Type II almost-homogeneous manifolds of cohomogeneity one

Daniel Guan

Vol. 253 (2011), No. 2, 383–422

DOI: 10.2140/pjm.2011.253.383

Abstract

This paper continues our investigation on the existence of extremal metrics of the general affine and type II almost-homogeneous manifolds of cohomogeneity one. It deals with the general type II cases with hypersurface ends: more precisely, with manifolds having certain ℂPⁿ × (ℂPⁿ)^∗- or ℂP²-bundle structures. In particular, we study the existence of Kähler–Einstein metrics on these manifolds and obtain new Kähler–Einstein manifolds as well as Fano manifolds without Kähler–Einstein metrics.

Keywords

almost-homogeneous manifold, Kähler–Einstein metric, Fano manifold, extremal metric, fourth-order differential equation, cohomogeneity one, fiber bundle, existence, Futaki invariant, geodesic stability

Mathematical Subject Classification 2000

Primary: 14M17, 32L05, 32M12, 53C10, 53C21

Secondary: 32Q20, 53C55, 53C26

Milestones

Received: 2 April 2009

Revised: 27 May 2011

Accepted: 11 July 2011

Published: 21 January 2012