Algebraic Geometry in String Theory Seminar

Speaker: Pierrick Bousseau, University of Georgia

Title: The KSBA moduli space of log Calabi-Yau surfaces

Abstract: The KSBA moduli space, introduced by Kollár–Shepherd-Barron, and Alexeev, is a natural generalization of “the moduli space of stable curves” to higher dimensions. It parametrizes stable pairs (X,B), where X is a projective algebraic variety satisfying certain conditions and B is a divisor such that K_X+B is ample. This moduli space is described concretely only in a handful of situations: for instance, if X is a toric variety and B=D+\epsilon C, where D is the toric boundary divisor and C is an ample divisor, it is shown by Alexeev that the KSBA moduli space is a toric variety. Generally, for a log Calabi-Yau variety (X,D) consisting of a projective variety X and an anticanonical divisor D, with B=D+\epsilon C where C is an ample divisor, it was conjectured by Hacking–Keel–Yu that the KSBA moduli space is still toric (up to passing to a finite cover). In joint work with Alexeev and Argüz, we prove this conjecture for all log Calabi-Yau surfaces. This uses tools from the minimal model program, log smooth deformation theory and mirror symmetry.