2005/03/16 by Eckart Viehweg, Kang Zuo, Viehweg, Eckart +1
Mathematics · #14D05 #14D07 #14G35 #14K10 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.math/0503339
openalex publication_date 2005/03/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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. The revised version corrects several mistakes and ambiguities, pointed out by the referee. Following suggestions of the referee the presentation of the results was improved.