A negatively curved Kähler threefold not covered by the ball

Abstract

We construct examples of three-dimensional compact Kahler manifolds with negative curvature, not covered by the ball. Our manifolds are obtained as a natural generalization of the two-dimensional examples discovered by Mostow and Siu, using their description in terms of monodromy covers of hypergeometric functions. Each example is obtained from a hypergeometric monodromy group in PU(3,1) that is not discrete but has finite local monodromy. We describe a manifold on which the group acts discretely, and check that it has a compact quotient with the above features. The examples are locally described as branched covers of the ball, with totally geodesic branch locus.