Extremal bounded holomorphic functions and an embedding theorem for arithmetic varieties of rank ? 2

Abstract

Let Ω be a bounded symmetric domain of rank ≥ 2, and Γ ⊂ Aut(Ω) be a torsion-free irreducible cocompact lattice, X := Ω/Γ. On the projective manifold X there is the canonical Kahler-Einstein metric, which is of nonpositive holomorphic bisectional curvature. In Mok [M1,2] we established a Hermitian metric rigidity theorem for such projective manifolds X, which in the case when Ω is irreducible says that any Hermitian metric of nonpositive curvature in the sense of Griffiths is necessarily a constant multiple of the Kahler-Einstein metric. As a consequence, we proved in the latter case that any nontrivial holomorphic mapping f : X → Z into a Hermitian manifold Z of nonpositive curvature in the sense of Griffiths is necessarily an isometric immersion totally geodesic with respect to the Hermitian connection on Z. The Hermitian metric rigidity theorem can be taken as a tool in establishing the statement that any f : X → Z is necessarily a holomorphic immersion, a statement which only concerns the complex structure of X and Z. The Hermitian metric rigidity theorem and its consequences were generalized by To [To] to be applicable to any torsion-free lattice Γ ⊂ Aut(Ω), where X := Ω/Γ may be noncompact.