Questions tagged [several-complex-variables]

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The boundary regularity of a Teichmüller domain

By a Teichmüller domain, I mean the Bers embedding of a Teichmüller space (of a compact oriented surface of finite type) in a complex space. It is known that the boundary of a Teichmüller domain is ...
Mahdi Teymuri Garakani's user avatar
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Residues and blow ups

On a 2-dimensional complex manifold consider two functions which are meromorphic with singularities along two divisors which meet at a point. There is a residue from these meromorphic functions (...
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Product of two circles and holomorphic functions

Assume that $f$ and $g$ are holomorphic functions in the unit disk having boundary values on the unit circle $T$ almost everywhere. Assume further that $$\int_0^{2\pi}\int_0^{2\pi}|f(e^{it})+g(e^{is})|...
AlphaHarmonic's user avatar
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Zeroes of entire function on $\mathbb C^n$

Let $n\ge 2$ be an integer and let $f$ be an entire function on $\mathbb C^n$. Let $A$ be a subset of $\mathbb R^n$ with positive $n$-dimensional Lebesgue measure. Then if $f$ vanishes at $A$, this ...
Bazin's user avatar
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Problem in understanding maximum principle for subharmonic functions

I am reading subharmonic functions and their properties from the book From Holomorphic Functions to Complex Manifolds by Grauert and Fritzsche. Let me first define what a subharmonic function is. ...
Anacardium's user avatar
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Fixed points free automorphisms of Teichmüller spaces

Let $\mathcal{T}_{g,n}$ be the Teichmüller space of a compact oriented surface of genus $g$ with $n$ marked points. Assume that $N:=3g-3+n>0$. Viewing $\mathcal{T}_{g,n}$ as a bounded domain in $\...
Mahdi Teymuri Garakani's user avatar
2 votes
1 answer
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An interior cone condition for Teichmuller spaces

Let $\mathcal{T}_{g,m}$ be the Teichmuller space of a compact oriented surface of genus $g$ with $m$ marked points. Consider it as a bounded domain in a complex space $\mathbb{C}^N$. Let $\xi$ be a ...
Mahdi Teymuri Garakani's user avatar
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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 ...
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Let $f: X \to D$ be a proper holomorphic submersion. Does a holomorphic form on $X$, closed on each fiber of $f$, have holomorphic coefficients?

Let $D$ be the unit disc in $\mathbb{C}^n$ and let $f: X \to D$ be a proper surjective holomorphic submersion, which is trivial as a smooth fiber bundle, with connected fiber $F$. We get an induced ...
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Complex geodesic coordinate, local ramified map, and the conic metric

Remark: I have asked this question in MSE, however, I got no responses. This is the reason I come to ask here. I am looking forward to some advices. Thanks in advance Let $X$ be a compact Kaehler ...
Invariance's user avatar
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Does Kobayashi isometry map preserve complex geodesics?

Let $\gamma_1, \gamma_2$ are real geodesics in a domain $D$ and these two real geodesics are lying in the same complex geodesics, the question is, are $f\left(\gamma_1\right)$ and $f\left(\gamma_2\...
Begginer-researcher's user avatar
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What is meant by saying that the Shilov boundary of the polydisc $\mathbb D^n$ is $\mathbb T^n\ $?

Let $A$ be a complex Banach algebra and $M_A$ be the space of all non-zero multiplicative linear functionals on $A$ equipped with the weak$^*$-topology. Let $\widehat A$ be the image of $A$ under the ...
Anacardium's user avatar
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Let $u$ be harmonic on domain $D\subset \mathbb R^d$, how far can we extend $u$ holomorphically?

If $u$ is harmonic then it is real analytic so then it can be extended locally holomorphically. I also know that if $u$ is harmonic on a ball in $\mathbb R^d$ we have that the radius of convergence is ...
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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 ...
GTA's user avatar
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pseudo inverse of a holomorphic multivariate injective map

Let $f:{\mathbb C}^n \rightarrow {\mathbb C}^N$, $N > n$, be holomorphic and injective on an open ball $B_n \subset {\mathbb C}^n$ such that the Jacobian matrices have full column rank at each ...
gil's user avatar
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A Hartogs analogue?

Let $0<r<1<R$, and $A:=\{z\in \ell^2: r<\|z\|_2<R\}$. For $n\in \mathbb{N}$, let $e_n$ denote the sequence in $\ell^2$ all of whose terms are zero, except the $n^{\text{th}}$ term, ...
Salla's user avatar
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Characterizing some similarity invariant homogeneous log-superharmonic functions of matrices

Let $L:M_n(\mathbb{C})^r\rightarrow[0,\infty)$ be a function that satisfies the following properties: $\log(L)$ is plurisubharmonic. $L$ is homogeneous in the sense that $L(\lambda A_1,\dots,\lambda ...
Joseph Van Name's user avatar
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A coradius of convergence - biggest open disk contained in the image of a power series?

