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. Given a polarized log Calabi-Yau variety (X,D,L), it was conjectured by Hacking-Keel-Yu that the KSBA moduli space of stable pairs , where C is a divisor in the linear system of L, is a toric variety (up to passing to a finite cover). We prove this conjecture for all log Calabi-Yau surfaces, using tools coming from the minimal model program, log smooth deformation theory and mirror symmetry. This is joint work with Valery Alexeev and Hülya Argüz.