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 bu...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.展开更多
In this paper, we consider minimal L^(2) integrals on the sublevel sets of plurisubharmonic functions on weakly pseudoconvex K?hler manifolds with Lebesgue measurable gain related to modules at boundary points of the ...In this paper, we consider minimal L^(2) integrals on the sublevel sets of plurisubharmonic functions on weakly pseudoconvex K?hler manifolds with Lebesgue measurable gain related to modules at boundary points of the sublevel sets, and establish a concavity property of the minimal L^(2) integrals. As applications, we present a necessary condition for the concavity degenerating to linearity, a concavity property related to modules at inner points of the sublevel sets, an optimal support function related to modules, a strong openness property of modules and a twisted version, and an effectiveness result of the strong openness property of modules.展开更多
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 ...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.展开更多
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-Jacob...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.展开更多
In this article,we consider a modified version of minimal L^(2) integrals on sublevel sets of plurisubharmonic functions related to modules at boundary points,and obtain a concavity property of the modified version.As...In this article,we consider a modified version of minimal L^(2) integrals on sublevel sets of plurisubharmonic functions related to modules at boundary points,and obtain a concavity property of the modified version.As applications,we give characterizations for the concavity degenerating to linearity on open Riemann surfaces and on fibrations over open Riemann surfaces.展开更多
In this paper, we prove that in a hyperconvex domain Ω in H^(n), if a non-negative Borel measure is dominated by a quaternionic Monge–Ampère measure, then it is a quaternionic Monge–Ampère measure of a fu...In this paper, we prove that in a hyperconvex domain Ω in H^(n), if a non-negative Borel measure is dominated by a quaternionic Monge–Ampère measure, then it is a quaternionic Monge–Ampère measure of a function in the class E(Ω).展开更多
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 com...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.展开更多
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 th...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.展开更多
In this article,we present the concavity of the minimal L^(2) integrals related to multiplier ideals sheaves on Stein manifolds.As applications,we obtain a necessary condition for the concavity degenerating to lineari...In this article,we present the concavity of the minimal L^(2) integrals related to multiplier ideals sheaves on Stein manifolds.As applications,we obtain a necessary condition for the concavity degenerating to linearity,a characterization for 1-dimensional case,and a characterization for the equality in 1-dimensional optimal L^(2) extension problem to hold.展开更多
Based on the results of (Wang 2001), we give some applications of division problem in spaces of entire functions of finite type. Especially, when p = 1 and H is the support functions of a bounded convex domain of C , ...Based on the results of (Wang 2001), we give some applications of division problem in spaces of entire functions of finite type. Especially, when p = 1 and H is the support functions of a bounded convex domain of C , our theorems extend the results of (Krivosheev 1991) and (Lelong 1986).展开更多
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...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.展开更多
In this paper, we survey some recent results on the existence of bounded plurisubharmonic functions on pseudoconvex domains, the Diederich-Forn^ess exponent and its relations with existence of domains with Levi-flat b...In this paper, we survey some recent results on the existence of bounded plurisubharmonic functions on pseudoconvex domains, the Diederich-Forn^ess exponent and its relations with existence of domains with Levi-flat boundary in complex manifolds.展开更多
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...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.展开更多
Let P_n={Z=(Z_1,…,Z_n)|Z_iZ_i^n<1, Z_i are 2×2 complex matrices},H_n={W=(W_1,…, W_n)|W_i=Z_iB,(Z_1,…,Z_n)∈P_n,B∈SL(2,C)}, D_n={W=(W_1,…,W_n)| W_i=AZ_iB,(Z_1,…,Z_n)∈P_n,A, B∈SL(2, C)}. Are H_n,D_n doma...Let P_n={Z=(Z_1,…,Z_n)|Z_iZ_i^n<1, Z_i are 2×2 complex matrices},H_n={W=(W_1,…, W_n)|W_i=Z_iB,(Z_1,…,Z_n)∈P_n,B∈SL(2,C)}, D_n={W=(W_1,…,W_n)| W_i=AZ_iB,(Z_1,…,Z_n)∈P_n,A, B∈SL(2, C)}. Are H_n,D_n domains of holomorphy? In the present paper, we prove that H_2, D_2 are domains of holomorphy by using the follow-ing proposition: H_2={W∈C^2[2×2]|W_1W_2~*∈P_1, |detW_1|<1, |detW_2|<1}.展开更多
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 v...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.展开更多
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.
