Determination of the pluripolar hull of graphs of certain holomorphic functions

Pluripolar Hulls and Fine Holomorphy

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.

Pluripolar graphs are holomorphic

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.

The image of a finely holomorphic map is pluripolar

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.

A Hartogs type extension theorem for generalized (N,k)-crosses with pluripolar singularities

The aim of this paper is to present an extension theorem for the functions separately holomorphic on generalized (N,k)-crosses with pluripolar singularities.

Cross theorem with singularities - pluripolar vs. analytic case

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.

An extension theorem for separately holomorphic functions with pluripolar singularities

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…

Continuity Properties of Finely Plurisubharmonic Functions and pluripolarity

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.

Proper discs in Stein manifolds avoiding complete pluripolar sets

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.

Graphs that are not complete pluripolar

Let D_1 be a subdomain of D_2 in the complex plane CC. Under very mild conditions on D_2 we show that there exist holomorphic functions f, defined on D_1 with the property that $f$ is nowhere extendible across the boundary of D_1, while the graph of f over D_1 is NOT complete pluripolar in D_2 times CC. This refutes a conjecture of Levenberg, Martin and Poletsky.

Shcherbina's Theorem for Finely Holomorphic Functions

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.

An extension theorem for separately meromorphic functions with pluripolar singularities

Let $D_j\subset\mathbb C^{n_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluriregular set, $j=1,...,N$. Put $$ X:=\bigcup_{j=1}^N A_1\times...\times A_{j-1}\times D_j\times A_{j+1}\times...\times A_N. $$ Let $M\subset X$ be relatively closed. For any $j\in\{1,...,N\}$ let $Σ_j$ be the set of all $(z',z'')\in(A_1\times...\times A_{j-1})\times(A_{j+1}\times...\times A_N)$ such that the fiber $M_{(z',\cdot,z'')}:=\{z_j\in\mathbb C^{n_j}: (z',z_j,z'')\in M\}$ is not pluripolar. Assume that $Σ_1,...,Σ_N$ are pluripolar. Put ${multline*} X':=\bigcup_{j=1}^N\{(z',z_j,z'')\in(A_1…

Equidistribution of non-pluripolar products associated with quasi-plurisubharmonic functions of finite energy

In this article, we consider currents given by the p-fold non-pluripolar product associated with a quasi-plurisubharmonic function of finite energy, and prove that normalized pull-backs of such currents converge to the Green (p, p)-current exponentially fast in the sense of currents for holomorphic endomorphisms of Pk.

紧凯勒流形上非多重极电流的等分布

Equidistribution for non-pluripolar currents on compact Kähler manifolds

本文研究了紧凯勒流形上具有简单上同调作用的满射全纯自同态，证明了非多重极电流在迭代拉回下的归一化序列收敛到格林电流，揭示了动力系统与多重位势理论之间的深层联系。

Pluri-polarity in almost complex structures

$J$-holomorphic curves are pluripolar, but they are not minus-infinity sets of pluri-subharmonic functions with logarithmic singularity.

Plurisubharmonic and holomorphic functions relative to the plurifine topology

A weak and a strong concept of plurifinely plurisubharmonic and plurifinely holomorphic functions are introduced. Strong will imply weak. The weak concept is studied further. A function f is weakly plurifinely plurisubharmonic if and only if f o h is finely subharmonic for all complex affine-linear maps h. As a consequence, the regularization in the plurifine topology of a pointwise supremum of such functions is weakly plurifinely plurisubharmonic, and it differs from the pointwise supremum at most on a pluripolar set. Weak plurifine plurisubharmonicity and weak plurifine holomorphy are preser…

