Some properties of maximal plurisubharmonic function

具有局部解析奇点的多次调和函数的局部到全局极大性

Local-to-global maximality for plurisubharmonic functions with locally analytic singularities

本文证明了具有局部解析奇点的局部极大多次调和函数在满足解析扩张界时即为极大，并通过Monge--Ampère测度消失条件刻画极大性。

Dirichlet Problems for Plurisubharmonic Functions on Compact Sets

In this paper we solve the Dirichlet problems for different classes of plurisubharmonic functions on compact sets in $\mathbb C^n$ including continuous, pluriharmonic and maximal functions.

Analyticity and propagation of plurisubharmonic singularities

A variant of Siu's analyticity theorem is proved for relative types of plurisubharmonic functions. Some results on propagation of plurisubharmonic singularities and maximality of pluricomplex Green functions with analytic singularities are derived.

Relative types and extremal problems for plurisubharmonic functions

We study properties of relative types of plurisubharmonic functions with respect to maximal plurisubharmonic weights. It is shown that they give a general form for upper semicontinuous, tropically additive functionals on plurisubharmonic singularities. We consider some extremal problems whose solutions are Green-like functions that give best possible bounds on a plurisubharmonic function, given the values of its types relative to some of (or all) the weights. An analyticity theorem is proved for the upperlevel sets for the types with respect to exponentially Hölder continuous weights, which le…

$\mathbb{C}^n$ 中有界拟 B-正则域上多重次调和函数的 Dirichlet 问题

Dirichlet problem for plurisubharmonic functions on bounded quasi B-regular domains in $\mathbb{C}^n$

本文研究了有界域上具有非连续非有界边界数据的极大多重次调和函数的存在性与唯一性，通过探讨 Perron-Bremermann 包络的连续性，实现了对可能上方无界的多重次调和函数的连续逼近。

复流形上局部多重次调和势的比较原理及其应用

Comparison principles for local plurisubharmonic potentials on complex manifolds and applications

本文证明了复流形上Cegrell类局部多重次调和势的加权比较原理，并应用于$\mathbb C^2$中具有特定奇异性的函数，得出极大性等价于Monge-Ampère测度消失且为局部性质。

On a convex level set of a plurisubharmonic function and the support of the Monge-Ampère current

In this paper, we study a geometric property of a continuous plurisubharmonic function which is a solution of the Monge-Ampère equation and has a convex level set. To prove our main theorem, we show a minimum principle of a maximal plurisubharmonic function. By using our results and Lempert's results, we show a relation between the supports of the Monge-Ampère currents and complex $k$-extreme points of closed balls for the Kobayashi distance in a bounded convex domain in $\mathbb{C}^{n}$.

Plurisubharmonic geodesics in spaces of non-Archimedean metrics of finite energy

Given a polarized projective variety (X,L) over any non-Archimedean field, assuming continuity of envelopes, we show that the space of finite-energy metrics on L is a geodesic metric space, where geodesics are given as maximal psh segments. Given two continuous psh metrics, we show that the maximal segment joining them is furthermore continuous.

Some properties of plurisubharmonic functions

Two properties of plurisubharmonic functions are proven. The first result is a Skoda type integrability theorem with respect to a Monge-Ampère mass with Hölder continuous potential. The second one says that locally, a p.s.h. function is $k$-Lipschitz outside a set of Lebesgue measure smaller that $c/k^2$.

Approximation of plurisubharmonic functions on complex varieties

We prove some results which give sufficient conditions so that pointwise approximation of negative plurisubharmonic functions on complex varieties by continuous plurisubharmonic ones is possible.

