Twist, elementary deformation and K/K correspondence in generalized geometry

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 [Formula: see text] 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 [Formula: see text] is toric, with emphasis on the four-dimensional case, and we apply it to deformations of the standard flat Kähler metric on [Formula: see text], the Fubini–Study metric on [Formula: see text] 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.

On generalized Gauduchon metrics

We study a class of Hermitian metrics on complex manifolds, recently introduced by Fu, Wang and Wu, which are a generalization of Gauduchon metrics. This class includes the class of Hermitian metrics for which the associated fundamental 2-form is ∂∂-closed. Examples are given on nilmanifolds, on products of Sasakian manifolds, on S1-bundles and via the twist construction introduced by Swann.