Complete Calabi-Yau metrics from Kahler metrics in D=4

Abstract

In the present work, a family of Calabi-Yau manifolds with a local Hamiltonian Killing vector is described in terms of a nonlinear equation whose solutions determine the local form of the geometries. The main assumptions are that the complex (3, 0)-form is of the form eik , where is preserved by the Killing vector, and that the space of the orbits of the Killing vector is, for fixed value of the momentum map coordinate, a complex 4-manifold, in such a way that the complex structure of the 4-manifold is part of the complex structure of the complex 3-fold. The family considered here include the ones considered in A. Fayyazuddin, Classical Quantum Gravity 24, 3151 (2007); O. P. Santillan, Classical Quantum Gravity 27, 155013 (2010); H. Lu, Y. Pang, and Z. Wang, Classical Quantum Gravity 27, 155018 (2010) as a particular case. We also present an explicit example with holonomy exactly SU(3) by use of the linearization introduced in A. Fayyazuddin, Classical Quantum Gravity 24, 3151 (2007), which was considered in the context of D6 branes wrapping a complex 1-cycle in a hyperkahler 2-fold.