Toric generalized Kähler structures. I

Abstract

This is a sequel of [20], which provides a general formalism for this paper. We mainly investigate thoroughly a subclass of toric generalized Kähler manifolds of symplectic type introduced by Boulanger [3]. We find that torus actions on such manifolds are all strong Hamiltonian in the sense of [20]. For each such a manifold, we prove that besides the ordinary complex structures J± underlying the biHermitian description, there is a third canonical complex structure J0, which makes the manifold toric Kähler. The other generalized complex structure besides the symplectic one is always a B-transform of a generalized complex structure induced from a J0-holomorphic Poisson structure β characterized by an anti-symmetric constant matrix. Stimulated by the above results, we introduce a generalized Delzant construction which starts from a Delzant polytope with d faces of codimension 1, the standard Kähler structure of Cd and an anti-symmetric d×d matrix. This construction is used to produce non-abelian examples of strong Hamiltonian actions.