A negatively curved Kähler threefold not covered by the ball
type: Item
instance of: scholarly article
title: A negatively curved Kähler threefold not covered by the ball
zbMATH Open document ID: 1075.53073
author: Martin Deraux
published in: Inventiones Mathematicae
publication date: 2005-06-13T00:00:00Z
review text: 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.
reviewed by: Vasile Oproiu
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 32J17
Mathematics Subject Classification ID: 32J27
Mathematics Subject Classification ID: 32Q15
zbMATH DE Number: 2175700
zbMATH Keywords: Kähler manifolds
zbMATH Keywords: universal cover
zbMATH Keywords: monodromy covers of hypergeometric functions
MaRDI profile type: Publication
full work available at URL: s00222-004-0414-z
OpenAlex ID: W2094532493
cites work: Fundamental groups of the spaces of regular orbits of the finite unitary reflection groups of dimension 2
cites work: Monodromy of hypergeometric functions and non-lattice integral monodromy
cites work: Commensurabilities among Lattices in PU (1,n). (AM-132)
cites work: On the universal cover of certain exotic Kähler surfaces of negative curvature
cites work: Quotients of the complex ball by discrete groups
cites work: On a remarkable class of polyhedra in complex hyperbolic space
cites work: Generalized Picard lattices arising from half-integral conditions
cites work: Braids, hypergeometric functions, and lattices
cites work: On Discontinuous Action of Monodromy Groups on the Complex n-Ball
cites work: A compact Kähler surface of negative curvature not covered by the ball
cites work: Finite Unitary Reflection Groups
cites work: Examples of non-positively curved Kähler manifolds
DOI: 10.1007/S00222-004-0414-Z
description: scientific article; zbMATH DE number 2175700