紧致三维流形的全纯包

Holomorphic Hulls for Compact 3-Manifolds

研究紧致实三维流形嵌入到C^3中沿复切线的全纯包结构，通过构造嵌入控制Bishop不变量，揭示椭圆、双曲和抛物点的不同局部包性质。

具有无界边界值的复Dirichlet问题

Complex Dirichlet problem with unbounded boundary values

研究无界边界数据的复Dirichlet问题，提出基于局部双全纯形变与障碍函数的内部连续性定理，并证明非单调收敛、稳定性与唯一性。

有界域间的全纯Bergman等距映射与Lu定理的推广

Holomorphic Bergman isometries between bounded domains and the extension of Lu's theorem

该研究探讨复流形间保持Bergman度量的全纯映射，得出等维情形下全纯Bergman等距映射的性质，推广Lu定理并给出相关应用。

加权全纯映射理想的单射包

The injective hull of ideals of weighted holomorphic mappings

本文引入并刻画了加权全纯映射赋范理想的单射包，描述了由复合与对偶生成的单射包，并基于 Ehrling 型不等式建立了算子理想闭单射包的刻画。

A general version of the Hartogs extension theorem for separately holomorphic mappings between complex analytic spaces

Using recent development in Poletsky theory of discs, we prove the following result: Let $X,$ $Y$ be two complex manifolds, let $Z$ be a complex analytic space which possesses the Hartogs extension property, let $A$ (resp. $B$) be a non locally pluripolar subset of $X$ (resp. $Y$). We show that every separately holomorphic mapping $f: W:=(A\times Y) \cup (X\times B)\longrightarrow Z$ extends to a holomorphic mapping $\hat{f}$ on $\hat{W}:=\left\lbrace(z,w)\in X\times Y:\ \widetildeω(z,A,X)+\widetildeω(w,B,Y)<1 \right\rbrace$ such that $\hat{f}=f$ on $W\cap \hat{W},$ where $\widetildeω(\cdot,A,…

On the size of attractors in $\mathbb{CP}^k$

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 $\mathbb{CP}^k$.

Cross theorems with singularities

We establish extension theorems for separately holomorphic mappings defined on sets of the form W\setminus M with values in a complex analytic space which possesses the Hartogs extension property. Here W is a 2-fold cross of arbitrary complex manifolds and M is a set of singularities which is locally pluripolar (resp. thin) in fibers.

Characterizations of Projective Hulls by Analytic Discs

The notion of the projective hull of a compact set in a complex projective space was introduced by Harvey and Lawson in 2006. In this paper we describe the projective hull by Poletsky sequences of analytic discs, in analogy to the known descriptions of the holomorphic and the plurisubharmonic hull.

Plurisubharmonic geodesics and interpolating sets

We apply a notion of geodesics of plurisubharmonic functions to interpolation of compact subsets of $C^n$. Namely, two non-pluripolar, polynomially closed, compact subsets of $C^n$ are interpolated as level sets $L_t=\{z: u_t(z)=-1\}$ for the geodesic $u_t$ between their relative extremal functions with respect to any ambient bounded domain. The sets $L_t$ are described in terms of certain holomorphic hulls. In the toric case, it is shown that the relative Monge-Ampère capacities of $L_t$ satisfy a dual Brunn-Minkowski inequality.

指数系统的完备性与凸包的周长

Completeness of exponential systems and the perimeter of the convex hull

本文提出复平面上两类函数空间中指数系统的新完备性条件尺度，基于函数定义域凸包周长和指数分布新特征进行表述。

An extension theorem for separately holomorphic functions with singularities

Let $D_j\subset\Bbb C^{k_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluripolar set, $j=1,...,N$. Put$$X:=\bigcup_{j=1}^N A_1\times...\times A_{j-1}\times D_j\times A_{j+1}\times...\times A_N\subset\Bbb C^{k_1+...+k_N}.$$Let $U$ be an open connected neighborhood of $X$ and let $M\varsubsetneq U$ be an analytic subset. Then there exists an analytic subset $\hat M$ of the `envelope of holomorphy' $\hat X$ of $X$ with $\hat M\cap X\subset M$ such that for every function $f$ separately holomorphic on $X\setminus M$ there exists an $\hat f$ holomorphic on $\hat X\setminus\hat…

