Generalized Kaehler-Einstein metrics and infinite-time singularity type of the Kaehler-Ricci flow
Abstract: Recent years have seen important progresses on geometric analysis aspect of semi-ample canonical line bundles. In this talk, we shall first recall Song-Tian's (possibly singular) generalized Kaehler-Einstein metrics on semi-ample canonical line bundles, and then we determine their metric asymptotics near singular points in Kodaira dimension one case, in its setting implying a conjecture of Song-Tian that the metric completion of the generalized Kaehler-Einstein metric is homeomorphic to the canonical model. Then we shall move to infinite-time singularity type (i.e. long-time Riemann curvature behaviors) of the Kaehler-Ricci flow (KRF) on compact Kaehler manifolds with semi-ample canonical line bundle and explain that these results provide an analytic viewpoint to classify the complex structures on the underlying manifolds. Finally, a precise relation between the singularity type of KRF and certain algebro-geometric properties of the singular fibers of the semi-ample fibration will be presented, which is a criterion for KRF developing type IIb singularities and may be seen as an evidence for the aforementioned classification viewpoint. Part of this talk is based on joint works with Frederick Fong.
张雅山，北京大学北京国际数学研究中心博士后，研究方向是复几何与几何分析，部分成果发表在Math.Ann, IMRN, JGA, MRL, AGAG等杂志。