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...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.展开更多
基金supported by the Agence Nationale de la Recherche grant“Convergence de Gromov-Hausdorff en géeométrie khlérienne”the European Research Council project“Algebraic and Khler Geometry”(Grant No.670846)from September 2015+1 种基金the Japan Society for the Promotion of Science Grant-inAid for Young Scientists(B)(Grant No.25800051)the Japan Society for the Promotion of Science Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers
文摘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.
基金supported by National Key R&D Program of China (Grant No. 2021YFA1003100)supported by National Natural Science Foundation of China (Grant Nos. 11825101, 11522101, and 11431013)+1 种基金supported by the Talent Fund of Beijing Jiaotong Universitysupported by China Postdoctoral Science Foundation (Grant Nos. BX20230402 and 2023M743719)。
文摘In this paper, we consider minimal L^(2) integrals on the sublevel sets of plurisubharmonic functions on weakly pseudoconvex K?hler manifolds with Lebesgue measurable gain related to modules at boundary points of the sublevel sets, and establish a concavity property of the minimal L^(2) integrals. As applications, we present a necessary condition for the concavity degenerating to linearity, a concavity property related to modules at inner points of the sublevel sets, an optimal support function related to modules, a strong openness property of modules and a twisted version, and an effectiveness result of the strong openness property of modules.
基金supported by China Postdoctoral Science Foundation (Grant No. BX2021015)supported by National Key R&D Program of China (Grant No. SQ2020YFA0712800)National Natural Science Foundation of China (Grant No. 11822101)。
文摘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.
文摘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.
基金supported by National Key R&D Program of China(Grant No.2021YFA1003100)supported by NSFC(Grant Nos.11825101,11522101 and 11431013)+1 种基金supported by the Talent Fund of Beijing Jiaotong Universitysupported by China Postdoctoral Science Foundation(Grant Nos.BX20230402 and 2023M743719)。
文摘In this article,we consider a modified version of minimal L^(2) integrals on sublevel sets of plurisubharmonic functions related to modules at boundary points,and obtain a concavity property of the modified version.As applications,we give characterizations for the concavity degenerating to linearity on open Riemann surfaces and on fibrations over open Riemann surfaces.
文摘In this paper, we prove that in a hyperconvex domain Ω in H^(n), if a non-negative Borel measure is dominated by a quaternionic Monge–Ampère measure, then it is a quaternionic Monge–Ampère measure of a function in the class E(Ω).
文摘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.
文摘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.
基金The first author was supported by NSFC-11825101,NSFC-11522101 and NSFC-11431013.
文摘In this article,we present the concavity of the minimal L^(2) integrals related to multiplier ideals sheaves on Stein manifolds.As applications,we obtain a necessary condition for the concavity degenerating to linearity,a characterization for 1-dimensional case,and a characterization for the equality in 1-dimensional optimal L^(2) extension problem to hold.
基金Supported by NSFC(60174007)and Shanxi Foundation of Science(20031002)
文摘Based on the results of (Wang 2001), we give some applications of division problem in spaces of entire functions of finite type. Especially, when p = 1 and H is the support functions of a bounded convex domain of C , our theorems extend the results of (Krivosheev 1991) and (Lelong 1986).
文摘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.
基金supported by NSF(Grant No.DMS 1500952)supported by NSF(Grant No.DMS 1700003)
文摘In this paper, we survey some recent results on the existence of bounded plurisubharmonic functions on pseudoconvex domains, the Diederich-Forn^ess exponent and its relations with existence of domains with Levi-flat boundary in complex manifolds.
文摘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.
文摘Let P_n={Z=(Z_1,…,Z_n)|Z_iZ_i^n<1, Z_i are 2×2 complex matrices},H_n={W=(W_1,…, W_n)|W_i=Z_iB,(Z_1,…,Z_n)∈P_n,B∈SL(2,C)}, D_n={W=(W_1,…,W_n)| W_i=AZ_iB,(Z_1,…,Z_n)∈P_n,A, B∈SL(2, C)}. Are H_n,D_n domains of holomorphy? In the present paper, we prove that H_2, D_2 are domains of holomorphy by using the follow-ing proposition: H_2={W∈C^2[2×2]|W_1W_2~*∈P_1, |detW_1|<1, |detW_2|<1}.
文摘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.
基金supported by National Natural Science Foundation of China (Grant Nos. 11571018 and 11331001)
文摘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.
基金supported by National Natural Science Foundation of China (Grant Nos. 11825101, 11522101 and 11431013)
文摘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.
基金supported by the National Natural Science Foundation of China(Nos.11431013,11825101,11522101,11688101)the National Key R&D Program of China(No.2021YFA1003100)。
文摘In the present article, the authors find and establish stability of multiplier ideal sheaves, which is more general than strong openness.
