Holomorphic retractions of bounded symmetric domains onto totally geodesic complex submanifolds

Abstract

<p>Given a bounded symmetric domain Ω the author considers the geometry of its totally geodesic complex submanifolds <em>S</em> ⊂ Ω. In terms of the Harish-Chandra realization Ω ⋐ ℂ^<em>n</em> and taking <em>S</em> to pass through the origin 0 ∈ Ω, so that <em>S</em> = <em>E</em> ⋂ Ω for some complex vector subspace of ℂ^<em>n</em>, the author shows that the orthogonal projection <em>ρ</em>: Ω → <em>E</em> maps Ω onto <em>S</em>, and deduces that <em>S</em> ⊂ 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 ��, 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 �� 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.<br></p>