The so-called Abreu-Guillemin theory classifies invariant Kähler metrics on toric symplectic manifolds in terms of certain types of singular metrics on the corresponding Delzant polytope. In a series of two talks, I will describe an extension of this theory to symplectic manifolds equipped with a singular Lagrangian fibration that only has elliptic singularities (also called a toric Lagrangian fibration).

In this first talk, I plan to (1) recall the classical Abreu-Guillemin theory, (2) explain how symplectic manifolds with toric Lagrangian fibrations can be viewed as the natural generalizations of toric symplectic manifolds in the context of Hamiltonian groupoid actions, and (3) sketch how, using this groupoid point of view, the Abreu-Guillemin theory naturally extends to such symplectic manifolds.

The provisional plan for the second talk is to explain the extension of the Abreu-Guillemin theory in more details and to discuss how this can be used to explicitly construct extremal Kähler metrics on some symplectic manifolds with toric Lagrangian fibrations.

This is based on arXiv:2401.02910 and ongoing joint work with Miguel Abreu and Rui Loja Fernandes.