This volume consists of 15 papers contributing to the Hayama Symposium on Complex Analysis in Several Variables XXIII, which was dedicated to the 100th anniversary of the creation of the Bergman kernel. The symposium took place in Hayama and Tokyo in July 2022. Each article is closely related to the Bergman kernel, covering topics in complex analysis, differential geometry, representation theory, PDE, operator theory, and complex algebraic geometry.
Specifically, some papers address the L2 extension operators from a newly opened viewpoint after solving Suita's conjecture for the logarithmic capacity. They are also continuations of quantitative solutions to the openness conjecture for the multiplier ideal sheaves. The study involves estimates for the solutions of the d-bar equations, focusing on the existence of compact Levi-flat hypersurfaces in complex manifolds.
The collection also reports progress on various topics, including the existence of extremal Kähler metrics on compact manifolds, Lp variants of the Bergman kernel, Wehrl-type inequalities, homogeneous Kähler metrics on bounded homogeneous domains, asymptotics of the Bergman kernels, and harmonic Szegő kernels and operators on the Bergman spaces and Segal-Bargmann spaces.
Some of the papers are written in an easily accessible way for beginners. Overall, this collection updates how a basic notion provides strong insights into the internal relationships between independently found phenomena.
S. Bao, Q. Guan, Zhitong Mi and Z. Yuan. Concavity property of minimal L^2 integrals with Lebesgue measurable gain VII—Negligible weights.- P. Blaschke and M. Engliš, M-harmonic Szego kernel on the ball.- Bo-Yong Chen, Y. Xiong and L. Zhang, Some aspects of the p-Bergman theory.- S. Finski, On semiclassical Ohsawa-Takegoshi extension theorem.- Y. Hashimoto, Balanced metrics for extremal Kahler metrics and Fano manifolds.- F. Haslinger, Unbounded operators on the Segal-Bargmann space.- T. Hisamoto, Asymptotic construction of the optimal degeneration for a Fano manifold.- Chin-Yu Hsiao and G. Marinescu, Semi-classical spectral asymptotics of Toeplitz operators on strictly pseudodonvex domains.- H. Ishi, On a concrete realization of simply connected complex domains admitting homogeneous Kahler metrics.- J. Kamimoto, The asymptotic behavior of the Bergman kernel on pseudoconvex model domains.- T. Ohsawa, Bundle-convexity and kernel asymptotics on a class of locally pseudoconvex domains.- Mei-Chi Shaw, The dbar-equation on the Hartogs triangles in C^2 and CP^2.- H. Tsuji, Dynamical systems of p-Bergman kernels.- G. Zhang, Wehrl-type inequalities for Bergman spaces on domains in C^d and completely positive maps.- X. Zhou , Converse of L^2 existence and extension of cohomology classes.
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