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.
751
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$-\log\delta_{\Omega}$ 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 ...
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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 ...
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60
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$n$-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 ...
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32
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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 ...
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Continuity of roots for holomorphic functions in several complex variables
Let $f(x,y)$ be a holomorphic function on a open $U \subset \mathbb{C}^2$ where $0 \in U$ and $f(0,0) = 0$. Suppose $y \mapsto f(0,y)$ is not identically zero near the origin and its order of ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
- 1,035
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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 $D \subset \mathbb C$ ...
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3
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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 ...
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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 ...
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25
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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 ...
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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 ...
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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 ...
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Maximal number of zeroes of analytic function on each fiber is finite
Let $f(z,w)$ be a holomorphic function on the polydisc $D\times D\subset \mathbb{C}^2$. Suppose that the zeroes of $f$ do not accumulate on $D\times\partial D$. Is it true that
$$ \text{max}_{z\in D}\#...
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Functions in disk algebra with summable series coefficients
We know that there are functions $f \in {A}(\mathbb{D}) = C(\overline{\mathbb{D}}) \cap \mathrm{Hol}(\mathbb{D})$ such that $f(z) = \sum_{k\geq0}a_k z^k$ on $\mathbb{D}$ with $a=(a_k)_{k\geq0} \notin \...
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Smooth bounded planar domain is tangent to the unit sphere
Let $D\subset\mathbb{C}$ be a smooth bounded planar domain. Let $p\in\partial {D}$ be any boundary point. Then My Teacher said $D$ is tangent to the unit sphere $S^{1}$ at $p$ up to any order, up to ...
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Does the Open Mapping Theorem hold for invertible holomorphic function of several variables?
Suppose there is a holomorphic function $\mathbf{f}:U\rightarrow V$ with several complex variables, where $U$ and $V$ are both open sets in $\mathbb{C}^n$. If the Jacobian matrix of $\mathbf{f}$ on $...
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Open mapping theorem $f:\mathbb{C}^n\to \mathbb{C}^n$
Is there any analogue of open mapping theorem for function $f:\Omega\subseteq\mathbb{C}^n\to \mathbb{C}^n$, $n>1$? I found some questions posted here(discussing similar problem) but couldn’t find ...
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Dimension of zero set of complex analytic function
Let $f$ be an analytic function on a domain $\Omega\subset \mathbb{C}^n$. I am trying to prove that the dimension of $Z_f=f^{-1}(0)$ is $n-1$. This is stated in this answer, but I do not understand ...
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1
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On the Hironaka resolution for a non-normal variety
The following is a well-known version of the famous Hironaka resolution.
Let $X$ be a compact complex space. Then there exists a finite sequence of blow-ups with smooth centers
$$
X_N \stackrel{\...
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1
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Is there a version of Parseval formula on $\mathbb{C}^n$?
I found in some article (analytic methods in algebraic geometry by Demailly) that the following formula holds
$\int_{B(0,r_0)\subset\mathbb{C}^n}\frac{|\sum a_\alpha z^\alpha|^2}{|z|^{2\gamma}}dV(z)=\...
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Cuts and discontinuities of multivalued complex functions with several variables
Can somebody recommend some literature regarding cuts and discontinuities of multivalued complex functions with several variables?
Thanks!
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1
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Is the fiber product $X\times_Y Z$ locally irreducible when $X$, $Y$, and $Z$ are all locally irreducible?
The space $X$ is called irreducible at $x$ if the stalk $\mathcal{O}_x$ of its structure sheaf is an integral domain.
The space $X$ is called locally irreducible if all points of $X$ are irreducible.
...
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Can we construct a compact complete pluripolar set?
We know that analytic sets are not relatively compact in $\mathbb{C}^n$ when $n\geq 2$. In fact, we can construct plurisubharmonic functions by holomorphic functions. So analytic subset belong to ...
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boundary of an bidisc as a disjoint union
I found in "Several complex variables" by Jaap Korevaar and Jan Wiegnerick (https://www.math.stonybrook.edu/~ebedford/PapersForM537/WiegerinckKorevaar.pdf) in Chapter 1.2 that the boundary ...
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Notation in Siu's paper
In the following paper of Siu Yum-Tong:https://www.ams.org/journals/proc/1967-018-05/S0002-9939-1967-0216032-3/S0002-9939-1967-0216032-3.pdf, which he proved that Cartan B implies A, he used the ...
user1055643
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Prove Metric Space is Complete
Define $\mathcal{H}_\lambda$ to be the space of functions $f$ holomorphic in $\mathbb{C}^{n−1}$, for which $$\|f\|_{\mathcal{H}_\lambda}^2 \equiv \int_{\mathbb{C}^{n−1}}|f(z)|^2 e^{-4\pi\lambda|z|^2}...
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How these two quantities compare in $\mathbb{C}^n$
Let $z=(z_1,…,z_n),w=(w_1,…,w_n) \in \mathbb{B}$ (the open unit ball in $\mathbb{C}^n$. I want to compare the quantities $A=|1-\left\langle z,w\right\rangle|$ and $|z-w|$ where $|z|^2=|z_1|^2+ \ldots +...
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Exterior algebra of the complex vector space $\Lambda^{p+q}V_\mathbb{C}$ and exterior products of $\Lambda^p V^{1,0}\otimes \Lambda^q V^{0,1}$
Let $(V,J)$ be an almost complex vector space. $V_\mathbb{C}$ is the complexification of $V$.
