Abstract

Ascher, DeVleming and Liu constructed a theory of variations of K-moduli of log Fano pairs, in which the coefficients of the divisors are allowed to change, introducing birational transformations on the K-moduli. The most natural example is K-moduli of smoothable log del Pezzo pairs formed by a del Pezzo surface and an anti-canonical divisor, a natural generalisation of the first description of K-moduli for del Pezzo surfaces given by Odaka-Spotti-Sun. Our case also implies analytic questions previously considered by Szekelyhidi on the existence of Kahler-Einstein metrics with conical singularities along a divisor on del Pezzo surfaces. For degrees 2, 3 and 4 we establish an isomorphism between the K-moduli spaces and variation of Geometric Invariant Theory compactifications. For degrees 2-9, we describe the wall-chamber structure of the K-moduli of these problems, including all K-polystable replacements. This is joint work with Theodoros Papazachariou and Junyan Zhao.