Log-terminal singularities and vanishing theorems

Abstract

Generalizing work of Smith and Hara, we give a new characterization of log-terminal singularities for finitely generated algebras over $\mathbb C$, 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 $Y\to X$ between affine $\mathbb Q$-Gorenstein varieties of finite type over $\mathbb C$, 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 $A\subset R$ is a Noether Normalization of a finitely generated $\mathbb C$-algebra $R$ and $S$ is a finitely generated $R$-algebra with log-terminal singularities, then the natural morphism $\operatorname{Tor}^A_i(M,R) \to \operatorname{Tor}^A_i(M,S)$ is zero, for every $A$-module $M$ and every $i\geq 1$. The final application is the Kawamata-Viehweg Vanishing Theorem for a connected projective variety $X$ of finite type over $\mathbb C$ 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 $\mathcal L$ on any GIT quotient $Y:=X//G$, each cohomology module $H^i(Y,\mathcal L)$ vanishes for $i>0$, and, if $\mathcal L$ is moreover big, then $H^i(Y,\mathcal L^{-1})$ vanishes for $i<\operatorname{dim}Y$.