearranged?

Let $\phi(\xi) :\mathbb{D} \to \mathbb{D}$ be holomorphic; $\mathbb{D}$ is the unit disk. Let $\phi(0) = 0$, and $0<\phi'(0) = \lambda <1$. Let $\vartheta(w,\xi) = \sum_{n=0}^\infty \phi^{\circ ...

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235 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+...

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419 views

### A question about openness theorem

The openness theorem says that:
If $\varphi$ be a negative plurisubharmonic function
in the unit ball $B(0,1)$ in $\mathbb{C}^{n}$ satisfying
$$
\intop_{B(0,1)}e^{-\varphi}<\infty,
$$
then there ...

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275 views

### Let $h(z) = g(f(z))$. If $f$ and $h$ are non-constant holomorphic function on domains in $\mathbb C^n$, then is $g$ holomorphic?

Suppose there exist functions $f,g,h$ such that $h(z) = g(f(z))$. If $f$ and $h$ are non-constant holomorphic function on domains in $\mathbb C^n,\, n>1$, then is $g$ holomorphic? We also assume ...

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278 views

### Equivalence of the term “Divisor”

Throughout my university education, I have studied some theory of Riemann surfaces, focusing particularly on Miranda's Algebraic curves and Riemann surfaces. My current studies however are in the ...

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### Can any plurisubharmonic function be represented as a sum of non-positive plurisubharmonic functions?

Let $f$ be a plurisubharmonic function, $f < 0$ in $\Omega$. Can we always find two negative plurisubharmonic functions $u$ and $v$ and real numbers $a,b\in(-1,1)$ such that
$$-f=(-u)^{a}+(-v)^{b}$...

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### What do we necessarily need for the image of a domain of holomorphy to be a domain of holomorphy

I posted this on Math.Stack.Exchange with no luck, so I thought it would be perhaps better suited for this site.
We recall that a domain of holomorphy is a domain in $\mathbb{C}^n$ that is ...

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274 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 ...

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122 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 \...

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### 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 ...

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176 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 (...

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172 views

### Modern 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 Haus Grauert.
Theorem. A proper holomorphic submersion with ...

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266 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:...

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### 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 ...

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### 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. ...

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### 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^{-...

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### Extending the projective action of several positive linear maps to a complex neighbourhood

I am currently reading a paper which, somewhat indirectly, asserts the following result:
Lemma: Let $\Delta \subset \mathbb{R}^d$ denote the simplex $\{(x_1,\ldots,x_d):\sum_{i=1}^d x_i=1\}$, let $...

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### 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. ...

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### Is the projection of a pseudoconvex domain necessarily pseudoconvex?

Is the projection of a pseudoconvex domain necessarily pseudoconvex?
I think that it is not necessarily true, but I cannot come up with an example.

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### 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 $\mathbb{C}$
$$\...

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168 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, ...

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### 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 ...

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271 views

### Bounded holomorphic functions in unbounded domain

Let $D$ be an unbounded pseudoconvex domain in $\mathbb{C}^2$. I would like to study the peak set of $D$.
1)Can the peak set of $D$ be empty? Or
2) Does $D$ always admit a nonconstant bounded ...

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### Continuity (and possibly smoothness) of a multivariable powerseries with positive coefficients bounded on a curve

Consider a multivariable power series with positive coefficients such that it is known to converge on a $C^\infty$ (bounded) curve of $\mathbb{R}^n$, where $n$ is the number of variables. In addition, ...

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### How to nominate a property of Kähler potential？

Let $u$ be a PSH function on $\mathbb{C}^n$. Assume the eigenvalues of $\partial\bar\partial u$ are comparable. Namely there is a constant $\Lambda>1$ such that $\max_{i}\mu_i(x)\leq\Lambda\min_{j}...

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### 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 ?

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### Does a bounded convex domain has one smooth boundary point?

In the study of analysis and geometry of a bounded domain, its boundary regularity is important. For example, it is known that a bounded convex domain has Lipschitz bounday. This implies that a ...

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428 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 $X_s$, ...

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### 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&...

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278 views

### Jensen formula in $\mathbb{C}^n$?

Let $f:\mathbb{C}\to\mathbb{C}$ be an entire function with zero set $X\subset \mathbb{C}$. Jensen's formula reads
$$
\log(|f(0)|)+\int_0^R\frac{|X\cap B_t(0)|}{t}dt = \frac{1}{2\pi}\int_0^{2\pi}\log(|...

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### 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
...

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133 views

### Domains of holomorphy and simply connected domains

Let $U\subset\mathbb{C}^n$ be a domain of holomorphy, we can say that $U$ is a simply connected domain?
Any hints would be appreciated.

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### To show there exists a unique function $u \in C^{1}(\mathbb{C^n})$ that satisfies $(\bar{\partial u})=f$

Assume $n \gt 1$. Let $f$ be a $(0,1)$ form in $\mathbb{C^n}$, with $C^1$-coefficients and compact support $K$, such that $\bar{\partial} f=0$. Let $\Omega_{0}$ be the unbounded component of $\mathbb{...

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347 views

### Hartogs's extension theorem

Let $(P,H)$ be a Euclidean Hartogs figure in $\mathbb{C}^n$, and
let $f:H\to \mathbb{C}^n$ be a holomorphic injective map. Then we know that $f$ extends holomorphically to the polydisc $P$, i.e. there ...