In the context of the Minimal Model Program, which strives to classify algebraic varieties, singularities are a natural concern. By the work of, among others, Caucher Birkar, the Minimal Model Program has seen much development over the complex numbers. In this talk, we investigate the singularities that appear in this context beyond the realm of the complex numbers, extending our focus to perfect fields of positive characteristic and excellent rings with perfect residue fields. We uncover the unique behavior of these singularities in characteristic p, closely linked to the breakdown of Kodaira-type vanishing theorems. Additionally, we explore how these questions are related to the moduli theory of varieties of general type.

This is based on joint work with F. Bernasconi and Zs. Patakfalvi, as well as joint work with Q. Posva.