Relative vanishing theorems for 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 that these vanishing and injectivity theorems hold for locally Moishezon (respectively, projective) morphisms of quasi-excellent algebraic spaces and semianalytic germs of complex-analytic spaces (respectively, quasi-excellent formal schemes and non-Archimedean analytic spaces), all in equal characteristic zero. We give many applications of our vanishing results. For example, we extend Boutot’s theorem to all Noetherian Q-algebras by showing that pseudo-rationality descends under pure maps of Q-algebras. This solves a conjecture of Boutot and answers a question of Schoutens. The proofs of this Boutot-type result and of our vanishing and injectivity theorems all use a new characterization of rational singularities using Zariski–Riemann spaces.