Toric generalized Kähler structures
I will report my recent work on toric generalized Kahler structures of symplectic type. This is an extension of the famous Abreu-Guillemin theory of toric Kahler structures, which characterizes a toric Kahler structure using a strictly convex function (called symplectic potential) on the image of the moment map. I will mainly focus on the special case of anti-diagonal ones introduced by L. Boulanger. For each such a manifold, I proved that besides the ordinary two complex structures $J_\pm$ associated to the biHermitian description, there is a third canonical complex structure $J_0$ underlying the geometry, which makes the manifold toric K$\ddot{a}$hler. 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. I will also explain how this theory can be used to construct new GK structures.