On polarizations of affine Calabi--Yau varieties

Abstract

Let $(X, Ω)$ be an affine Calabi--Yau variety, by which we mean a klt affine variety $X$ with a non-vanishing algebraic volume form $Ω$. In connection with complete Calabi--Yau metrics of Euclidean volume growth on $(X, Ω)$, we study algebraic aspects of polarizations of $X$ given by a negative valuation. We show that such a valuation is finitely generated with a klt degeneration if and only if $X$ admits a dlt log Fano type compactification, generalizing a compactification considered by Tian--Yau. We further give a valuative criterion detecting K-semistability of the degeneration by a negative valuation.