Holomorphic Retractions of Bounded Symmetric Domains onto Totally Geodesic Complex Submanifolds

Abstract

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 ⋐ ℂⁿ and taking S to pass through the origin 0 ∈ Ω, so that S = E ⋂ Ω for some complex vector subspace of ℂⁿ, the author shows that the orthogonal projection ρ: Ω → E maps Ω onto S, and deduces that S ⊂ Ω is a holomorphic isometry with respect to the Carathéodory 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 ${{\cal H}_g}$, a result which was crucial for proving the nonexistence of totally geodesic complex suborbifolds of dimension ≥ 2 on the open Torelli locus of the Siegel modular variety ${{\cal A}_g}$ 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.