Abstract. We pursue the symplectic description of toric Kähler manifolds. There exists a general local classification of metrics on toric Kähler manifolds equipped with Hamiltonian 2-forms due to Apostolov, Calderbank and Gauduchon (ACG). We derive the symplectic potential for these metrics. Using a method due to Abreu, we relate the symplectic potential to the canonical potential written by Guillemin. This enables us to recover the moment polytope associated with metrics and we thus obtain global information about the metric. We illustrate these general considerations by focusing on six-dimensional Ricci-flat metrics and obtain Ricci-flat metrics associated with real cones over L^pqr and Y^pq manifolds. The metrics associated with cones over Y^pq manifolds turn out to be partially resolved with two blow-up parameters taking special (non-zero) values. For a fixed Y^pq manifold, we find explicit metrics for several inequivalent blow-ups parametrized by a natural number k in the range 0 < k < p. We also show that all known examples of resolved metrics such as the resolved conifold and the resolution of also fit the ACG classification.

PACS numbers: 04.20.−q, 04.20.Jb

Print publication: Issue 24 (21 December 2007)
Received 1 September 2007, in final form 31 October 2007
Published 29 November 2007