Good moduli spaces for boundary polarized Calabi-Yau surface pairs

TL;DR

This paper constructs a new projective moduli space for boundary polarized Calabi-Yau surface pairs, linking existing moduli theories and providing geometric interpretations for special cases like K3 surfaces with automorphisms.

Contribution

It introduces a novel moduli space for boundary polarized CY surface pairs, bridging KSBA and K-moduli spaces, and offers a modular interpretation for certain K3 surface compactifications.

Findings

Constructed a projective moduli space for boundary polarized CY pairs.

Established a wall crossing between KSBA and K-moduli spaces.

Provided a modular interpretation for K3 surfaces with automorphisms.

Abstract

We construct projective asymptotically good moduli spaces parametrizing boundary polarized CY surface pairs, which are projective slc Calabi-Yau pairs $(X,D)$ such that $D$ is ample and $X$ has dimension two. The moduli space provides a wall crossing between certain KSBA and K-moduli spaces and is the ample model of the Hodge line bundle. In the case of K3 surfaces with a non-symplectic automorphism, the moduli space gives a modular interpretation for the Baily--Borel compactification.