Twist, elementary deformation, and KK correspondence in generalized complex geometry

TL;DR

This paper introduces new operations in generalized complex geometry, applies twist constructions to generate new structures, and recovers the Kahler-Kahler correspondence through these methods, especially in four dimensions.

Contribution

It defines conformal change and elementary deformation in generalized complex geometry and applies twist constructions to produce new generalized Kahler structures, including a generalized KK correspondence.

Findings

Established conditions for Courant integrability of twisted structures

Constructed new generalized Kahler manifolds from Hamiltonian Killing vector fields

Reproduced the Kahler-Kahler correspondence via generalized geometry methods

Abstract

We define the operations of conformal change and elementary deformation in the setting of generalized complex geometry. Then we apply Swann's twist construction to generalized (almost) complex and Hermitian structures obtained by these operations and establish necessary and sufficient conditions for the Courant integrability of the resulting twisted structures. In particular, we associate to any appropriate generalized Kahler manifold (M, G, \mathcal J ) with a Hamiltonian Killing vector field a new generalized Kahler manifold, depending on the choice of a pair of non-vanishing functions and compatible twist data. We study this construction when (M, G, \mathcal J) is (diagonal) toric, with emphasis on the four dimensional case. In particular, we apply it to deformations of the standard flat Kahler metric on C^{n}, the Fubini-Study Kahler metric on CP^{2} and the so called admissible Kahler metrics on Hirzebruch surfaces. As a further application, we recover the KK (Kahler-Kahler) correspondence, which is obtained by specializing to the case of an ordinary Kahler manifold.