A negatively curved Kähler threefold not covered by the ball
The author presents a generalization to the dimension \(3\) of the Mostow-Siu construction to produce an infinite family of compact negatively curved Kähler surfaces whose universal cover is not biholomorphic to the ball. He obtains examples of three-dimensional compact Kähler manifolds with negative curvature, not covered by the ball. The obtained examples are presented in terms of monodromy covers of hypergeometric functions, using the techniques developed by Deligne and Mostow. The finite local monodromy condition can be translated into a purely numerical condition, and one can list all choices of exponents for hypergeometric functions that yield only finite local monodromy. In dimension 3 or higher, the list is finite. There are nine groups with finite monodromy, all in dimension three. It turns out that only three of these nine groups yield smooth compact negatively curved manifolds. Each example is obtained from a hypergeometric monodromy group in PU\((3,1)\) that is not discrete but has finite local monodromy. The main result is stated in Theorem 3.3, where one presents the construction as well as the main features of the obtained examples.
monodromy covers of hypergeometric functions
scientific article; zbMATH DE number 2175700