Two Calabi-Yau theorems for degenerations of compact K\"ahler manifolds

Abstract

We discuss two closely related Calabi-Yau theorems for degenerations of compact K\"ahler manifolds. The first is a Calabi-Yau theorem for big test configurations, that generalizes a result in [WN24]. It follows from recent joint work with Mesquita-Piccione [MW25], but is here given a more direct proof. The second result is a Calabi-Yau theorem for a wider class of degenerations, formulated in the language of non-Archimedean K\"ahler geometry. It was first proved in the algebraic setting by Boucksom-Jonsson [BJ22], building on earlier work of Boucksom-Favre-Jonsson [BFJ15], while the general K\"ahler case was established in [MW25]. Our main focus here is on the connection between these results and the theory of big cohomology classes and their volumes.