Let $f \in \mathbb{C}\{z_1,\dots,z_n\}$ be non-constant with $f(0) = 0$, where $n \geq 1$, and let $D$ be its domain of convergence. Recall that for $n=1$ this is just some open disk $\mathbb{D}_r(0)$ ...
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Do we have a Grauert-Fischer theorem for non-trivial families?

This question is related to my previous question. Let $X$ be a compact complex manifolds and $\Delta\in \mathbb{C}^n$ be a small neighborhood of $0$. A family of deformations of $X$ over $\Delta$ is a ...
Zhaoting Wei's user avatar
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Inverse of Bochner–Martinelli formula

Suppose that $f$ is a holomorphic function on a domain $D$ in $\mathbb{C}^n$, $\partial D$ is smooth, and $f$ is $C^1$ on $\partial D$. Then, the Bochner-Martinelli formula states that $f(z) = \int_{\...
Chicken feed's user avatar
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Abelian subgroup of the automorphism group of $\mathbb C^n$

Let $Aut(\mathbb C^n)$ be the automorphism group of $\mathbb C^n$, i.e., the group of all biholomorphic maps of $\mathbb C^n \to \mathbb C^n$. Suppose $T$ is a finitely dimensional torus which is a ...
Adterram's user avatar
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1-convex and holomorphically convex

A complex manifold $M$ is called $1$-convex if there exists a smooth, exhaustive, plurisubharmonic function that is strictly plurisubharmonic outside a compact set of $M$. Can we prove that if $M$ is $...
Adterram's user avatar
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Is there a dense set of Lipschitz functions in $H^\infty(U)$, each of which maps $(1,0,\ldots,0)$ to 1, where $U$ is the unit ball in $\mathbb{C}^N$?

Let $U$ be the open unit ball in $\mathbb{C}^N$, let $A(U)$ be the algebra of functions analytic on $U$ and continuous on $\bar U$, and let $u=(1,0,\ldots,0)$. Let $\mathcal{B}=\{f\in H^\infty (U): \|...
David Walmsley's user avatar
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Characterization of a "complex" hull?

This is a complex continuation of my previous question. There Iosif Pinelis showed that the so obtained closure from taking the intersection of the preimages of the linear functionals indeed coincides ...
M.G.'s user avatar
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Estimate of minimum of the Poisson integrals corresponding to a convergent Hausdorff sequence of smooth bounded domains from below

Let $\{\Omega_{j}\}_{j\in\mathbb{N}}$ be a sequence of smooth bounded domains in $\mathbb{C}^{n}$ such that $\Omega_{j}$ converges to a smooth bounded domain $\Omega$ in the sense that the defining ...
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Determinant of the conormal bundle

Let $Y$ be a smooth submanifold of codimension $r$ in a complex manifold $X$. By virtue of the adjunction formula, we always have the isomorphism $$K_Y\simeq (K_X\otimes \det N_Y){\,|\,}_Y.$$ Recall ...
Invariance's user avatar
3 votes
1 answer
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Factorization of an analytic function in $\mathbb{C}^n$

Let $\Omega$ be an open subset of $\mathbb{C}^n$ and let $f$ be analytic in $\Omega$. Assume $P\in\mathbb{C}[z_1,\ldots,z_n]$ is a polynomial whose irreducible factors are all of multiplicity one. If $...
user111's user avatar
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Local integrability of $\log|f(x)|$ in several variables

If $f(z)$ is an analytic function in a complex neighborhood (in $\mathbb C^n$) of a real point $x^0 \in \mathbb R^n \subset \mathbb C^n$, then $\log|f(x)|$ is integrable over some neighborhood $U \...
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harmonic envelope of holomorphy

Let $D$ be a (real) domain in $\mathbb R^n=\mathbb R^n+i\lbrace 0 \rbrace\subset \mathbb C^n$. Then, due to P. Lelong, there exists a maximal (complex) domain $\tilde D\subset\mathbb C^n$, $D=\tilde D\...
Peter Pflug's user avatar
1 vote
2 answers
209 views

A characterization of plurisubharmonic functions

Let $\Omega\subset \mathbb{C}^n$ be an open subset. Let $u\colon \Omega\to [-\infty,+\infty)$ be an upper semi-continuous function. Recall that $u$ is called plurisubharmonic (psh) if its restriction ...
asv's user avatar
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Determine the coefficient of the exceptional divisor

Consider the following setting: suppose that $X$ is a smooth variety and let $f:X\rightarrow \Delta$ be a smooth morphism outside the origin $0$. Let the central fiber $X_0$ be a reduced (Cartier) ...
Invariance's user avatar
2 votes
0 answers
184 views

Maximum modulus principle for vector valued functions of several complex variables

In the following paper: Shub and Smale, "On the Existence of Generally Convergent Algorithms", Journal of Complexity 2, 2-11 (1986), trying to understand Lemma 2 on page 4. Paraphrased, ...
user125930's user avatar
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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
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Analogous tensor product operation for reflexive sheaf

Suppose now $(X,\mathcal O_X)$ is a normal complex space, and $\mathcal F$ is a coherent analytic sheaf on it. Product the reflexive sheaf $$\mathcal F^{[p]}:=(\mathcal F^{\otimes p})^{**},$$ where $\...
Invariance's user avatar
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
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Global sections of a line bundle on a reducible complex space

Let $S$ be a reducible compact complex analytic space, thus we have the decomposition $S=\bigcup_{i=1}^n {V_i}$ where $V_i$ is the irreducible component of $S$. Let $L$ be a line bundle on $S$, I ...
Invariance's user avatar
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L2 estimate on strongly pseudoconvex complex manifold

Suppose $(X,g,I)$ is a Hermitian (non kahler) complex manifold with small torsion, small derivative of torsion and small curvature. Let $\varphi$ be smooth PSH function satisfying $\sqrt{-1}\bar\...
xin fu's user avatar
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What can be said about cluster sets for power series of two variables?

