Moduli of products of stable varieties

TL;DR

This paper investigates the structure of the moduli space of products of stable varieties, establishing that the product map is finite étale and often an isomorphism, thus advancing the understanding of moduli spaces in algebraic geometry.

Contribution

It proves that the product map between moduli spaces of stable varieties is finite étale and often an isomorphism, extending previous work to higher dimensions.

Findings

The product map between moduli spaces is finite étale.

The product map is often an isomorphism.

The results generalize Van Opstall's work to higher dimensions.

Abstract

We study the moduli space of a product of stable varieties over the field of complex numbers, as defined via the minimal model program. Our main results are: (a) taking products gives a well-defined morphism from the product of moduli spaces of stable varieties to the moduli space of a product of stable varieties, (b) this map is always finite \'etale, and (c) this map very often is an isomorphism. Our results generalize and complete the work of Van Opstall in dimension 1. The local results rely on a study of the cotangent complex using some derived algebro-geometric methods, while the global ones use some differential-geometric input.