Log-terminal singularities and vanishing theorems

Original · EN

Generalizing work of Smith and Hara, we give a new characterization of log-terminal singularities for finitely generated algebras over C, in terms of purity properties of ultraproducts of characteristic p Frobenii. The first application is a Boutôt-type theorem for log-terminal singularities: given a pure morphism Y→ X between affine Q-Gorenstein varieties of finite type over C, if Y has at most a log-terminal singularities, then so does X. The second application is the Vanishing for Maps of Tor for log-terminal singularities: if A⊂ R is a Noether Normalization of a finitely generated C-algebra R and S is a finitely generated R-algebra with log-terminal singularities, then the natural morphism Torᵃᵢ(M,R) → Torᵃᵢ(M,S) is zero, for every A-module M and every i≥ 1. The final application is the Kawamata-Viehweg Vanishing Theorem for a connected projective variety X of finite type over C whose affine cone has a log-terminal vertex (for some choice of polarization). As a smooth Fano variety has this latter property, we obtain a proof of the following conjecture of Smith for quotients of smooth Fano varieties: if G is the complexification of a real Lie group acting algebraically on a projective smooth Fano variety X, then for any numerically effective line bundle L on any GIT quotient Y:=X//G, each cohomology module Hⁱ(Y,L) vanishes for i>0, and, if L is moreover big, then Hⁱ(Y,L⁻¹) vanishes for i<dimY.