On the product property of the pluricomplex Green function
Armen Edigarian
证明了复格林函数对任意整环$D_1\subset\Bbb C^n$和$D_2\subset\Bbb C^m$都具有乘积性质$g_{D_1\times D_2}=\max\{ g_{D_1},g_{D_2}\}$.
We prove that the pluricomplex Green function has the product property $g_{D_1\times D_2}=\max\{ g_{D_1},g_{D_2}\}$ for any domains $D_1\subset\Bbb C^n$ and $D_2\subset\Bbb C^m$.
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In this paper we prove the basic facts for pluricomplex Green functions on manifolds. The main goal is to establish properties of complex manifolds that make them analogous to relatively compact or hyperconvex domains in Stein manifolds. The final version to appear in JGA.
We prove a version of the Schwarz lemma for holomorphic mappings from the unit disk into the symmetric product of a Riemann surface. Our proof is function-theoretic and self-contained. The main novelty in our proof is the use of the pluricomplex Green's function. We also prove several other Schwarz lemmas using this function.
Continuity of pluricomplex Green functions with hypersurface poles on certain B-regular domains
本文证明了一类有界(B)-正则域（含所有(B)-正则可凸化域）上具有超曲面极点的多重复格林函数的连续性。
Łojasiewicz exponent and pluricomplex Green function on algebraic sets
该文研究代数集上的多重复Green函数，通过映射的Łojasiewicz指数与增长指数建立紧集及其原像的Green函数估计关系，给出显式例子与增强版Bernstein-Walsh不等式，为相关多项式逼近研究提供理论框架。
In this paper, we explore the existence of pluricomplex Green functions for Stein manifolds from a functional analysis point of view. For a Stein manifold $M$, we will denote by $O(M)$ the Fréchet space of analytic functions on $M$ equipped with the topology of uniform convergence on compact subsets. In the first section, we examine the relationship between the existence of pluricomplex Green functions and the diametral dimension of $O(M)$. This led us to consider negative plurisubharmonic functions on $M$ with a nontrivial relatively compact sublevel set (semi-proper). In section 2, we charac…
Pluricomplex Green's functions and Fano manifolds
本文证明Fano流形若无Kahler-Einstein度量，则连续性方法中的Kahler势子收敛到具有解析奇点的函数，满足齐次复Monge-Ampere方程，证实Tian-Yau预期。
Entire pluricomplex Green functions and Lelong numbers of projective currents
设$T$是复射影空间${\Bbb P}^n$上的一个正的二维（1，1）单位质量闭流.证明了当$T$的Lelong数大于$α$时，点集$V_α(T)$包含在复直线中，若$α\geq2/3$，则点集$V_α(T)$包含在复直线中;若$1/2\leqα<2/3$，则点集$|V_α(T)\setminus L|\leq1$包含在复直线$L$中.我们还证明了在2维中，如果$2/5\leqα<1/2$，那么$|V_α(T)\setminus C|\leq1$对于某些圆锥$C$。
Let V_E be the pluricomplex Green function associated to a compact subset E of C^N. The well known Hölder Continuity Property of E means that there exist constants B > 0, 0< c =< 1 such that V_E(z) =< B dist(z,E)^c. The main result of this paper says that this condition is equivalent to a Vladimir Markov type inequality, i.e. || D^αP ||_E =< M^{|α|} (deg P)^{m|α|} (|α|!)^{1-m} ||P||_E, where m,M>0 are independent of the polynomial P of N variables. We give some applications of this equivalence and we present its generalization related to a notion of a fit majorant. Moreover, as a consequence o…
A Formula for the Pluricomplex Green Function of the Bidisk
本文利用双圆盘上 Lempert 函数与多复变 Green 函数相等的结果，分区域推导等权两极点 Green 函数，并据此将对称双圆盘 Carathéodory 度量中的上确界化为四次多项式求根。
This is the first paper in a series of investigation of the pluripotential theory on Teichmüller space. The main purpose of this paper is to give an alternative approach to the Krushkal formula of the pluricomplex Green function on Teichmüller space. We also show that Teichmüller space carries a natural stratified structure of real-analytic submanifolds defined from the structure of singularities of the initial differentials of the Teichmüller mappings from a given point. We will also give a description of the Levi form of the pluricomplex Green function using the Thurston symplectic form via …
It is shown that, on a compact Kahler manifold with boundary, the singularities of the pluricomplex Green's function with multiple poles can be prescribed to be of the form $\log\sum_{j=1}^n|f_j(z)|^2$ at each pole, where $f_j(z)$ are arbitrary local holomorphic functions with the pole as their only common zero. The proof is a combination of blow-ups and recent a priori estimates for the degenerate complex Monge-Ampere equation, and particularly the $C^1$ estimates away from a divisor.
