Rationally connected and rationally simply connected manifolds

Speaker: Prof. Jason Starr (Stony Brook University)

Time: 2011-06-10 14:00-15:00

Venue: 理科一号楼1114（数学所活动）

Abstract: There are analogues in complex algebraic geometry of path connectedness and simple connectedness replacing continuous maps from the closed unit interval with holomorphic maps from the complex projective line. And there are analogues of the theorem that a topological fiber bundle over an r-dimensional base has a continuous section if the the fiber is (r-1)-connected, i.e., the first r homotopy groups vanish (r=1 by Graber-Harris-S, r=2 by A. J. de Jong - Xuhua He - S). One application is the proof by de Jong - He - S of the "split case" of Serre's Conjecture II for function fields of complex surfaces: an algebraic principal bundle over a complex surface has a rational / meromorphic section if the structure group is simply connected and semisimple. (The split case completes the full proof following earlier, monumental work by Merkurjev-Suslin, Bayer-Parimala, Chernousov and Ph. Gille).

Time allowing, I will also discuss applications to fixed points of algebraic actions of Z/a x Z/b on projective manifolds, to the "weak approximation conjecture" of Hassett-Tschinkel, and to finiteness of quantum K-theory following A. Buch, et al. This will be a broad audience lecture; no particular background will be assumed.