Armen Edigarian, Said El Marzguioui, Jan Wiegerinck
We prove that the image of a finely holomorphic map on a fine domain in $\mathbb{C}$ is pluripolar subset of $\mathbb{C}^{n}$. We also discuss the relationship between pluripolar hulls and finely holomorphic function.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We prove that every bounded finely plurisubharmonic function can be locally (in the pluri-fine topology) written as the difference of two usual plurisubharmonic functions. As a consequence finely plurisubharmonic functions are continuous with respect to the pluri-fine topology. Moreover we show that -infinity sets of finely plurisubharmonic functions are pluripolar, hence graphs of finely holomorphic functions are pluripolar.
We prove that the graph of a continuous function $f$, defined on a domain of ${\mathbb C}^n$, is pluripolar if and only if $f$ is holomorphic.
We study the pluripolar hull of the graph of a holomorphic function f, defined on a domain D in the complex plane outside a polar set A of D. This leads to a theorem that describes under what conditions f is nowhere extendable over A, while the graph of f over D \ A is NOT complete pluripolar.
Let A be a closed polar subset of a domain D in the complex plane C. We give a complete description of the pluripolar hull in D X C of the graph of a holomorphic function defined on D A. To achieve this, we prove for pluriharmonic measure certain semi-continuity properties and a localization principle.
We prove an analogue of Sadullaev's theorem concerning the size of the set where a maximal totally real manifold can meet a pluripolar set. The manifold has to be of class C-1 only. This readily leads to a version of Shcherbina's theorem for C-1 functions f that are defined in a neighborhood of certain compact sets K in the complex plane. If the graph of f on K is pluripolar, then f satisfies the Cauchy Riemann equations in the closure of the fine interior of K.
The aim of this paper is to present an extension theorem for the functions separately holomorphic on generalized (N,k)-crosses with pluripolar singularities.
We prove that in the extension theorem for separately holomorphic functions on an $N$-fold cross with singularities the case of analytic singularities follows from the case of pluripolar singularities.
We show that if the graph of a bounded analytic function in the unit disk $\mathbb D$ is not complete pluripolar in $\mathbb C^2$ then the projection of the closure of its pluripolar hull contains a fine neighborhood of a point $p \in \partial \mathbb D$. On the other hand we show that if an analytic function $f$ in $\mathbb D$ extends to a function $\mathcal{F}$ which is defined on a fine neighborhood of a point $p \in \partial \mathbb D$ and is finely analytic at $p$ then the pluripolar hull of the graph of $f$ contains the graph of $\mathcal{F}$ over a smaller fine neighborhood of $p$. We g…
Let $D_j\subset\Bbb C^{n_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluriregular set, $j=1,...,N$. Put $$ X:=\bigcup_{j=1}^N A_1\times...\times A_{j-1}\times D_j\times A_{j+1}\times ...\times A_N\subset\Bbb C^{n_1}\times...\times\Bbb C^{n_N}=\Bbb C^n. $$ Let $U\subset\Bbb C^n$ be an open neighborhood of $X$ and let $M\subset U$ be a relatively closed subset of $U$. For $j\in\{1,...,N\}$ let $Σ_j$ be the set of all $(z',z'')\in(A_1\times...\times A_{j-1}) \times(A_{j+1}\times...\times A_N)$ for which the fiber $M_{(z',\cdot,z'')}:=\{z_j\in\Bbb C^{n_j}\: (z',z_j,z'')\in M…
We prove the following theorem: Let X be a Stein manifold of dimension at least 2 and Y a closed complete pluripolar subset of X. Given a point p in the complement of Y there is a proper holomorphic map f from the unit disc to X such that f(0)=p and the image of f avoids Y.
We study connectedness in the pluri-fine topology on $\CC^n$ and obtain the following results. If $Ω$ is a pluri-finely open and pluri-finely connected set in $\CC^n$ and $E\subset\CC^n$ is pluripolar, then $Ω\setminus E$ is pluri-finely connected. The proof hinges on precise information about the structure of open sets in the pluri-fine topology: Let $Ω$ be a pluri-finely open subset of $\CC^{n}$. If $z$ is any point in $Ω$, and $L$ is a complex line passing through $z$, then obviously $Ω\cap L$ is a finely open neighborhood of $z$ in $L$. Now let $C_L$ denote the finely connected component o…
Examples by Poletsky and the author and by Zwonek show the existence nowhere extendable holomorphic functions with the property that the pluripolar hull of their graphs is much larger than the graph of the respective functions and contains multiple sheets. We will explain this phenomenon by fine analytic continuation of the function over part of a Cantor-type set involved. This gives more information on the hull, and allows for weakening and effectiveness of the conditions in the original examples.
The pluripolar hull of a pluripolar set E in $\mathbb{P}^n$ is the intersection of all complete pluripolar sets in $\mathbb{P}^n$ that contain $E$. We prove that the pluripolar hull of each compact pluripolar set in $\mathbb{P}^n$ is $F_σ$. The convergence set of a divergent formal power series $f(z_{0}, \dots,z_{n})$ is the set of all "directions" $ξ\in\mathbb{P}^{n}$ along which $f$ is convergent. We prove that the union of the pluripolar hulls of a countable collection of compact pluripolar sets in $\mathbb{P}^n$ is the convergence set of some divergent series $f$. The convergence sets on $…
We show that fine domains in $\mathbf{C}$ with the property that they are Euclidean $F_σ$ and $G_δ$, are in fact fine domains of existence for finely holomorphic functions. Moreover \emph{regular} fine domains are also fine domains of existence. Next we show that fine domains such as $\mathbf{C}\setminus \mathbf{Q}$ or $\mathbf{C}\setminus (\mathbf{Q}\times i\mathbf{Q})$, more specifically fine domains $V$ with the properties that their complement contains a non-empty polar set $E$ that is of the first Baire category in its Euclidean closure $K$ and that $(K\setminus E)\subset V$, are NOT fine…
$J$-holomorphic curves are pluripolar, but they are not minus-infinity sets of pluri-subharmonic functions with logarithmic singularity.
Let $f$ be a holomorphic endomorphism of $\mathbb{CP}^k$ having an attracting set $A$. In this paper, we address the question of the "size" of $A$ in a pluripolar sense. We introduce a conceptually simple framework to have non-algebraic attracting sets. We prove that adding a dimensional condition, these sets support a closed positive current with bounded quasi-potential (which answers a question from T.C. Dinh). Therefore, they are not pluripolar. Moreover, the examples are abundant on $\mathbb{CP}^k$.
We prove that for a bounded domain in $\mathbb C^n$ with the Bergman metric of constant holomorphic sectional curvature being biholomorphic to a ball is equivalent to the hyperconvexity or the exhaustiveness of the Bergman-Calabi diastasis. By finding its connection with the Bergman representative coordinate, we give explicit formulas of the Bergman-Calabi diastasis and show that it has bounded gradient. In particular, we prove that any bounded domain whose Bergman metric has constant holomorphic sectional curvature is Lu Qi-Keng. We also extend a theorem of Lu towards the incomplete situation…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。