Limits of log canonical thresholds. (English) Zbl 1186.14007

de Fernex, Tommaso; Mustaţă, Mircea

Ann. Sci. Éc. Norm. Supér. (4) 42, No. 3, 491-515 (2009).

This article is about a basic analytic invariant of singularities, the log canonical threshold. Let \(\mathcal{T}_n\) denote the set of all log canonical thresholds of pairs \((X,Y)\) consisting of a smooth \(n\)-dimensional variety \(X\) defined over an algebraically closed field of characteristic zero, and a closed subscheme \(Y\) of \(X\). In this article it is proved that: (1) \(\mathcal{T}_n\) is closed in \(\mathbb R\); (2) every limit of a sequence in \(\cup_{n\leq m}\mathcal{T}_n\) is rational number; (3) the set of accumulation points from above of \(\mathcal{T}_n\) is exactly \(\mathcal{T}_{n-1}\). The last result has been conjectured by J. Kollár. The ACC Conjecture, due to V. Shokurov, predicts that \(\mathcal{T}_n\) has no accumulation points from below. In this article, this statement is reduced to the task of showing that 1 is not an accumulation point from below of any \(\mathcal{T}_n\), and reinterpreted as a semicontinuity property of log canonical thresholds of power series.
The proofs rely on non-standard analysis: to a sequence of ideals \({\mathbf a}_m\subset k[x_1,\dots ,x_n]\) with \(\text{lct} ({\mathbf a}_m)\) converging to \(c\), the authors attach an ideal \(\mathbf a\) in \(K[x_1,\dots ,x_n]\) with \(\text{lct}({\mathbf a})=c\), where \(K\) is a much larger field. This, together with the fact that \(\mathcal{T}_n\) is invariant under the change of the field of definition, are crucial points of the proof.
This article brought fresh ideas leading to the recent proof of the ACC Conjecture for \({\mathcal T}_n\) by the authors together with L. Ein [Shokurov’s ACC conjecture for log canonical thresholds on smooth varieties, arXiv:0905.3775], based on the following improvement of (3) of J. Kollár [Which powers of holomorphic functions are integrable? arXiv:0805.0756]: all accumulation points of \(\mathcal{T}_n\) form \(\mathcal{T}_{n-1}\). A proof of the general ACC Conjecture, where the ambient spaces \(X\) are allowed to have some mild singularities, is known to have implications to the Minimal Model Program.

Reviewer: Nero Budur (Notre Dame)

MSC:

14B05	Singularities in algebraic geometry
14E30	Minimal model program (Mori theory, extremal rays)

Keywords:

log canonical threshold; multiplier ideals; ultrafilter; resolution of singularities; ACC Conjecture