Department Colloquium

Title

Moduli and stability of varieties

Abstract

The classification of algebraic varieties naturally leads to moduli questions: how do varieties of a given type vary in families, and how can they be compactified? A basic organizing principle is the positivity of the canonical bundle, which leads to the three broad classes of varieties of general type, Fano varieties, and Calabi–Yau varieties. In dimension one, this picture is already familiar from the theory of moduli of curves.

In this talk, we will discuss the higher-dimensional story. For varieties of general type, we will review the KSBA theory and its use of stable varieties to construct compact moduli spaces. For Fano varieties, we will discuss K-stability and the resulting theory of K-moduli. We will then turn to Calabi–Yau varieties, where the moduli problem is closely related to Hodge theory and mirror symmetry, and discuss recent progress toward compactifications, as well as some open problems and future directions.