Holomorphic Retractions of Bounded Symmetric Domains onto Totally Geodesic Complex Submanifolds

摘要

Given a bounded symmetric domain Ω the author considers the geometry of its totally geodesic complex submanifolds S■Ω.In terms of the Harish-Chandra realization Ω and taking S to pass through the origin 0∈Ω,so that S=E∩Ω for some complex vector subspace of C^(n),the author shows that the orthogonal projectionρ:Ω→E maps Ω onto S,and deduces that S■Ω is a holomorphic isometry with respect to the Caratheodory metric.His first theorem gives a new derivation of a result of Yeung’s deduced from the classification theory by Satake and Ihara in the special case of totally geodesic complex submanifolds of rank 1 and of complex dimension≥2 in the Siegel upper half plane Hg,a result which was crucial for proving the nonexistence of totally geodesic complex sub or bifolds of dimension≥2 on the open Torelli locus of the Siegel modular variety Ag by the same author.The proof relies on the characterization of totally geodesic submanifolds of Riemannian symmetric spaces in terms of Lie triple systems and a variant of the Hermann Convexity Theorem giving a new characterization of the Harish-Chandra realization in terms of bisectional curvatures.