Relative vanishing theorems for $\mathbf{Q}$-schemes

Abstract

We prove the relative Grauert-Riemenschneider vanishing, Kawamata-Viehweg vanishing, and Kollár injectivity theorems for proper morphisms of schemes of equal characteristic zero, solving conjectures of Boutot and Kawakita. Our proof uses the Grothendieck limit theorem for sheaf cohomology and Zariski-Riemann spaces. We also show these vanishing and injectivity theorems hold for locally Moishezon (resp. projective) morphisms of quasi-excellent algebraic spaces admitting dualizing complexes and semianalytic germs of complex analytic spaces (resp. quasi-excellent formal schemes admitting dualizing complexes, rigid analytic spaces, Berkovich spaces, and adic spaces locally of weakly finite type over a field), all in equal characteristic zero. We give many applications of our vanishing results. For example, we extend Boutot's theorem to all Noetherian $\mathbf{Q}$-algebras by showing that if $R \to R'$ is a cyclically pure map of $\mathbf{Q}$-algebras and $R'$ is pseudo-rational, then $R$ is pseudo-rational. This solves a conjecture of Boutot and affirmatively answers a question of Schoutens. The proof of this Boutot-type result uses a new characterization of pseudo-rationality and rational singularities using Zariski-Riemann spaces. This characterization is also used in the proofs of our vanishing and injectivity theorems and is of independent interest.