复射影簇对上的完备常数量曲率凯勒度量与代数稳定性

The famous Yau-Tian-Donaldson conjecture in the case of Fano manifolds about Kahler-Einstein metric and K-stability has been proved, yet leaving the more general problem about constant scalar curvature Kahler (CSCK) metrics and K-stability of positive line bundles over complex projective manifolds open. Although it has been proved that the existence of CSCK metrics implies K-Stability, the opposite direction remains widely open, hence standing as a leading problem in both fields of differential geometry and algebraic geometry. Following the frame work built up by Donaldson for the purpose of solving the Kahler-Einstein metric case, we consider questions in the more general setting, which is a log pair of complex projective manifolds (X,D). These questions include conical CSCK metrics, logarithmic K-stability, logarithmic Chow-Stability and relations between these concepts. Based on the collaborative work by the applicant and his collaborator on dimension one, this project will study log pairs of complex projective manifolds in the case of complete CSCK metric, which corresponds to the conical metric with angle 0. More specifically, this project will study whether the existence of complete CSCK metric can imply almost Chow-stability, whether one can deform almost Chow-stability to Chow-stability, whether balancing metrics, corresponding to Chow-stability, can converge to complete CSCK metrics, and the model of degeneration for Chow-semistability. This project will also study the Bergman kernels on log pairs of complex projective manifolds with singular Kahler metrics.

关于Fano流形上的凯勒-爱因斯坦度量的存在性与K-稳定性的邱-田-唐纳森猜想的解决催生了更广泛的关于复射影流形上正线丛的常数量曲率凯勒(CSCK)度量的存在性与其K稳定性的问题。尽管CSCK度量若存在已被证明能推出K稳定性，反过来的问题还远未解决。沿着唐纳森为解决凯勒-爱因斯坦情形而开创的框架性架构，我们考虑更广泛的复射影簇对的情形. 这包括其上的锥角CSCK度量，对数K稳定性，对数周稳定性以及这些概念之间的关系。在申请人与其合作者在一维的工作的基础上，本项目将着重研究复射影簇对的对应于锥角为零，即完备CSCK度量的情形。具体的，本项目将研究复射影簇对上的完备CSCK度量能否推出其几乎周稳定性，几乎周稳定能否渐变为周稳定，对应于周稳定的平衡度量能否收敛到完备CSCK度量，周半稳定的退化模型等问题。因其密切相关性，本项目还将研究复射影簇对上奇异凯勒度量的伯格曼核。

沿着唐纳森为解决著名的邱-田-唐纳森猜想而开创的框架性架构，我们考虑更广泛的复射影簇对上的锥角CSCK度量和其对数K稳定性这两个概念和它们之间的关系。本项目着重研究具有完备庞加莱类CSCK度量的复射影簇对和用全纯向量丛上的值分布理论构造奇异度量的方法。我们证明了一维黎曼面的情形和全空间是除子上的正线丛的射影完备化的这种情形时复射影簇对上的完备常数量曲率凯勒度量能推出代数K-半稳定性。这两个结果对我们研究更一般的情形提供了重要的思路和方法，也是该领域的重要成果。我们还证明了一维时的庞加莱类常曲率度量的Bergman kernel的渐近展开和高维光滑度量情形下的对数Bergman kernel的渐进展开，这是该领域关于此类奇异度量的Bergman kernel渐近展开的最新的成果. 我们进一步利用一维庞加莱类常曲率度量的Bergman kernel的渐进展开，证明了一列收敛的双曲黎曼面的多典则嵌入的收敛性，该成果在黎曼面模空间完备化理论中有重要意义。我们还计算了一般全纯向量丛的全纯截面的零点的平均值，为该领域的进一步发展打下了坚实的基础。