The Runge approximation theorem for generalized polynomial hulls

It is known from the Runge approximation theorem that every function which is holomorphic in a neighborhood of a compact polynomially convex set $K\subset \complexes^{n}$ can be approximated uniformly on $K$ by analytic polynomials. We shall here prove the same result when the role of the polynomially convex hull $\hat {K}$ is played by the generalized polynomial hull $h_{q}(K)$ introduced by Basener and which can be defined, for each integer $q\in {0,...,n-1}$, by $h_{q}(K)=\displaystyle\bigcup_{P\in \complexes [z_{1},...,z_{n}]} A_{P}$ where $A_{P}=\{z\in \complexes^{n}: |P(z)|\leq δ_{K}(P,z…

Projective Hulls and Characterizations of Meromorphic Functions

We give conditions characterizing holomorphic and meromorphic functions in the unit disk of the complex plane in terms of certain weak forms of the maximum principle. Our work is directly inspired by recent results of John Wermer, and by the theory of the projective hull of a compact subset of complex projective space developed by Reese Harvey and Blaine Lawson.

Wermer type sets and extension of CR functions

For each $n\geq2$ we construct an unbounded closed pseudoconcave complete pluripolar set $\mathcal E$ in $\mathbb C^n$ which contains no analytic variety of positive dimension (we call it a \textit{Wermer type set}). We also construct an unbounded strictly pseudoconvex domain $Ω$ in $\mathbb C^n$ and a smooth $CR$ function $f$ on $\partialΩ$ which has a single-valued holomorphic extension exactly to the set $\barΩ\setminus\mathcal E$.}

Bergman代表坐标、常全纯曲率及Carathéodory定理的多维推广

Bergman representative coordinate, constant holomorphic curvature and a multidimensional generalization of Carathéodory's theorem

本文利用Bergman代表坐标和Calabi的diastasis，将Lu的定理推广至度量不完整且具常全纯曲率的有界伪凸域，并多维推广了Carathéodory定理，给出双全纯球边界为拓扑球面的充分条件。

Bergman-Calabi diastasis and Kähler metric of constant holomorphic sectional curvature

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…

Representing de Rham cohomology classes on an open Riemann surface by holomorphic forms

Let $X$ be a connected open Riemann surface. Let $Y$ be an Oka domain in the smooth locus of an analytic subvariety of $\mathbb C^n$, $n\geq 1$, such that the convex hull of $Y$ is all of $\mathbb C^n$. Let $\mathscr O_*(X, Y)$ be the space of nondegenerate holomorphic maps $X\to Y$. Take a holomorphic $1$-form $θ$ on $X$, not identically zero, and let $π:\mathscr O_*(X,Y) \to H^1(X,\mathbb C^n)$ send a map $g$ to the cohomology class of $gθ$. Our main theorem states that $π$ is a Serre fibration. This result subsumes the 1971 theorem of Kusunoki and Sainouchi that both the periods and the div…

Holomorphic injective extensions of functions in P(K) and algebra generators

We present necessary and sufficient conditions on planar compacta $K$ and continuous functions $f$ on $K$ in order that $f$ generates the algebras $P(K), R(K), A(K)$ or $C(K)$. We also unveil quite surprisingly simple examples of non-polynomial convex compacta $K\subseteq\mathbb C$ and $f\in P(K)$ with the property that $f \in P(K)$ is a homeomorphism, but for which $f^{-1}\notin P(f(K))$. As a consequence, such functions do not admit injective holomorphic extensions to the interior of the polynomial convex hull $\widehat K$. On the other hand, it will be shown that the restriction $f^*|_G$ of…

