A metric question on compactified complex ball quotients

Tsz Ho Ip, Purdue University

Abstract

A complex ball quotient can be smoothly compactified to be a projective algebraic manifold. It is interesting to distinguish the differences between the category of Riemannian metrics and Kahler metrics on this compactified space. The present thesis is a result in this direction. We show that there exists a projective algebraic manifold which supports a Riemannian metric with non-positive Riemannian sectional curvature but no Kahler metric with the same curvature condition. We first show that the canonical line bundle is positive on the compatifying divisors. With respect to the metric of the compactified manifold, the Ricci tensor is zero along these divisors. We can then show that it is impossible for a Kahler metric on such a manifold to have non-positive holomorphic bisectional curvature. On the other hand, if the metric on the compactified manifold is of non-positive Riemannian sectional curvature, then the holomorphic bisectional curvature must be non-positive, because it can be written as a sum of the Riemannian sectional curvature. Hence, this metric is not Kahler.

Degree

Ph.D.

Advisors

Yeung, Purdue University.

Subject Area

Applied Mathematics|Mathematics

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