Toric Kahler geometry: October 5 – 9, 2015

Organized by Xiuxiong Chen, Simon Donaldson

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Following the seminal ideas of Calabi, the existence of extremal and contant scalar curvature is a central problem in Kahler geometry. Over the past 30 years most attention has been paid to the special case of Kahler-Einstein metrics, but there is now renewed interest in the more general situation, relating the existence to algebro-geometric stability conditions. In this case, the metrics are solutions of 4th. (or higher) order fully-nonlinear PDE and the existence theory is difficult, so comparatively little is known in the way of general existence results.
This workshop will focus on the special case of toric manifolds. In this case one can transform the problem to a 4th. order PDE on a polytope. Techniques from convex analysis and real Monge-Amp`ere theory can be brought into play and the PDE is similar to the maximal equation arising in affine differential geometry. These special features have allowed substantial advances. Donaldson showed that stability implies the existence of constant scalar curvature metrics in dimension 2. The converse is a result of B. Zhou and X.H. Zhu. The existence results in dimension 2 were extended to extremal metrics by Li, Chen and Sheng, who introduced new techniques from affine differential geometry. In other directions there has been work on the Calabi flow and connections with algebraic geometry. The workshop will allow some of the leading people in this area to review these various developments, looking towards further progress, for example in extending the known results to higher dimensions and simplifying the lengthy proofs in dimension 2. The workshop is part of the 3-month SCGP program Moduli spaces and singularities in algebraic and Riemannian geometry, and there are many connections between the topics of the workshop and the program.