A long $\mathbb C^2$ without holomorphic functions

In this paper we construct for every integer $n>1$ a complex manifold of dimension $n$ which is exhausted by an increasing sequence of biholomorphic images of $\mathbb C^n$ (i.e., a long $\mathbb C^n$), but it does not admit any nonconstant holomorphic or plurisubharmonic functions. Furthermore, we introduce new biholomorphic invariants of a complex manifold $X$, the stable core and the strongly stable core, that are based on the long term behavior of hulls of compact sets with respect to an exhaustion of $X$. We show that every compact polynomially convex set $B\subset \mathbb C^n$ which is t…

Finite interpolation with minimum uniform norm in C^n

Given a finite sequence $a:={a_1, ..., a_N}$ in a domain $Ω\subset C^n$, and complex scalars $v:={v_1, ..., v_N}$, consider the classical extremal problem of finding the smallest uniform norm of a holomorphic function verifying $f(a_j)=v_j$ for all $j$. We show that the modulus of the solutions to this problem must approach its least upper bound along a subset of the boundary of the domain large enough to contain the support of a measure whose hull contains a subset of the original $a$ large enough to force the same minimum norm. Furthermore, all the solutions must agree on a variety which als…

紧集或区域上函数空间中指数系统的完备性及其面积

Completeness of exponential systems in spaces of functions on a compact or domain and their area

本文提出了复平面上指数系统在紧集连续函数Banach代数和单连通有界域全纯函数空间中的新完备性条件，条件由凸包的Euclidean面积和指数系统指标分布的新特征共同表述。

复 Monge-Ampère 方程的拟有界解

Quasibounded solutions to the complex Monge-Ampère equation

本文研究 B-正则区域上允许奇异无界边界数据的复 Monge-Ampère Dirichlet 问题，提出多复变拟有界函数并证明 Błocki–Cegrell 类中解的存在唯一性。

一致代数与特殊簇

Uniform algebras and distinguished varieties

本文揭示了特殊簇与双圆盘上一致代数的联系，证明了真多项式映射像域上Samuelsson-Wold结果的类似结论，并描述了相应多项式凸包。

关于Ohsawa-Takegoshi $L^2$延拓定理与多次调和函数的可去奇点

On the Ohsawa-Takegoshi $L^2$ extension theorem and removable singularities of plurisubharmonic functions

本文推广了Ohsawa-Takegoshi $L^2$延拓定理至Stein流形上扭曲典范丛的全纯截面，并建立了多次调和函数在紧完备多极集上的Hartogs型延拓定理，填补了文献空白。

关于对称空间的Bergman度量

On the Bergman metric of symmetric spaces

本文研究了具有局部对称Bergman度量的有界域，证明了其在完备或伪凸条件下与对称域的biholomorphic关系。

New solution of a problem of Kolmogorov on width asymptotics in holomorphic function spaces

Given a domain $D$ in $\mathbb{C}^n$ and $K$ a compact subset of $D$, the set $\mathcal{A}_K^D$ of all restrictions of functions holomorphic on $D$ the modulus of which is bounded by $1$ is a compact subset of the Banach space $C(K)$ of continuous functions on $K$. The sequence $(d_m(\mathcal{A}_K^D))_{m\in \mathbb{N}}$ of Kolmogorov $m$-widths of $\mathcal{A}_K^D$ provides a measure of the degree of compactness of the set $\mathcal{A}_K^D$ in $C(K)$ and the study of its asymptotics has a long history, essentially going back to Kolmogorov's work on $ε$-entropy of compact sets in the 1950s. In …

$\mathbb{C}^2$ 中某些多项式多面体上的一致逼近

Uniform approximation on certain polynomial polyhedra in $\mathbb{C}^2$

本文扩展了$\mathbb{C}^2$中多项式多面体的Samuelsson-Wold二分法，证明了由多项式和多重调和函数生成的一致代数的结构性质，并计算了特定多重调和函数图像的多项式凸包，同时给出了低边界正则性区域上的一致逼近定理。