On compact splitting complex submanifolds of quotients of bounded symmetric domains

We study compact complex submanifolds S of quotient manifolds X = Ω/Γ of irreducible bounded symmetric domains by torsion free discrete lattices of automorphisms, and we are interested in the characterization of the totally geodesic submanifolds among compact splitting complex submanifolds S ? X, i.e., under the assumption that the tangent sequence over S splits holomorphically. We prove results of two types. The first type of results concerns S ? X which are characteristic complex submanifolds, i.e., embedding Ω as an open subset of its compact dual manifold M by means of the Borel embedding, the non-zero (1, 0)-vectors tangent to S lift under a local inverse of the universal covering map π: Ω → X to minimal rational tangents of M. We prove that a compact characteristic complex submanifold S ? X is necessarily totally geodesic whenever S is a splitting complex submanifold. Our proof generalizes the case of the characterization of totally geodesic complex submanifolds of quotients of the complex unit ball B ⁿ obtained by Mok (2005). The proof given here is however new and it is based on a monotonic property of curvatures of Hermitian holomorphic vector subbundles of Hermitian holomorphic vector bundles and on exploiting the splitting of the tangent sequence to identify the holomorphic tangent bundle T _S as a quotient bundle rather than as a subbundle of the restriction of the holomorphic tangent bundle T _X to S. The second type of results concerns characterization of total geodesic submanifolds among compact splitting complex submanifolds S ? X deduced from the results of Aubin (1978) and Yau (1978) which imply the existence of Kähler-Einstein metrics on S ? X. We prove that compact splitting complex submanifolds S ? X of sufficiently large dimension (depending on Ω) are necessarily totally geodesic. The proof relies on the Hermitian-Einstein property of holomorphic vector bundles associated to T _S , which implies that endomorphisms of such bundles are parallel, and the construction of endomorphisms of these vector bundles by means of the splitting of the tangent sequence on S. We conclude with conjectures on the sharp lower bound on dim(S) guaranteeing total geodesy of S ? X for the case of the type-I domains of rank 2 and the case of type-IV domains, and examine a case which is critical for both conjectures, viz. on compact complex surfaces of quotients of the 4-dimensional Lie ball, equivalently the 4-dimensional type-I domain dual to the Grassmannian of 2-planes in ?⁴.