Discrepancies of non-\(\mathbb Q\)-Gorenstein varieties
The author investigates the singularities of non-\(\mathbb Q\)-Gorenstein varieties. The case of \(\mathbb Q\)-Gorenstein singularities has been extensively studied in the context of the minimal model program and is by now well understood. A new approach extending the definitions of canonical singularities, klt singularities and multiplier ideals to the non-\(\mathbb Q\)-Gorenstein setting was introduced in the recent paper [\textit{T. de Fernex} and \textit{C. D. Hacon}, Compos. Math. 145, No. 2, 393--414 (2009; Zbl 1179.14003)]. In the paper under review, the author shows that: 1) the relative canonical divisor of a klt singularity always has rational valuations and gives examples of a non-klt non-\(\mathbb Q\)-Gorenstein singularity whose relative canonical divisor has irrational valuations, 2) there are canonical singularities that are not klt, 3) canonical models have canonical singularities, and 4) if \(X\) has a klt or an isolated singularity then the jumping numbers are discrete.
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