Log canonical singularities and complete moduli of stable pairs

TL;DR

This paper constructs a complete moduli space for stable log pairs assuming the log Minimal Model Conjecture, extending the concept of moduli spaces for curves to higher dimensions, and proves semi log canonical singularities for certain stable pairs.

Contribution

It provides a new construction of a complete moduli space of stable log pairs in arbitrary dimensions, generalizing the moduli space of pointed stable curves, under the assumption of the log Minimal Model Conjecture.

Findings

Construction of a complete moduli space of stable log pairs.

Proof that stable quasiabelian pairs have semi log canonical singularities.

Extension of moduli space concepts from curves to higher-dimensional pairs.

Abstract

1) Assuming log Minimal Model Conjecture, we give a construction of a complete moduli space of stable log pairs of arbitrary dimension generalizing directly the space M_{g,n} of pointed stable curves. Each stable pair has semi log canonical singularities. 2) We prove that a stable quasiabelian pair, defined by author and I.Nakamura as the limit of abelian varieties with theta divisors, has semi log canonical singularities.