The pluricomplex Green function of the Monge-Ampère equation for $(n-1)$-plurisubharmonic functions and form type $k$-Hessian equations
本文引入$(n-1)$-多次调和函数的Monge-Ampère方程的多次复Green函数，通过求解Dirichlet问题并建立先验估计证明其$C^{1,\alpha}$正则性，且$k$-Hessian方程的奇异解光滑。
Pluricomplex Green and Lempert functions for equally weighted poles
对于$\mathbb C^n$中的一个域$Ω$，具有极点$a_1, ...,a_N \in Ω$的复格林函数被定义为$G(z):=\sup \{u(z): u\in PSH_-(Ω), u(x)\le \log \|x-a_j\|+C_j \text{when} x \to a_j, j=1,...,N \}$。当单位球中只有一个极点或两个极点时，它等于由$L_S (z) :=\inf \{\sum^N_{j=1}ν_j\log|ζ_j|: \exists ϕ\in \mathcal {O}(\mathbb D,Ω), ϕ(0)=z, ϕ(ζ_j)=a_j, j=1,...,N \}$从解析圆盘定义到$Ω$的Lempert函数。众所周知，我们总是有$L_S (z) \ge G_S(z)$。在更一般的情况下，我们允许加权极点，有一个反例，平等由于卡莱赫德和Wiegerinck，与$Ω$等于bidisk。在这里，我们展示了一个反例，只使用四个不同的平等加权极点的双圆盘。为了做到这一点，我们首先定义了一个更一般的概念，Lempert函数的“多重性”，类似于广义格林函数的Lelong和Rashkovskii，然后我们展示了如何在一些例子中，这可以实现为一个限制的定期Lempert函数时，极点趋于对方。最后，从一个例子，其中$L_S (z) > G_S(z)$在多极的情况下，我们推断，不同的…
In this paper we study Hölder continuity of the pluricomplex Green function with logarithmic growth at infinity of a smooth generic submanifold of $\C^n$. In particular we prove that the pluricomplex Green function of any $C^2$-smooth generic compact submanifold of $\C^n$ (without boundary) is Lipschitz continuous in $\C^n$.
Let $S_ε$ be a set of $N$ points in a bounded hyperconvex domain in $C^n$, all tending to 0 as$ε$ tends to 0. To each set $S_ε$ we associate its vanishing ideal $I_ε$ and the pluricomplex Green function $G_ε$ with poles on the set. Suppose that, as $ε$ tends to 0, the vanishing ideals converge to $I$ (local uniform convergence, or equivalently convergence in the Douady space), and that $G_ε$ converges to $G$, locally uniformly away from the origin; then the length (i.e. codimension) of $I$ is equal to $N$ and $G \ge G_I$. If the Hilbert-Samuel multiplicity of $I$ is strictly larger than $N$, t…
The pluricomplex Poisson kernel for strongly convex domains
设$D$是$n$维复空间中的有界强凸区域.固定一点$p\in \partial D$，考虑在$p$上具有单极点的齐次复Monge-Ampere方程的解.证明了这样的解具有单位圆盘上经典泊松核的许多性质，因而称之为极点位于$p$的$D$的复泊松核.特别是，我们讨论的极值性质（如经典的Phragmen-Lindelof定理的推广），与多复格林函数的$D$的关系，在相关的叶理和边界表现和再生公式多重次调和函数方面的唯一性。
Using pluricomplex Green functions we introduce a compactification of a complex manifold $M$ invariant with respect to biholomorphisms similar to the Martin compactification in the potential theory. For this we show the existence of a norming volume form $V$ on $M$ such that all negative plurisubharmonic functions on $M$ are in $L^1(M,V)$. Moreover, the set of such functions with the norm not exceeding 1 is compact. Identifying a point $w\in M$ with the normalized pluricomplex Green function with pole at $w$ we get an imbedding of $M$ into a compact set and the closure of $M$ in this set is th…
Plurisubharmonic extremal functions, Lelong numbers and coherent ideal sheaves
引入了一类新的复复格林函数，它沿复流形X的复子空间A具有对数极点。它是X上最大的负多重次调和函数，其Lelong数至少为log max{ |f_1|，.，|f_m| }，其中f_1，.，f_m是A的理想层的局部生成元。具有单个对数极点或有限个加权极点的复格林函数是我们构造的一个非常特殊的情况。我们给出了这个函数的几个等价定义，并研究了它的性质，包括边界行为，连续性和唯一性。这是基于和扩展我们以前的工作，圆盘泛函和他们的信封。
The pluricomplex Poisson kernel for convex finite type domains
研究有界凸有限型区域上齐次复蒙日 - 安培方程的解 \(\Omega_\xi\)，基于度量和缩放技术，证明其性质，包括与多复变格林函数关系等，它是单位圆盘泊松核推广，还可用于强凸域，过程中建立了小林距离新估计。
We extend the upper estimates obtained by M. Carlehed and B.-Y. Chen about the ratio of the classical and pluricomplex Green functions to the case of $\mathcal C^2$-smooth locally $\mathbb C$-convexifiable domains of finite type. We also give some lower estimates. In order to obtain those results, and because it is of independent interest, we refine and unify some classical estimates about the Kobayashi distance and the Lempert function in such domains.
