Log-terminal singularities and vanishing theorems via non-standard tight closure

• Published in: Journal of algebraic geometry 2005-04, Vol.14 (2), p.357-390
• Main Author: Schoutens, Hans
• Format: Article
• Language: English
• ISSN: 1056-3911; 1534-7486
• DOI: 10.1090/S1056-3911-04-00395-9

Generalizing work of Smith and Hara, we give a new characterization of log-terminal singularities for finitely generated algebras over C \mathbb C , in terms of purity properties of ultraproducts of characteristic p p Frobenii. As a first application we obtain a Boutot-type theorem for log-terminal singularities: given a pure morphism Y → X Y\to X between affine Q \mathbb Q -Gorenstein varieties of finite type over C \mathbb C , if Y Y has at most log-terminal singularities, then so does X X . The second application is the Vanishing for Maps of Tor for log-terminal singularities: if A ⊆ R A\subseteq R is a Noether Normalization of a finitely generated C \mathbb C -algebra R R and S S is an R R -algebra of finite type with log-terminal singularities, then the natural morphism Tor i A ⁡ ( M , R ) → Tor i A ⁡ ( M , S ) \operatorname {Tor}^A_i(M,R) \to \operatorname {Tor}^A_i(M,S) is zero, for every A A -module M M and every i ≥ 1 i\geq 1 . The final application is Kawamata-Viehweg Vanishing for a connected projective variety X X of finite type over C \mathbb 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 G is the complexification of a real Lie group acting algebraically on a projective smooth Fano variety X X , then for any numerically effective line bundle L \mathcal L on any GIT quotient Y := X / / G Y:=X/\!/G , each cohomology module H i ( Y , L ) H^i(Y,\mathcal L) vanishes for i > 0 i>0 , and, if L \mathcal L is moreover big, then H i ( Y , L − 1 ) H^i(Y,\mathcal L^{-1}) vanishes for i > dim ⁡ Y i>\operatorname {dim}Y .