Rui Fernandes (Urbana-Champaign)

Title:

Invariant Kähler metrics for toric fibrations

 

Abstract:

In the late 1990s, Guillemin and Abreu described all invariant, compatible Kähler metrics for a compact symplectic toric manifold, in terms of singular Hessian metrics in the associated Delzant polytope. Abreu's work also encompasses a fourth-order nonlinear PDE expressing the condition for an invariant Kähler metric to be extremal in the sense of Calabi. Subsequently, Donaldson developed various estimates for solutions to Abreu's equation, sparking a series of subsequent research works in the subject.

In this talk, I will discuss (extremal) invariant Kähler metrics for any Lagrangian fibration admitting only elliptic singularities. It turns out that such fibrations are precisely the Hamiltonian spaces of toric actions of symplectic torus bundles, a special type of symplectic groupoid. This allows us to extend the Abreu-Guillemin theory to a much larger class and to provide many more examples of invariant (extremal) Kähler metrics. For example, this includes the case of complex ruled surfaces over elliptic curves, as studied by Apostolov et al. This presentation is based on ongoing joint work with Miguel Abreu (IST-Lisbon) and Maarten Mol (Max Planck-Bonn).