Moduli of products of stable varieties

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 étale; 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.