A characterization of varieties whose universal cover is a bounded symmetric domain without ball factors

TL;DR

This paper characterizes certain complex algebraic varieties with universal covers that are bounded symmetric domains, using holomorphic endomorphisms and geometric properties of associated tensors and cones.

Contribution

It provides two novel characterizations of these varieties based on the existence of specific holomorphic endomorphisms and geometric conditions of the Mok characteristic cone.

Findings

Characterizations in terms of holomorphic endomorphisms of tensor products.

Description of the Mok characteristic cone and its projective scheme.

Conditions under which the projective scheme is a union of skew projective varieties.

Abstract

We give two characterizations of varieties whose universal cover is a bounded symmetric domain without ball factors in terms of the existence of a holomorphic endomorphism \s of the tensor product T\otimes T' of the tangent bundle T with the cotangent bundle T'. To such a curvature type tensor \s one associates the first Mok characteristic cone S, obtained by projecting on T the intersection of ker (\s) with the space of rank 1 tensors. The simpler characterization requires that the projective scheme associated to S be a finite union of projective varieties of given dimensions and codimensions in their linear spans which must be skew and generate.