无界域上 $q$-多次调和函数的 Perron-Bremermann 包络

The Perron-Bremermann envelope for $q$-plurisubharmonic functions on unbounded domains in $\mathbb{C}^n$

研究无界域上$q$-多次调和函数的Perron-Bremermann包络，证明在边界有$r$-峰值点且有界类型的条件下，有界连续边界函数可延拓为极大有界连续函数，满足$q$-多次调和与$(n-q-1)$-多次超调和性质。

Hénon-Sibony映射的混合性与中心极限定理：多次调和可观测函数

Mixing and CLT for Hénon-Sibony maps: plurisubharmonic observables

本文证明复Hénon映射及更一般的Hénon-Sibony映射的最大熵测度对多次调和可观测函数具有指数混合性和中心极限定理。

A cup product lemma for continuous plurisubharmonic functions

A version of Gromov's cup product lemma in which one factor is the (1,0)-part of the differential of a continuous plurisubharmonic function is obtained. As an application, it is shown that a connected noncompact complete Kaehler manifold that has exactly one end and admits a continuous plurisubharmonic function that is strictly plurisubharmonic along some germ of a 2-dimensional complex analytic set at some point has the Bochner-Hartogs property; that is, the first compactly supported cohomology with values in the structure sheaf vanishes.

Plurisubharmonic Approximation and Boundary Values of Plurisubharmonic Functions

We study the problem of approximating plurisubharmonic functions on a bounded domain $Ω$ by continuous plurisubharmonic functions defined on neighborhoods of $\barΩ$. It turns out that this problem can be linked to the problem of solving a Dirichlet type problem for functions plurisubharmonic on the compact set $\barΩ$ in the sense of Poletsky. A stronger notion of hyperconvexity is introduced to fully utilize this connection, and we show that for this class of domains the duality between the two problems is perfect. In this setting, we give a characterization of plurisubharmonic boundary valu…

On the approximation of weakly plurifinely plurisubharmonic functions

In this note, we study the approximation of singular plurifinely plurisubharmonic function $u$ defined on a plurifinely domain $Ω$. Under some conditions, we prove that $u$ can be approximated by an increasing sequence of plurisubharmonic functions defined on Euclidean neighborhoods of $Ω$.

Tropical analysis of plurisubharmonic singularities

Some results on singularities of plurisubharmonic functions are put into the context of tropical mathematics.

Local Plurisubharmonic Defining Functions on the Boundary

Necessary conditions for a domain $Ω\subset \mathbb C^n$ admitting a local plurisubharmonic defining function on the boundary are given. In tandem, we give an algorithm to construct a local plurisubharmonic defining function on the boundary when one exists. In some cases we show that the necessary conditions are also sufficient.

On some weighted energy classes of plurisubharmonic functions

In this paper we study the relation between the weighted energy class $\mathcal{E}_χ$ introduced by S. Benelkouchi, V. Guedj and A. Zeriahi recently with the classes $\mathcal{E}$ and $\mathcal{N}$ studied by Cegrell. Moreover, we establish a generalized comparison principle for the operator $\text{M}_χ$ and, as an application, we prove a slight version of existence of solutions of Monge-Ampère type equation in the class $\mathcal{E}_χ(H,Ø)$.

On the conditional plurisubharmonic envelopes of bounded functions

In this paper, we extend some recent results of Guedj-Lu-Zeriahi about psh envelopes of bounded functions on bounded domains in $\mathbb{C}^n$. We also present a result on the regularity of psh envelopes.

极值全纯亚调和函数与 Fujii-Seo 确定性在希尔伯特空间中的应用

Maximal Plurisubharmonic Functions and Fujii-Seo Determinants in Hilbert spaces

本文提出利用 Fujii-Seo 正则化确定性描述极值全纯亚调和函数，定义 Fujii-Seo 确定性密度并研究其与 Levi 形的谱端点关系，证明极值性与该密度为零等价，建立全局退化条件和比较原理。

Kähler流形上的全阶指数混合：（拟）多次调和可观测函数

Exponential mixing of all orders on Kähler manifolds: (quasi-)plurisubharmonic observables

本文证明紧致Kähler流形上具有简单上同调作用的全纯自同构的最大熵测度对所有d.s.h.可观测函数具有全阶指数混合性，并由此推导出中心极限定理。

Some achieved wedge products of positive currents

In this paper, we study the existence of the current gT for positive plurisubharmonic currents T and unbounded plurisubharmonic functions g.

