Crepant Property of Fujiki-Oka Resolutions for Gorenstein Abelian Quotient Singularities

Abstract

We show a sufficient condition for Fujiki-Oka resolutions of Gorenstein abelian quotient singularities to be crepant in all dimensions by using Ashikaga's continuous fractions. Moreover, we prove that all three dimensional Gorenstein abelian quotient singularities possess a crepant resolution as a corollary. This alternative proof of existence is really reasonable comparing with the results ever known.

Hilbert desingularizations for three dimensional canonical cyclic quotient singularities

“…We proved that all Fujiki-Oka resolutions are crepant for any three dimensional semiisolated Gorenstein quotient singularity in [20].…”

“…Naturally, the definition of this resolution can be extended to the cyclic quotient singularities. Let G ⊂ GL(n, C) be a cyclic subgroup and H be a component of a decomposition by cyclic subgroups of G. If the singularity C n /H is semi-isolated, then we have the Fujiki-Oka resolution ( Y H , FO 1 ) and the toric partial resolution (Y G , φ) satisfying the following diagram: We proved the existence of a crepant iterated Fujiki-Oka resolution for any three dimensional Gorenstein abelian quotient singularity [20]. For the case of (iii) and (iv), we have convenient subgroup to find Hilbert iterated Fujiki-Oka resolutions.…”

“…It is known that there exists an iterated Fujiki-Oka resolution for Gorenstein abelian quotient singularities in all dimensions by Lemma 4.2 in [20]. Naturally, the definition of this resolution can be extended to the cyclic quotient singularities.…”

Numerical and kodaira dimensions of cotangent bundles

“…One can get information on this decomposition from the values of the dimensions of H 0,p (A/G) = (H 0,p (A)) G and χ(O A/G ). The local existence of crepant resolutions is proved for all Sl(3) quotients when n = 3 (see the references in [42]).…”