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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



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

Author: Hans Schoutens
Journal: J. Algebraic Geom. 14 (2005), 357-390
Published electronically: December 30, 2004
MathSciNet review: 2123234
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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. As a first application we obtain a Boutot-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 log-terminal singularities, then so does $X$. The second application is the Vanishing for Maps of Tor for log-terminal singularities: if $A\subseteq R$ is a Noether Normalization of a finitely generated $\mathbb C$-algebra $R$ and $S$ is an $R$-algebra of finite type 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 Kawamata-Viehweg Vanishing 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 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 $\mathcal L$ 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 $i<\operatorname {dim}Y$.

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Hans Schoutens
Affiliation: Department of Mathematics, NYC College of Technology, City University of New York, New York, New York 11201
MR Author ID: 249272

Received by editor(s): January 28, 2004
Received by editor(s) in revised form: April 21, 2004
Published electronically: December 30, 2004
Additional Notes: Partially supported by a grant from the National Science Foundation and by visiting positions at Paris VII and at the Ecole Normale Supérieure.