Holomorphic minorants of plurisubharmonic functions

Let $φ$ be a plurisubharmonic function on a pseudoconvex domain $D \subset \mathbb C^n$. We show that there exists a nonzero holomorphic function $f$ on $D$ such that some local mean value of $φ$ with logarithmic additional terms majorizes $\log |f|$. A similar problem is discussed for a locally integrable function on $D$ in terms of balayage of positive measures. Key words: holomorphic function, plurisubharmonicity, minorant, balayage, Jensen inequality, mean value in the ball.

关于具有对称性的多次调和函数的剩余Monge-Ampère质量，II

On the residual Monge-Ampère mass of plurisubharmonic functions with symmetry, II

本文利用Sasakian几何，对具有圆对称性和孤立奇点的多次调和函数，建立了剩余Monge-Ampère质量与Lelong数及最大方向Lelong数之间的估计，部分解决了零质量猜想。

On plurisubharmonic defining functions for pseudoconvex domains in $\mathbb{C}^2$

We investigate the question of existence of plurisubharmonic defining functions for smoothly bounded, pseudoconvex domains in $\mathbb{C}^2$. In particular, we construct a family of simple counterexamples to the existence of plurisubharmonic smooth local defining functions. Moreover, we give general criteria equivalent to the existence of plurisubharmonic smooth defining functions on or near the boundary of the domain. These equivalent characterizations are then explored for some classes of domains.

Radó- type theorem for subharmonic and plurisubharmonic functions

We observe that a recent result by Gardiner and Sjödin, solving a problem of Král on subharmonic functions, can be easily generalized to yield a somewhat stronger result. This can be combined with a viscosity technique of ours, which we slightly improve, to obtain Radó- type theorems for plurisubharmonic functions. Finally, we study the Borel complexity of the critical set and the set where the gradient does not exist finitely of subharmonic functions and general real valued functions.

$q$-多次调和函数子族的Bergman-Shilov边界

The Bergman-Shilov boundary for subfamilies of $q$-plurisubharmonic functions

本文为紧致Hausdorff空间上某些上半连续函数子类引入Bergman-Shilov边界，证明其存在唯一性，建立与峰值点及Bishop型定理的联系，推广Bychkov定理，并证明光滑伪凸域上$q$-多次调和函数Shilov边界的部分具有$q$维复子流形叶状结构。

Bergman kernel and oscillation theory of plurisubharmonic functions

Based on Harnack's inequality and convex analysis we show that each plurisubharmonic function is locally BUO (bounded upper oscillation) with respect to polydiscs of finite type but not for arbitrary polydiscs. We also show that each function in the Lelong class is globally BUO with respect to all polydiscs. A dimension-free BUO estimate is obtained for the logarithm of the modulus of a complex polynomial. As an application we obtain an approximation formula for the Bergman kernel that preserves all directional Lelong numbers. For smooth plurisubharmonic functions we derive a new asymptotic id…

关于$\u2102^2$中具有对称性的多重次调和函数的剩余蒙日-安培质量

On the residual Monge-Ampère mass of plurisubharmonic functions with symmetry in $\mathbb{C}^2$

本文研究$\u2102^2$中带孤立奇点的对称多重次调和函数的剩余蒙日-安培质量，证明$S^1$-不变函数原点勒洛数为零时剩余质量为零，解答了特殊情形下的零质量猜想，并给出更一般的剩余质量估计。

Extremal omega-plurisubharmonic functions as envelopes of disc functionals

For each closed, positive (1,1)-current ωon a complex manifold X and each ω-upper semicontinuous function ϕon X we associate a disc functional and prove that its envelope is equal to the supremum of all ω-plurisubharmonic functions dominated by ϕ. This is done by reducing to the case where ωhas a global potential. Then the result follows from Poletsky's theorem, which is the special case ω=0. Applications of this result include a formula for the relative extremal function of an open set in X and, in some cases, a description of the ω-polynomial hull of a set.

On sharp constants in Paley problem for plurisubharmonic functions of lower order $ρ>1$

In 1999 Khabibullin established the best estimate in Paley problem for a plurisubharmonic function $u$ of finite lower order $0\leqρ\leq 1$. For $ρ>1$ obtaining a sharp estimate has remained an open question. In this work we solve this problem. We also provide some estimates for the types of the characteristic functions $T(r,u)$ and $M(r,u)$.

Symmetrization of plurisubharmonic functions on the Fano manifolds

Given a compact complex manifold $Y$ with a negative line bundle $L\rightarrow Y$, we study the Schwarz-type symmetrization on the total space of $L$. We prove that this symmetrization does not increase the Monge-Ampère energy for the fibrewise $S^{1}$-invariant plurisubharmonic functions in the "unit ball" under some assumptions. As an application we generalize the sharp Moser-Trudinger inequality on the unit ball.

A note on multiplier ideal sheaves on complex spaces with singularities

The goal of this note is to present some recent results of our research concerning multiplier ideal sheaves on complex spaces and singularities of plurisubharmonic functions. We firstly introduce multiplier ideal sheaves on complex spaces (\emph{not} necessarily normal) via Ohsawa's extension measure, as a special case of which, it turns out to be the so-called Mather-Jacobian multiplier ideals in the algebro-geometric setting. As applications, we obtain a reasonable generalization of (algebraic) adjoint ideal sheaves to the analytic setting and establish some extension theorems on Kähler mani…

Extension of Plurisubharmonic Functions in the Lelong Class

Let $X$ be an algebraic subvariety of $\mathbb C^n$ and $\bar X$ be its closure in $\mathbb P^n.$ In their paper \cite{CGZ} Coman-Guedj-Zeriahi proved that any plurisubharmonic function with logarithmic growth on $X$ extends to a plurisubharmonic function with logarithmic growth on $\mathbb C^n$ when the germs $(\bar X,a)$ in $\mathbb P^n$ are irreducible for all $a\in \bar X\setminus X.$ In this paper we consider $X$ for which the germ $(\bar X,a)$ is reducible for some $a\in \bar X\setminus X$ and we give a necessary and sufficient condition for $X$ so that any plurisubharmonic function with…

依赖参数的多重次调和函数的解析性定理

Analyticity theorems for parameter-dependent plurisubharmonic functions

研究依赖参数的（拟）多重次调和函数的水平集解析性，证明 Lelong 数上水平集之并通常多重极，并给出复奇点指数次水平集及相对 Bergman 核对数极点的解析性条件与非解析反例。

无限维伪凸域上的光滑多次调和穷竭函数

Smooth plurisubharmonic exhaustion functions on pseudo-convex domains in infinite dimensions

本文首次在典型Hilbert空间的伪凸域上显式构造光滑多次调和穷竭函数，服务于无限维复分析中的$L^{2}$方法。

驯服的最大权、相对类型与赋值

Tame maximal weights, relative types and valuations

本文引入驯服最大权（Zhou 权），利用 Tian 函数建立相对类型的积分表达式，证明其满足热带性质，从而构成赋值，并用于度量奇异性及推广多重复 Green 函数。

具有不连续边界行为的多重次调和函数

Plurisubharmonic functions with discontinuous boundary behavior

本文研究带不连续边界数据的复 Monge–Ampère Dirichlet 问题，在 b-多重极不连续集与 B-正则域条件下证明解的存在唯一性及正则性，并说明高维中这些条件并非必要。

