主 题: Rationally connected and rationally simply connected manifolds
报告人: Prof. Jason Starr (State Univ. of New York at Stony Brook )
时 间: 2011-06-10 14:00-15:00
地 点: 理科一号楼1114（数学所活动）
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.
