Existence of crepant resolutions. (English) Zbl 1388.14051

Oguiso, Keiji (ed.) et al., Higher dimensional algebraic geometry. In honour of Professor Yujiro Kawamata’s sixtieth birthday. Proceedings of the conference, Tokyo, Japan, January 7–11, 2013. Tokyo: Mathematical Society of Japan (MSJ) (ISBN 978-4-86497-046-4/hbk). Advanced Studies in Pure Mathematics 74, 185-202 (2017).
Summary: Let \(G\) be a finite subgroup of SL\((n,\mathbb{C})\), then the quotient \(\mathbb{C}^n/G\) has a Gorenstein canonical singularity. If \(n=2\) or \(3\), it is known that there exist crepant resolutions of the quotient singularity. In higher dimension, there are many results which assume existence of crepant resolutions. However, few examples of crepant resolutions are known.
In this paper, we will show several trials to obtain crepant resolutions and give a conjecture on existence of crepant resolutions.
For the entire collection see [Zbl 1388.14012].


14E16 McKay correspondence
14C05 Parametrization (Chow and Hilbert schemes)
14L30 Group actions on varieties or schemes (quotients)
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies