Indicators for Plurisubharmonic Functions of Logarithmic Growth

Total Masses of Mixed Monge-Ampere Currents

Monge-Ampere currents generated by plurisubharmonic functions of logarithmic growth are studied. Upper bounds for their total masses are obtained in terms of growth characteristics of the functions. In particular, this gives a plurisubharmonic version of D.Bernstein's theorem on the number of zeros of polynomial mappings in terms of the Newton polyhedra, and a representation for Demailly's generalized degrees of plurisubharmonic functions.

Polya's inequalities, global uniform integrability and the size of plurisubharmonic lemniscates

First we prove a new inequality comparing uniformly the relative volume of a Borel subset with respect to any given complex euclidean ball $\B \sub \C^n$ with its relative logarithmic capacity in $\C^n$ with respect to the same ball $\B$. An analoguous comparison inequality for Borel subsets of euclidean balls of any generic real subspace of $\C^n$ is also proved. Then we give several interesting applications of these inequalities. First we obtain sharp uniform estimates on the relative size of \psh lemniscates associated to the Lelong class of \psh functions of logarithmic singularities at in…

Plurisubharmonically separable complex manifolds

Let $M$ be a complex manifold and $PSH^{cb}(M)$ be the space of bounded continuous plurisubharmonic functions on $M$. In this paper we study when functions from $PSH^{cb}(M)$ separate points. Our main results show that this property is equivalent to each of the following properties of $M$: (1) the core of $M$ is empty. (2) for every $w_0\in M$ there is a continuous plurisubharmonic function $u$ with the logarithmic singularity at $w_0$. Moreover, the core of $M$ is the disjoint union of 1-pseudoconcave in the sense of Rothstein sets $E_j$ with the following Liouville property: every function f…

Plurisubharmonic functions and nef classes on complex manifolds

We prove the existence of plurisubharmonic functions with prescribed logarithmic singularities on complex 3-folds equipped with a nef class of positive volume. We prove the same result for rational classes on Moishezon n-folds.

Multi-circled singularities, Lelong numbers, and integrability index

By comparing Green functions of multi-circled plurisubharmonic singularities in the n-domensional complex space to their indicators, we obtain formulas for the higher Lelong numbers and integrability index for such singularities and extend Howald's result on multiplier ideals for monomial ideals to multi-circled singularities. This also leads to an elementary proof of the relations between the k-th Lelong numbers and the integrability index. For k=1 and arbitrary plurisubharmonic functions the inequality is due to Skoda, and for k=n and any plurisubharmonic function with isolated singularity t…

A log canonical threshold test

In terms of log canonical threshold, we characterize plurisubharmonic functions with logarithmic asymptotical behaviour.

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。

Logarithmic potentials on $\mathbb{P}^n$

We study the projective logarithmic potential $\mathbb{G}_μ$ of a Probability measure $μ$ on the complex projective space $\mathbb{P}^{n}$. We prove that the Range of the operator $μ\longrightarrow \mathbb{G}_μ$ is contained in the (local) domain of definition of the complex Monge-Ampère operator acting on the class of quasi-plurisubharmonic functions on $\mathbb{P}^n$ with respect to the Fubini-Study metric. Moreover, when the measure $μ$ has no atom, we show that the complex Monge-Ampère measure of its Logarithmic potential is an absolutely continuous measure with respect to the Fubini-Study…

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。

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。

指数位于紧凸集中的多项式及相关加权极值函数——基础结果

Polynomials with exponents in compact convex sets and associated weighted extremal functions -- Fundamental results

本文综述指数限制在紧凸集mS内的分次多项式环P^S(C^n)相关多下调和理论，研究其与增长性由S对数支撑函数决定的全局多下调和函数类L^S(C^n)的关系，介绍二者加权极值函数的性质及与S性质的关联。

Meromorphic functions of finite $φ$-order and linear Askey-Wilson divided difference equations

The growth of meromorphic solutions of linear difference equations containing Askey-Wilson divided difference operators is estimated. The $φ$-order is used as a general growth indicator, which covers the growth spectrum between the logarithmic order $ρ_{\log}(f)$ and the classical order $ρ(f)$ of a meromorphic function $f$.

Meromorphic functions of finite $φ$-order and linear $q$-difference equations

The $φ$-order was introduced in 2009 for meromorphic functions in the unit disc, and was used as a growth indicator for solutions of linear differential equations. In this paper, the properties of meromorphic functions in the complex plane are investigated in terms of the $φ$-order, which measures the growth of functions between the classical order and the logarithmic order. Several results on value distribution of meromorphic functions are discussed by using the $φ$-order and the $φ$-exponent of convergence. Instead of linear differential equations, the applications in the complex plane lie i…