Time：14:00-16:00, Friday, November 15 2024
Venue: E10-212
Speaker：Kang Zuo, Wuhan University
Biography: Kang Zuo, professor at Wuhan University, has worked in Heidelberg University, the Chinese University of Hong Kong and the University of Mainz. His research interests include Hodge theory in algebraic geometry, moduli spaces, and arithmetic geometry. He has made great achievements in the representation of fundamental groups of complex algebraic varieties, hyperbolicity of moduli spaces, characterization of Shimura subvarieties and other important problems. His articles have been published in top mathematical magazines such as Invent. Math.，Duke Math.J.，JEMS，Crelle's Journal.
Title: Finiteness of families of higher dimensional varieties
Abstract: We introduce notions of Moduli spaces of polarized smooth varieties, Kodaira-Spencer deformation map, and variation of pola Hodge structures. We recall the original Shafarevich conjecture on the finiteness of families of curves over fixed bases and with fixed degeneration loci. This conjecture has been proven by Parshin and Arakelov. We present a proof using the positivity on the moduli space of smooth projective curves.
We then discuss the Shafarevich program on a moduli space M of higher dimensional varieties with good minimal models.
We propose a conjecture on the Bombieri-Lang type finiteness of the set of maps into M. The crucial part of the conjecture is about the distribution of loci of non-rigid maps into M. We show the conjecture holds for those moduli spaces carrying local Torelli theorem and such that the Mumford-Tate group is absolutely simple and the Mumford-Tate domain is not a bounded symmetry domain of rank >1. We discuss various examples of loci of non-rigid map into moduli spaces of Calabi-Yau manifolds. The talk is based a joint paper with K. Chen, TZ. Hu and RR. Sun and a joint project with RR. Sun and CL. Yu.
