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Singularities of pairs via jet schemes. (English) Zbl 0998.14009

From the paper: For an arbitrary scheme \(W\), the \(m\)-th jet scheme \(W_m\) parametrizes morphisms \(\text{Spec} \mathbb{C} [t]/(t^{m+1})\to W\). Main result:
If \(X\) is a smooth variety, \(Y\subset X\) a closed subscheme, and \(q>0\) a rational number, then:
(1) The pair \((X,q\cdot Y)\) is log canonical if and only if \(\dim Y_m\leq (m+1) (\dim X-q)\), for all \(m\).
(2) The pair \((X,q\cdot Y)\) is Kawamata log terminal if and only if \(\dim Y_m< (m+1)(\dim X-q)\), for all \(m\).
The main technique we use in the proof of this result is motivic integration, a technique due to Kontsevich, Batyrev, and Denef and Loeser. As a consequence of the above result, we obtain a formula for the log canonical threshold:
Corollary. If \(X\) is a smooth variety and \(Y\subset X\) is a closed subscheme, then the log canonical threshold of the pair \((X,Y)\) is given by \(c(X,Y)= \dim X-\sup_{m\geq 0} {\dim Y_m\over m+1}\).
We apply this corollary to give simpler proofs of some results on the log canonical threshold proved by J.-P. Demailly and J. Kollár [Ann. Sci. Éc. Norm. Supér., IV. Sér. 34, No. 4, 525-556 (2001; Zbl 0994.32021)] using analytic techniques.

MSC:

14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14B05 Singularities in algebraic geometry
14E30 Minimal model program (Mori theory, extremal rays)

Citations:

Zbl 0994.32021
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References:

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