We know from the Young Person's Guide of Miles Reid that every quotient singularity is log-terminal in 2 dimensional case. Is there a similar result holds in dimensions $\geq$ 3. If so, what is the exact reference for this?
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$\begingroup$ As pointed by Karl: In fact we are looking at quotients by finte cyclic groups. $\endgroup$– user17385Commented Aug 25, 2011 at 19:31
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$\begingroup$ By a theorem of Viehweg, finite quotient singularities on a variety over an algebraically closed field are rational. Is is true that rational singularities are log-terminal? $\endgroup$– Ariyan JavanpeykarCommented Aug 25, 2011 at 22:43
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1$\begingroup$ Rational singularities are not generally log terminal. Log terminal singularities are rational. Perhaps I should point out that there are a number of generalizations to that result you mention, including what's sometimes called Boutot's theorem (one form of which is: a summand of a ring with rational singularities has rational singularities, see further generalizations also by Sandor Kovacs). $\endgroup$– Karl SchwedeCommented Aug 26, 2011 at 3:30
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I guess it depends on what groups you are quotienting out by (finite groups, reductive groups?), anyway, for example see
- The cone of curves on algebraic varieties by Y. Kawamata, Prop. 1.7 and the following discussion in the finite group case.
For some generalities to non-finite groups, see
- Pure subrings of regular rings are pseudo-rational by Hans Schoutens, Theorem B
You can find some discussion I'm sure in other standard sources such as Flips and Abundance for Algebraic Threefolds by Koll\'ar et. al. Also see Kollar-Mori.
answered Aug 25, 2011 at 18:34
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$\begingroup$ In fact we are looking at quotients by finte cyclic groups. $\endgroup$ Commented Aug 25, 2011 at 20:53