Narrow Results

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