Green functions with singularities along complex spaces

Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields

We prove the existence of non-positively curved Kähler-Einstein metrics with cone singularities along a given simple normal crossing divisor on a compact Kähler manifold, under a technical condition on the cone angles, and we also discuss the case of positively-curved Kähler-Einstein metrics with cone singularities. As an application we extend to this setting classical results of Lichnerowicz and Kobayashi on the parallelism and vanishing of appropriate holomorphic tensor fields.

Kähler-Einstein metrics with mixed Poincaré and cone singularities along a normal crossing divisor

Let X be a Kähler manifold and D be a R-divisor with simple normal crossing support and coefficients between 1/2 and 1. Assuming that K_X+D is ample, we prove the existence and uniqueness of a negatively curved Kahler-Einstein metric on X\D having mixed Poincaré and cone singularities according to the coefficients of D. As an application we prove a vanishing theorem for certain holomorphic tensor fields attached to the pair (X,D).

沿奇异叶形结构的可统一叶形投影结构

Uniformizable foliated projective structures along singular foliations

研究了紧复流形上由曲线构成的叶形的可统一投影结构，发现奇点类型对结构存在性有严格限制，并证明了某些条件下叶形无法具有强可统一的投影结构。

Blow-up rate of the scalar curvature along the conical Kähler-Ricci flow with finite time singularities

We investigate the scalar curvature behavior along the normalized conical Kähler-Ricci flow $ω_t$, which is the conic version of the normalized Kähler-Ricci flow, with finite maximal existence time $T<\infty $. We prove that the scalar curvature of $ω_t$ is bounded from above by $C/(T-t)^2$ under the existence of a contraction associated to the limiting cohomology class $[ω_T]$. This generalizes Z. Zhang's work to the conic case.

Pluripotential theory on Teichmüller space I -- Pluricomplex Green function

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 …

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…

Schwartz指数与沿具有孤立奇点的超曲面的对数叶状结构的留数

The Schwartz index and the residue of logarithmic foliations along a hypersurface with isolated singularities

本文证明了对数叶状结构沿孤立奇点超曲面的Baum-Bott型公式，用Schwartz指数表达留数，并推广了Poincaré问题的障碍结果至任意偶数维射影空间。

Conic singularities metrics with prescribed Ricci curvature: the case of general cone angles along normal crossing divisors

Let $X$ be a non-singular compact Kähler manifold, endowed with an effective divisor $D= \sum (1-β_k) Y_k$ having simple normal crossing support, and satisfying $β_k \in (0,1)$. The natural objects one has to consider in order to explore the differential-geometric properties of the pair $(X, D)$ are the so-called metrics with conic singularities. In this article, we complete our earlier work \cite{CGP} concerning the Monge-Ampère equations on $(X, D)$ by establishing Laplacian and ${\mathscr C}^{2,α, β}$ estimates for the solution of this equations regardless to the size of the coefficients $0…

关于双Floquet-Bloch变换及针对周期结构的Green函数远场渐近性的注记

A note on double Floquet-Bloch transforms and the far-field asymptotics of Green's functions tailored to periodic structures

提出一种通过复形变积分曲面绕过奇点的方法，用于高效估计周期结构Green函数的远场渐近性，并给出示例及处理Dirac锥点等退化情形的公式。

具有超平面奇点的积分的一个留数公式

A residue formula for integrals with hyperplane singularities

本文提出了沿仿射超平面奇点积分的新留数公式，借助受全正性启发的实矩阵稳定性概念，为相关积分计算提供了新工具。

Deformations of polynomials, boundary singularities and monodromy

We study the topology of polynomial functions by deforming them generically. We explain how the non-conservation of the total ``quantity'' of singularity in the neighbourhood of infinity is related to the variation of topology in certain families of boundary singularities along the hyperplane at infinity.

Singularities of Nonconfluent Hypergeometric Functions in Several Variables

The paper deals with singularities of nonconfluent hypergeometric functions in several variables. Typically such a function is a multi-valued analytic function with singularities along an algebraic hypersurface. We describe such hypersurfaces in terms of amoebas and the Newton polytopes of their defining polynomials. In particular, we show that all $\mathcal{A}$-discriminantal hypersurfaces (in the sense of Gelfand, Kapranov and Zelevinsky) have solid amoebas, that is, amoebas with the minimal number of complement components.

Degeneracy of entire curves into higher dimensional complex manifolds

Pursuing McQuillan's philosophy in proving the Green-Griffiths conjecture for certain surfaces of general type, we deal with the algebraic degeneracy of entire curves tangent to holomorphic foliations by curves. Inspired by the recent work of Paun and Sibony, we study the intersection of Ahlfors current with the tangent bundle of the foliation, and derive some consequences. In particular, we introduce the definition of weakly reduced singularities for foliations by curves, which requires less work than the exact classification for foliations. Finally we discuss the strategy to prove the Green-…

Green functions for killed random walks in the Weyl chamber of Sp(4)

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…

Continuity of the Green function in meromorphic families of polynomials

We prove that along any marked point the Green function of a meromorphic family of polynomials parameterized by the punctured unit disk explodes exponentially fast near the origin with a continuous error term.