Characterizations of plurisubharmonic functions

We give characterizations of(quasi-)plurisubharmonic functions in terms of L^(p)-estimates of■and Lp-extensions of holomorphic functions.

A Class of Plurisubharmonic Functions of Positively Homogeneous of Order ρ

In this paper, we construct and discuss some plurisubharmonic functions of positively homogeneous of oder ρ and some weight system, which will be used in the study of the solution of Dirichlet problem for the complex Monge\|Ampère equations and the Division problem in spaces of entire function.

Optimal regularity of plurisubharmonic envelopes on compact Hermitian manifolds

In this paper, we prove the C^(1,1)-regularity of the plurisubharmonic envelope of a C^(1,1) function on a compact Hermitian manifold. We also present the examples to show this regularity is sharp.

Nadel-type multiplier ideal sheaves on complex spaces with singularities

In this article,we introduce multiplier ideal sheaves on complex spaces with singularities(not necessarily normal)via Ohsawa’s extension measure,as a special case of which,it turns out to be the socalled Mather-Jacobian multiplier ideals in the algebro-geometric setting.Inspired by Nadel’s coherence and Guan-Zhou’s strong openness properties of the multiplier ideal sheaves,we discuss similar properties for the generalized multiplier ideal sheaves.As applications,we obtain a reasonable generalization of(algebraic)adjoint ideal sheaves to the analytic setting and establish some extension theorems on K?hler manifolds from singular hypersurfaces.Relying on our multiplier and adjoint ideals,we also give characterizations for several important classes of singularities of pairs associated with plurisubharmonic functions.

Trace inequalities, isocapacitary inequalities, and regularity of the complex Hessian equations

In this paper, we study the relations between trace inequalities(Sobolev and Moser-Trudinger types), isocapacitary inequalities, and the regularity of the complex Hessian and Monge-Amp`ere equations with respect to a general nonnegative Borel measure. We obtain a quantitative characterization for these relations through the properties of the capacity-minimizing functions.

A general extension theorem for cohomology classes on non reduced analytic subspaces

The main purpose of this paper is to generalize the celebrated L^2 extension theorem of Ohsawa and Takegoshi in several directions: The holomorphic sections to extend are taken in a possibly singular hermitian line bundle, the subvariety from which the extension is performed may be non reduced, the ambient manifold is K¨ahler and holomorphically convex, but not necessarily compact.

Strong openness of multiplier ideal sheaves and optimal L^2 extension

In this paper, we reveal that our solution of Demailly's strong openness conjecture implies a matrix version of the conjecture; our solutions of two conjectures of Demailly-Koll′ar and Jonsson-Mustat?a implies the truth of twisted versions of the strong openness conjecture; our optimal L^2 extension implies Berndtsson's positivity of vector bundles associated to holomorphic fibrations over a unit disc.

Invariant holomorphic extension in several complex variables Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

Two fundamental problems on the invariant holomorphic extensions have been posed, which are naturally arose from our solution of the extended future tube conjecture and closely and deeply related to the general theory of Stein manifolds due to Cartan-Serre. In this paper, the relationship is presented between the two problems, the motivation of considering the problems, and the methods to approach the problems. We have also posed some questions and conjectures related to this two problems.

ON THE EXTENDED MATRIX DISC CONJECTURE

Let Pn={Z=（Z1,…,Zn）|ZiZin【1, Zi are 2×2 complex matrices},Hn={W=（W1,…, Wn）|Wi=ZiB,（Z1,…,Zn）∈Pn,B∈SL（2,C）}, Dn={W=（W1,…,Wn）| Wi=AZiB,（Z1,…,Zn）∈Pn,A, B∈SL（2, C）}. Are Hn,Dn domains of holomorphy? In the present paper, we prove that H2, D2 are domains of holomorphy by using the follow-ing proposition: H2={W∈C2[2×2]|W1W2*∈P1, |detW1|【1, |detW2|【1}.

On the Cegrell Classes Associated to a Positive Closed Current

The aim of this paper is to study the operatoron■ on some classes of plurisubharmonic (psh) functions, which are not necessary bounded, where T is a positive closed current of bidimension (q, q) on an open set ? of C^n. The author introduces two classes F_p^T (?) and■ and shows first that they belong to the domain of definition of the operator■. Then the author proves that all functions that belong to these classes are C_T-quasi-continuous and that the comparison principle is valid for them.

On interpolation for Hormander’s algebras

We discuss some recent results on interpolation problems for weighted Hrmander’s algebras of holomorphic functions in several complex variables, and also give a sharp estimate on counting functions of interpolating varieties.

Concavity of minimal L^(2) integrals related to multiplier ideal sheaves on weakly pseudoconvex Kahler manifolds

In this paper,we present the concavity of the minimal L^(2)integrals related to multiplier ideal sheaves on the weakly pseudoconvex Kahler manifolds,which implies the sharp effectiveness results of the strong openness conjecture and a conjecture posed by Demailly and Kollar(2001)on weakly pseudoconvex Kahler manifolds.We obtain the relation between the concavity and the L^(2)extension theorem.

Stability of Multiplier Ideal Sheaves

In the present article, the authors find and establish stability of multiplier ideal sheaves, which is more general than strong openness.