On minimal kernels and Levi currents on weakly complete complex manifolds

A complex manifold $X$ is \emph{weakly complete} if it admits a continuous plurisubharmonic exhaustion function $ϕ$. The minimal kernels $Σ_X^k, k \in [0,\infty]$ (the loci where are all $\mathcal{C}^k$ plurisubharmonic exhaustion functions fail to be strictly plurisubharmonic),introduced by Slodkowski-Tomassini, and the Levi currents, introduced by Sibony, are both concepts aimed at measuring how far $X$ is from being Stein. We compare these notions, prove that all Levi currents are supported by all the $Σ_X^k$'s, and give sufficient conditions for points in $Σ_X^k$ to be in the support of so…

A conjecture on some estimates for integrals

A conjecture concerning some pairs of interfering estimates for some integrals is formulated in three equivalent versions. Its importance for the the Paley problem for plurisubharmonic functions and for certain classes of extremal problems for entire functions of several variables is declared.

与多次下调和函数相关的全纯函数的一个性质

A property of holomorphic functions related to plurisubharmonic functions

本文利用 $\overline{\partial}-$ 方程的 $L^2-$ 估计和强开性猜想，证明了在伪凸域中，当权重函数在 $L^1$ 范数下接近时，存在相应的全纯函数在相对紧开子集的 $L^2$ 范数下接近原函数。

复 Monge-Ampere 测度的预定极限与聚点集

Prescribed Limits and Cluster Sets of Complex Monge-Ampere Measures

本研究针对 Bedford 问题，刻画了有界多重次调和序列的 Monge-Ampère 测度在模糊拓扑下的聚点集，并构造逼近序列实现预定测度极限，推广了经典结果。

Pluricomplex charge at weak singularities

Let $u$ be a plurisubharmonic function, defined on a neighbourhood of a point $x,$ such that the complex Monge-Ampère operator is well-defined on $u.$ Suppose also that $u$ has a weak singularity, in the sense that the Lelong number of $u$ at $x$ vanish. Is it true that the residual mass of the measure $(dd^c{u})^n$ vanish at $x$? To our knowledge there is no known example that falsifies the posed question. In this paper some partial results are obtained. We find that for a significant subset of plurisubharmonic functions with well defined Monge-Ampère mass vanishing Lelong number does implies…

Toric plurisubharmonic functions and analytic adjoint ideal sheaves

In the first part of this paper, we study the properties of some particular plurisubharmonic functions, namely the toric ones. The main result of this part is a precise description of their multiplier ideal sheaves, which generalizes the algebraic case studied by Howald. In the second part, almost entirely independent of the first one, we generalize the notion of the adjoint ideal sheaf used in algebraic geometry to the analytic setting. This enables us to give an analogue of Howald's theorem for adjoint ideals attached to monomial ideals. Finally, using the local Ohsawa-Takegoshi-Manivel theo…

关于二变量八元数 Monge-Ampère 方程和与八元数 plurisubharmonic 函数相关的 pluripotential 理论

On octonionic Monge-Ampère equation and pluripotential theory associated to octonionic plurisubharmonic functions of two variables

本文研究了二变量八元数 plurisubharmonic 函数的 Monge-Ampère 方程和 pluripotential 理论，解决了八元数非结合性问题，并建立了相关函数的正则性与容量性质。

Parabolic Stein Manifolds

An open Riemann surface is called parabolic in case every bounded subharmonic function on it reduces to a constant. Several authors introduced seemingly different analogs of this notion for Stein manifolds of arbitrary dimension. In the first part of this note we compile these notions of parabolicity and give some immediate relations among them. In section 3 we relate some of these notions to the linear topological type of the Fréchet space of analytic functions on the given manifold. In sections 4 and 5 we look at some examples and show, for example, that the complement of the zero set of a W…

Subharmonicity properties of the Bergman kernel and some other functions associated to pseudoconvex domains