The Lempert function for a set of poles in a domain of $\mathbb C^n$ at a point $z$ is obtained by taking a certain infimum over all analytic disks going through the poles and the point $z$, and majorizes the corresponding multi-pole pluricomplex Green function. Coman proved that both coincide in the case of sets of two poles in the unit ball. We give an example of a set of three poles in the unit ball where this equality fails.
We investigate the limits of the ideals of holomorphic functions vanishing on three points in $\C^2$ when all three points tend to the origin, and what happens to the associated pluricomplex Green functions. This is a continuation of the work of Magnusson, Rashkovskii, Sigurdsson and Thomas, where those questions were settled in a generic case.
Let us have a family of ideals of holomorphic functions vanishing at N distinct points of a complex manifold, all tending to a single point. As is known, convergence of the ideals does not guarantee the convergence of the pluricomplex Green functions to the Green function of the limit ideal; moreover, the existence of the limit of the Green functions was unclear. Assuming that all the powers of the ideals converge to some ideals, we prove that the Green functions converge, locally uniformly away from the limit pole, to a function which is essentially the upper envelope of the scaled Green func…
We continue the study of convergence of multipole pluricomplex Green functions for a bounded hyperconvex domain of $\mathbb C^n$, in the case where poles collide. We consider the case where all poles do not converge to the same point in the domain, and some of them might go to the boundary of the domain. We prove that weak convergence will imply convergence in capacity; that it implies convergence uniformly on compacta away from the poles when no poles tend to the boundary; and that the study can be reduced, in a sense, to the case where poles tend to a single point. Furthermore, we prove that…
Polynomials with exponents in compact convex sets and associated weighted extremal functions -- Generalized product property
本文推广了Siciak关于两个集合的极值函数关系的结果，探讨了在紧凸集增长条件下Siciak-Zakharyuta函数的广义乘积性质及其在加权情况下的局限性。
Let $1\leq m\leq n$ be two fixed integers. Let $Ω\Subset \mathbb C^n$ be a bounded $m$-hyperconvex domain and $\mathcal A \subset Ω\times ]0,+ \infty[$ a finite set of weighted poles. We define and study properties of the $m$-subharmonic Green function of $Ω$ with prescribed behaviour near the weighted set $A$. In particular we prove uniform continuity of the exponential Green function in both variables $(z,\mathcal A)$ in the metric space $\bar Ω\times \mathcal F$, where $\mathcal F$ is a suitable family of sets of weighted poles in $Ω\times ]0,+ \infty[$ endowed with the Hausdorff distance. …
We study the limits of pluricomplex Green functions with four poles tending to the origin in a hyperconvex domain, and the (related) limits of the ideals of holomorphic functions vanishing on those points. Taking subsequences, we always assume that the directions defined by pairs of points stabilize as they tend to $0$. We prove that in a generic case, the limit of the Green functions is always the same, while the limits of ideals are distinct (in contrast to the three point case). We also study some exceptional cases, where only the limits of ideals are determined. In order to do this, we est…
$C^{1,1}$ Regularity Global Estimates for Homogeneous Complex Hessian Equations on Punctured Domains
本文通过引入复线性向量场生成的辅助函数，将全局实 Hessian 估计归结到边界，证明了穿孔域上齐次复 Hessian 方程 Green 函数的 $C^{1,1}$ 正则性，并推广至复 Monge-Ampère 情形和外部问题。
We prove a generalization of Dunham Jackson's famous approximation inequality to the case of compact sets in the complex plane admitting both upper and lower bounds for their Green's functions, i.e. the well known Holder Continuity Property (HCP) and the less known but crucial Lojasiewicz-Siciak inequality (LS). Moreover, we show that (LS) is a necessary condition for our Jackson type inequality.
