Title: Invariant Kähler metrics for toric actions
Abstract: In the late 1990s, Guillemin and Abreu described all invariant, compatible Kähler metrics for symplectic toric manifolds. They used singular Hessian metrics in the associated Delzant polytopes. Abreu's work also includes a fourth-order nonlinear PDE expressing the condition for an invariant Kähler metric to be extremal, in the sense of Calabi. Later, Donaldson developed various estimates for solutions to Abreu's equation, sparking a series of subsequent research works in the subject.
In this talk, I'll discuss invariant Kähler metrics for toric actions of symplectic torus bundles. This extends the theory to non-toric manifolds and allows us to discover many more examples of invariant (extremal) Kähler metrics, including, for instance, 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).
