Toric generalized K$\ddot{a}$hler structures. I

Abstract: This is a sequel of \cite{Wang}, which provides a general formalism for this paper. We mainly investigate thoroughly a subclass of toric generalized K$\ddot{a}$hler manifolds of symplectic type introduced by Boulanger in \cite{Bou}. We find torus actions on such manifolds are all \emph{strong Hamiltonian} in the sense of \cite{Wang}. For each such a manifold, we prove that besides the ordinary two complex structures $J_\pm$ associated to the biHermitian description, there is a \emph{third} canonical complex structure $J_0$ underlying the geometry, which makes the manifold toric K$\ddot{a}$hler. We find the other generalized complex structure besides the symplectic one is always a B-transform of a generalized complex structure induced from a $J_0$-holomorphic Poisson structure $\beta$ characterized by an anti-symmetric constant matrix. Stimulated by the above results, we introduce a \emph{generalized Delzant construction} which starts from a Delzant polytope with $d$ faces of codimension 1, the standard K$\ddot{a}$hler structure of $\mathbb{C}^d$ and an anti-symmetric $d\times d$ matrix. This construction is used to produce non-abelian examples of strong Hamiltonian actions.