We prove that some holomorphic functions on the moduli space of tori have only simple zeros. Instead of computing the derivative with respect to the moduli parameter $τ$, we introduce a conceptual proof by applying Painlevé VI\ equation. As an application of this simple zero property, we obtain the smoothness of all the degeneracy curves of trivial critical points for some multiple Green function.
Sets non-thin at \infty in \Bbb C ^m, and the growth of sequences of entire functions of genus zero
在本文中，我们定义的概念，非薄在$\infty$如下：让$E$是一个子集的$\Bbb C^m$。对于任何$R>0$，定义$E_R=E\cap \{z\in \Bbb C ^m :|z|\leq R\}$。如果对于所有$z\in \Bbb C^m$，\lim_{R\to\infty} V_{E_R}（z）=0，则我们说$E$在$\infty$处是非瘦的，其中$V_E$是$E$的复格林函数。在$\infty$上的非瘦定义具有良好的性质：如果$E\subset \Bbb C^m$在$\infty$上是非瘦的，并且$A$是多极的，则$E\backslash A$在$\infty$上是非瘦的，如果$E\subset \Bbb C^m$和$F\subset \Bbb C^n$是在$\infty$上非瘦的闭集，则$E\times F\subset \Bbb C^m\times \Bbb C^n$在$\infty$上是非瘦的（参见引理\ref{Lem1}）。然后我们研究了$\infty$集上的非瘦集的性质，并将其推广到\cite{mul-yav}和\cite{trong-tuyen}中。
In 1962 J. Siciak introduced polynomial extremal function. The Siciak extremal function is one of the most important tool to investigate derivatives of polynomials. W. Pleśniak in the circle of papers (most important joint with W. Pawłucki) showed deep connections between behavior of Siciak extremal function near compact $K$ and bounds for derivatives of polynomials. In 1990 W. Pleśniak introduced condition (P) which is equivalent to Markov property of compact $K$. In the same paper there was stated a problem which property of Siciak's extremal function are necessary to Markov's property. A st…
Linearisation d'endomorphismes holomorphes de CP(k) et caracterisation des exemples de Lattes par leur mesure de Green
设f是CP（k）的全纯自同态，其度大于2.我们证明了如果f的Green测度不是奇异的，则f是刚性的：这是一个Lattes例子。的证明依赖于线性化的f的迭代，沿典型的轨道。这一性质使我们能够正则化格林电流，并证明其刚性。--我们的蒙特罗斯认为，如果绿色的测量值不是单一的，那么它就非常坚硬：这是一个拿铁咖啡的例子。该演示是基于典型长轨道迭代线性化的一种特性。这一点我们可以通过格林报的“正规化”来实现，也可以通过严格的规定来实现。
Polynomials with exponents in compact convex sets and associated weighted extremal functions -- The Siciak-Zakharyuta theorem
本文将经典Siciak-Zakharyuta定理推广到指数限制在紧凸集$mS$中的多项式空间，证明加权极值函数等式成立当且仅当$S$的有理点稠密。
We consider a family of random walks killed at the boundary of the Weyl chamber of the dual of $\rm{Sp}(4)$, which in addition satisfies the following property: for any $n\geq 3$, there is in this family a walk associated with a reflection group of order $2n$. Moreover, the case $n=4$ corresponds to a process which appears naturally by studying quantum random walks on the dual of $\rm{Sp}(4)$. For all the processes belonging to this family, we find the exact asymptotic of the Green functions along all infinite paths of states as well as that of the absorption probabilities along the boundaries…
Analyticity and positivity of Green's functions without Lorentz
本文研究了在自发破缺洛伦兹对称性理论中，迟滞二点格林函数的微因果性和正性对性质的约束，推导了格林函数的解析性及谱密度的积分约束，并验证了在非零化学势或温度背景下的相关性质。
Reconstructing the thermal Green functions at real times from those at imaginary times
通过利用的解析性和边界值性质的热格林函数，从KMS条件在时间和能量的复杂变量，我们处理的一般（非微扰）的问题，恢复的热函数在实时从相应的功能在虚时间，介绍了作为主要对象的松原形式主义。我们所依赖的关键属性是这样一个事实，即能量变量中延迟和高级函数的傅立叶变换必须是时间相关器的傅立叶系数的“唯一卡尔森解析插值”，后者是在离散松原虚能量处分别在上半平面和下半平面中进行的。从傅立叶系数视为“数据集”，然后我们开发了一种方法的基础上Pollaczek多项式明确构建其解析插值。
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