Let $D$ be a pseudoconvex domain in $\C^k_t\times\Cn_z$ and let $ϕ$ be a plurisubharmonic function in $D$. For each $t$ we consider the $n$-dimensional slice of $D$, $D_t=\{z; (t,z)\in D\}$, let $ϕ^t$ be the restriction of $ϕ$ to $D_t$ and denote by $K_t(z,ζ)$ the Bergman kernel of $D_t$ with the weight function $ϕ^t$. Generalizing a recent result of Maitani and Yamaguchi (corresponding to $n=1$ and $ϕ=0$) we prove that $\log K_t(z,z)$ is a plurisubharmonic function in $D$. We also generalize an earlier results of Yamaguchi concerning the Robin function and discuss similar results in the setti…

无穷远处的log canonical阈值

Log canonical thresholds at infinity

本文研究了 plurisubharmonic 函数在无穷远处的 log canonical 阈值问题，通过 toric 情况下的显式公式和多项式逼近方法，探讨了 $e^{-u} \in L^p$ 的范围及 tame 多项式映射的应用。

Polynomial approximation, local polynomial convexity, and degenerate CR singularities -- II

We provide some conditions for the graph of a Hoelder-continuous function on \bar{D}, where \bar{D} is a closed disc in the complex plane, to be polynomially convex. Almost all sufficient conditions known to date --- provided the function (say F) is smooth --- arise from versions of the Weierstrass Approximation Theorem on \bar{D}. These conditions often fail to yield any conclusion if rank_R(DF) is not maximal on a sufficiently large subset of \bar{D}. We bypass this difficulty by introducing a technique that relies on the interplay of certain plurisubharmonic functions. This technique also a…

On the generalized squeezing functions and Fridman invariants of special domains

The main purpose of this paper is to study the generalized squeezing functions and Fridman invariants of some special domains. As applications, we give the precise form of generalized squeezing functions and Fridman invariants of various domains such as n-dimensional annuli. Furthermore, we provide domains with non-plurisubharmonic generalized squeezing function or Fridman invariant.

Liouville-type theorems outside of small exceptional sets for functions of finite order

We prove that convex functions of finite order on the real line and subharmonic functions of finite order on finite dimensional real space, bounded from above outside of some set of zero relative Lebesgue density, are bounded from above everywhere. It follows that subharmonic functions of finite order on the complex plane, entire and plurisubharmonic functions of finite order, and convex or harmonic functions of finite order bounded from above outside some set of zero relative Lebesgue density are constant.

带边界值的复Monge-Ampère方程：一种变分方法

Complex Monge-Amp{è}re equations with boundary values: a variational approach

针对复空间中有界超凸域上带边界值的复Monge-Ampère方程，引入具有限Φ-能量的多重次调和函数类，用变分方法刻画其Monge-Ampère测度的可积性条件，推广相关结果并通过对数变换得到实方程的类似结论。

Pluripotential-theoretic stability thresholds

Given a compact polarized manifold $(X,L)$, we introduce two new stability thresholds in terms of singularity types of global quasi-plurisubharmonic functions on $X$. We prove that in the Fano setting, the new invariants can effectively detect the K-stability of $X$. We study some functionals of geodesic rays in the space of Kähler potentials by means of the corresponding test curves. In particular, we introduce a new entropy functional of quasi-plurisubharmonic functions and relate the radial entropy functional to this new entropy functional.

Valuations and plurisubharmonic singularities

We extend to higher dimensions some of the valuative analysis of singularities of plurisubharmonic (psh) functions developed by the last two authors. Following Kontsevich and Soibelman we describe the geometry of the space V of all normalized valuations on C[x_1,...,x_n] centered at the origin. It is a union of simplices naturally endowed with an affine structure. Using relative positivity properties of divisors living on modifications of C^n above the origin, we define formal psh functions on V, designed to be analogues of the usual psh functions. For bounded formal psh functions on V, we def…