Title：Degenerations of hyperkahler metrics on K3 surfaces I, II

Speaker：孙 崧  （教授，UC Berkeley & Stony Brook）

Time：2018年8月15日、16日      下午  16:00-17:00

Room：东区管理科研楼  数学科学学院1318室

Abstract：A K3 surface is a simply connected compact complex surface with trivial canonical bundle. Moduli space of K3 surfaces has been extensively studied in algebraic geometry and it can be characterized in terms of the period map by the Torelli theorem. The differential geometric significance is that every K3 surface admits a hyperkahler metric (a metric whose holonomy group is SU(2)), which is in particular Ricci-flat. The understanding of limiting behavior of a sequence of hyperkahler K3 surfaces gives prototype for more general questions concerning Ricci curvature in Riemannian geometry.  

In these two talks I will survey what is known on this, and explain a new glueing construction, joint with Hans-Joachim Hein, Jeff Viaclovsky and Ruobing Zhang, that shows a multi-scale collapsing phenomenon, and discuss possible connection with degeneration of polarized K3 surfaces in algebraic geometry. 

The first talk will be an elementary introduction to the Gibbons-Hawking construction, which gives a way of building hyperkahler four manifolds with U(1) symmetry in terms of a positive harmonic function in the three space. This yields many interesting examples, as well as provides models for more complicated geometric constructions.