Symplectic potentials and resolved Ricci-flat ACG metrics

TL;DR

This paper develops a symplectic framework for toric Kahler manifolds, derives explicit Ricci-flat metrics in six dimensions, and connects these to known resolved geometries like the conifold and orbifold resolutions.

Contribution

It provides a symplectic potential approach to classify and construct Ricci-flat ACG metrics, including new explicit examples with blow-up parameters.

Findings

Derived symplectic potentials for ACG metrics.

Constructed explicit Ricci-flat metrics over Y^{pq} manifolds.

Unified known resolved geometries within the ACG classification.

Abstract

We pursue the symplectic description of toric Kahler manifolds. There exists a general local classification of metrics on toric Kahler manifolds equipped with Hamiltonian two-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 blowup parameters taking special (non-zero)values. For a fixed Y^{pq} manifold, we find explicit metrics for several inequivalent blow-ups parametrised 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 C^3/Z_3 also fit the ACG classification.