Kyoto Journal of Mathematics
- Kyoto J. Math.
- Volume 57, Number 2 (2017), 395-434.
On symplectic resolutions of a -dimensional quotient by a group of order
We provide a construction of symplectic resolutions of a -dimensional quotient singularity obtained by an action of a group of order . The existence of such resolutions is known by a result of Bellamy and Schedler. Our explicit construction is obtained via geometric invariant theory (GIT) quotients of the spectrum of a ring graded in the Picard group generated by the divisors associated to the conjugacy classes of symplectic reflections of the group in question. As a result we infer the geometric structure of these resolutions and their flops. Moreover, we represent the group in question as a group of automorphisms of an abelian -fold so that the resulting quotient has singularities with symplectic resolutions. This yields a new Kummer-type symplectic -fold.
Kyoto J. Math., Volume 57, Number 2 (2017), 395-434.
Received: 24 December 2015
Revised: 17 March 2016
Accepted: 23 March 2016
First available in Project Euclid: 9 May 2017
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Primary: 14E15: Global theory and resolution of singularities [See also 14B05, 32S20, 32S45]
Secondary: 14E30: Minimal model program (Mori theory, extremal rays) 14L30: Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17] 14L24: Geometric invariant theory [See also 13A50] 14C20: Divisors, linear systems, invertible sheaves 53C26: Hyper-Kähler and quaternionic Kähler geometry, "special" geometry
Donten-Bury, Maria; Wiśniewski, Jarosław A. On $81$ symplectic resolutions of a $4$ -dimensional quotient by a group of order $32$. Kyoto J. Math. 57 (2017), no. 2, 395--434. doi:10.1215/21562261-3821846. https://projecteuclid.org/euclid.kjm/1494295224