Log-terminal singularities and vanishing theorems
| dc.creator | Schoutens, Hans | |
| dc.date | 2003-03-15 | |
| dc.date.accessioned | 2026-07-07T04:56:05Z | |
| dc.date.available | 2026-07-07T04:56:05Z | |
| dc.description | 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. The first application is a Boutôt-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 a log-terminal singularities, then so does $X$. The second application is the Vanishing for Maps of Tor for log-terminal singularities: if $A\subset R$ is a Noether Normalization of a finitely generated $\mathbb C$-algebra $R$ and $S$ is a finitely generated $R$-algebra 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 the Kawamata-Viehweg Vanishing Theorem 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 smooth Fano variety has this latter property, we obtain a proof of the following conjecture of Smith for quotients of smooth Fano varieties: 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 $i<\operatorname{dim}Y$. | |
| dc.identifier | https://arxiv.org/abs/math/0303189 | |
| dc.identifier | http://arxiv.org/abs/math/0303189 | |
| dc.identifier | J. Algebraic Geom. 14 (2005), 357-390 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/66798 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Commutative Algebra | |
| dc.subject | Rings and Algebras | |
| dc.subject | 14F17, 14B05, 13H10 | |
| dc.title | Log-terminal singularities and vanishing theorems | |
| dc.type | text |