I'm still trying to prove the continuity of a function $u$ which can be interpreted as the restriction of a power series of two variables, which I haven't managed to approach the right way yet. To ...
Raphael B's user avatar
2 votes
1 answer
197 views

Bishop's compactness theorem and convergence of analytic subset

Let $V_i$ be a sequence of $k$ dimensional analytic subsets in $\mathbb C^N$. Suppose that the volume of $V_i$ is uniformly bounded, then Bishop's compactness theorem says that $V_i$ will convergence ...
xin fu's user avatar
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1 answer
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Can a power series of several variables be discontinuous on a compact set if it converges in every point of this set?

Say we have a power series of two variables, with an associated function $f$ defined as $$ \begin{split} f(x, y) =\, & \sum_{n,m} a_{n,m}x^ny^m,\\ & a_{n,m} \geq 0 \quad \forall n, m \in\...
Raphael B's user avatar
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70 views

When a strictly positive log pluriharmonic function $g$ is equal to the norm of holomorphic function?

Suppose $V$ is a local analytic variety (singular). Suppose $g$ a strictly positive log pluriharmonic function on $V$, i.e. $\log g$ is pluriharmonic. I wonder when $g=|f|^2$, where $f$ is a ...
xin fu's user avatar
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Characterization of elements of Hardy Space

Let $\Omega\subset\mathbb{C}^n$ be a $C^{\infty}$ bounded domain. Let $H^2(\partial\Omega)$ denote the Hardy space, and $S(.,.)$ denote its Szego Kernel. We know that $$ \forall f\in H^2(\partial\...
Naruto's user avatar
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Do Szego Kernel in one variable by fixing another variable in a $C^{\infty}$ bounded domain is Bounded?

Let $\Omega\subset\mathbb{C}^n$ be any $C^{\infty}$ bounded domain. Let $ S(.,.)$ denotes the Szego Kenel of Holomorphic Hardy Space $H^2(\partial\Omega)$. Then for $w\in\Omega$ do $S(.,w)$ is a ...
Naruto's user avatar
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On the definition of Cauchy transform [closed]

I have seen two different definitions of the Cauchy transform of a smooth function one is with respect to the line integral (for eg. in. the book "The Cauchy transform and potential theory")...
naruto's user avatar
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1 answer
215 views

Extension of a Szegő Kernel to the boundary

Let $\Omega\subset\mathbb{C}^n$ be any smooth bounded pseudoconvex domain. Let $S$ denote the Szegő kernel of $\Omega$. Recall: the Szegő kernel is a kernel of the Szegő projection $P: L^{2}(\partial\...
Naruto's user avatar
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Show that this holomorphic function can be extended to $D_{2}((0,0) ;(2,2))$ [closed]

consider a domain in $C^{2}$:$\Omega=D_{2}((0,0) ;(1,2)) \cup\left\{(z, w) \in \mathbb{C}^{2}:|z|<2 \text { and } 1<|w|<2\right\}$ and $f \in \operatorname{Hol}(\Omega)$, I want to show that ...
吴yuer's user avatar
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A question on the proof of Bedford-Taylor theorem in Demailly's book

I am trying to understand a proof of the Bedford-Taylor theorem on the weak convergence of Monge-Ampere operators of decreasing sequences of plurisubharmonic functions. I am reading a proof in the ...
asv's user avatar
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Starlike sets in $\mathbb{C}^n$

Let $S$ be a bounded domain in $\mathbb{C}^n$. $S$ is called starlike about the point $x_0\in S$ if for every point of $S$, the segment of the straight line from the point to $x_0$ lies in $S$. If $S$ ...
user332912's user avatar
2 votes
0 answers
31 views

Relation between polynomial convexity and Runge-Stein neighborhood basis

I am searching for some reference about the relation between polynomial convexity and Runge-Stein neighborhood basis for a compact set $K$ inside $\Bbb C^n$. I read on this paper, Remark 3.1, that ...
Joe's user avatar
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Number of roots of a Vandermonde like complex determinant

I am originally interested in the determinant $$ \left|\begin{array}{cccc}\exp(i\lambda_1\cdot x_1) & \exp(i\lambda_2\cdot x_1) & ... & \exp(i\lambda_n\cdot x_1) \\\exp(i\lambda_1\cdot x_2)...
kaleidoscop's user avatar
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