Log-terminal Singularities and Vanishing Theorems via Non-standard Tight Closure

Generalizing work of Smith and Hara, we give a new characterization of logterminal singularities for finitely generated algebras over C, in terms of purity properties of ultraproducts of characteristic p Frobenii. As a first application we obtain a Boutot-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 an R-algebra of finite type with log-terminal singularities, then the natural morphism Tori (M, R) → Tori (M, S) is zero, for every A-module M and every i ≥ 1. The final application is Kawamata-Viehweg Vanishing 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 corollary, we obtain a proof of the following conjecture of Smith: 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 < dim Y .