Journal of Algebraic Geometry

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

Author(s): Hans Schoutens
Journal: J. Algebraic Geom. 14 (2005), 357-390.
Posted: December 30, 2004
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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

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

has at most log-terminal singularities, then so does

. 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

and

is an

-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

-module

and every $i\geq 1$ . The final application is Kawamata-Viehweg Vanishing for a connected projective variety

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

is the complexification of a real Lie group acting algebraically on a projective smooth Fano variety

, then for any numerically effective line bundle $\mathcal L$ on any GIT quotient

, each cohomology module $H^i(Y,\mathcal L)$ vanishes for

, 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
Email: hschoutens@citytech.cuny.edu

PII: S 1056-3911(04)00395-9
Received by editor(s): January 28, 2004
Received by editor(s) in revised form: April 21, 2004
Posted: 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.

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