Arakelov inequalities and the uniformization of certain rigid Shimura varieties
by
Kang Zuo
FB Mathematik, Universitaet Mainz
Coauthors: Eckart Viehweg

Let Y be a non-singular projective manifold with an ample canonical sheaf, and let V be a rational variation of Hodge structures of weight one on Y with Higgs bundle E(1,0) + E(0,1), coming from a family of Abelian varieties. If Y is a curve the Arakelov inequality says that the difference of the slope of E(1,0) and the one of E(0,1) is is smaller than or equal to the degree of the canonical sheaf. We prove a similar inequality in the higher dimensional case. If the latter is an equality, as well as the Bogomolov inequality for E(1,0) or for E(0,1), one hopes that Y is a Shimura variety, and V a uniformizing variation of Hodge structures. This is verified, in case the universal covering of Y does not contain factors of rank >1. Part of the results extend to variations of Hodge structures over quasi-projective manifolds.

Paper reference: arXiv:math.AG/0503339

Date received: March 29, 2005