I have the following questions:
$1.$ A $k$-forms on $V_\mathbb{C}$ is a $k$-$\mathbb{C} $ linear and ...
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Converse of Hurwitz theorem in several complex variables
Let $\phi_{n}: \overline{\Omega} \to \mathbb{C}^{n}$ be e sequence of holomorphic map converges to a $\mathcal{C}^{2}$ smooth differmorphism $f: \overline{\Omega} \to f(\overline{\Omega})$, with $f \...
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Question regarding an inequality of a function which is holomorphuc in first variable and anti-holomorphic in second variable
Let $D$ be a domain in $\mathbb{C}^{n}$, where $n>1$. Let $F:D×D\to\mathbb{C}$ be a function defined as: $F(z,w)$ is holomorphic function in first variable $z$ and $F(z,w)$ is anti-holomorphic in ...
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On the Weierstrass Preparation Theorem and Weierstrass Division Theorem
Denote by $R_{n}=\mathbb{C}\{z_{1},\cdots, z_{n}\}$ the ring of convergent power series. We say that
$f\in \mathbb{C}\{z_{1},\cdots, z_{n}\}$ is $z_n$-general of order $m$ if there is an $h\in\mathbb{...
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Cauchy transform in polydisc
It is well known that for $p\in(1,\infty)$ and any $f\in L^p(\mathbb{T})$ function $Cf(z)=\int\limits_{\mathbb{T}}\frac{f(\zeta)}{1-\bar\zeta z}dm(\zeta)$ (where $m$ is normalized Lebesgue measure on ...
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Princples of analytic continuation in several complex variables
I was studying several complex variables and am little bit confused about analytic continuation here. My questions are as follows:
(i) why we need connected set here? I understand a connected set ...
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Existance of holomorphic extension of funcion in $O(\mathbb{C}^2 - \mathbb{R}^2)$ to $\mathbb{C}^2$ [duplicate]
Consider $\mathbb{R}^2$ in $\mathbb{C}^2$ (natural embedding). Need show if $f \in O(\mathbb{C}^2 - \mathbb{R}^2)$ then it extends to a holomorphic function of $\mathbb{C}^2$.
The hint that has been ...
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1
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Why should analytic classes sit inside $H^{p,p}(X)$ for various values of $p>0$?
I am reading these notes notes by Popa. I did some reading of this post but I didn't exactly answer my question.
It is claimed in example 4.5 that analytic classes sit inside $H^{p,p}(X)$ for various ...
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Analytic Continuation in Several Complex Variables
I was trying to use the following theoerem to prove analytic continuation. Here it is:
Let $f: \Omega \to \mathbb{C}^m$ be holomorphic where $\Omega \subseteq \mathbb{C}^n$ is open. If $\Omega$ is ...
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Theorem 2.4.5 of Hormander's book
Hi I was trying to understand Theoerem 2.4.5 of Hormander's book "An Introduction to Complex Analysis in Several Variables". It shows the existence and uniqueness of power series expansion ...
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Holomorphic function on several variables
A map $f:U\ni(z_1,\dots,z_n)\mapsto f(z_1,\dots,z_n)\in\mathbb C$ (with $U\subseteq\mathbb C^n$ open) is called holomorphic if $f$ is real continuously differentiable and for every $i\in\{1,\dots,n\}$,...
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Difference between strongly and strictly pseudoconvex domains in $\mathbb{C}^n$
Could anyone help me find the difference between strictly and strongly pseudoconvex domains in $\mathbb{C}^n$? I managed to find in the literature only the definition of strictly pseudoconvex domains (...
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1
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Generalized Liouville Theorem for several complex variables
I am trying to prove the following theorem:
Let $f:\mathbb{C}^{n}\to \mathbb{C}$ be a holomorphic function, and let, for $z\in \mathbb{C}^{n}$, there exists $A$ and $k$ constants, such that,
$$
|f(z)|...
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Applications of Pluripotential Theory in real world
I am reading for a math PhD with research in Pluripotential Theory (a subfield in Several Complex Variables). I particularly do study and develop theory related to extremal functions associated with a ...
- 5,705
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Preimage of domain of holomorphy is also a domain of holomorphy
I'm working on an exercise concerning about domain of holomorphy from Krantz's Function Theory of Several Complex Variables:
Let $\Omega \subset \mathbb{C}^{n}$ be a domain of holomorphy and let $\...
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Why Second Cousin problem is generalization of the Weierstrass’ Theorem
The Weierstrass’ problem in one complex variable asked, if there is holomorphic function with the pre-assigned zeros in $\Bbb{C}$? This is proved using the Weierstrass infinite product.
The second ...
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2
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How to prove the partial derivative of a holomorphic function is holomorphic and obtained by differentiating the series term by term?
Let $U\subset\mathbb{C}^n$ be open. Suppose $f:U\rightarrow\mathbb{C}$ is holomorphic, then by the definition, $f$ can be expressed by an absolutely and uniformly convergent power series,
$$f(\mathbf{...
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1
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Hartog's theorem why continuity of $f$ is needed in the first step?
I was reading Paul Garrett's note about Hartogs’ Theorem . In the first step to prove separate analyticity implies joint.
We assume $f$ is a separate analytic function on a polydisk $U\subset \Bbb{C}^...
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0
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Why the $\alpha$-th term in the sum of series does not exceed a bound?
Recently, I read Steven G. Krantz's book of Function theory of Several Complex Variables. I am confusing about a proof of a proposition. The proposition is expressed as follows:
If $f:\Omega\subset\...
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