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

Author:
Hans Schoutens

Journal:
J. Algebraic Geom. **14** (2005), 357-390

DOI:
https://doi.org/10.1090/S1056-3911-04-00395-9

Published electronically:
December 30, 2004

MathSciNet review:
2123234

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Abstract |
References |
Additional Information

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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Additional Information

**Hans Schoutens**

Affiliation:
Department of Mathematics, NYC College of Technology, City University of New York, New York, New York 11201

MR Author ID:
249272

Email:
hschoutens@citytech.cuny.edu

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.