Jesús Martinez Garcia (Essex) 21 May 2020 "The moduli continuity method for log Fano pairs" The moduli continuity method, pioneered by Odaka, Spotti and Sun, allows us to explicitly provide algebraic charts of the Gromov-Hausdorff compactification of (possibly singular) Kähler-Einstein metrics. Assuming we can provide a homeomorphism to some 'known' algebraic compactification (customarily, a GIT one) the method allows us to determine which Fano varieties (or more generally log Fano pairs) are K-polystable in a given deformation family. In this talk we provide the first examples of compactification of the moduli of log Fano pairs for the simplest deformation family: that of projective space and a hypersurface, and mention related results for cubic surfaces. This is joint work with Patricio Gallardo and Cristiano Spotti.
