Session 29. Quaternion-Kähler manifolds and related structures in Riemannian and algebraic geometry

On \(81\) symplectic resolutions of a \(4\)-dimensional quotient by a group of order \(32\)

Maria Donten-Bury, Uniwersytet Warszawski, Poland
In a joint project with Jarosław Wiśniewski we study the symplectic quotient singularity \(\mathbb{C}^4/G\) where \(G\) is a certain matrix group with 32 elements, generated by Dirac matrices. The existence of a symplectic resolution of this singularity was proved by Bellamy and Schedler in [3] by non-constructive methods based on Poisson deformations. We give a construction of all its symplectic resolutions using the theory of the Cox rings, see [1]. The structure of the Cox ring of a resolution \(X\) of \(\mathbb{C}^4/G\) can be determined without knowing any explicit description of \(X\). Then one may obtain all the resolutions as GIT quotients of the spectrum of the ring \(\mathrm{Cox}(X)\). A motivation for this work is a possibility of using the results in the framework of the generalized Kummer construction, see [2]. This might lead to finding new compact hyperkähler manifolds.
References
  1. I. Arzhantsev, U. Derenthal, J. Hausen, A. Laface, Cox Rings , arXiv:1003.4229 [math.AG] (2010).
  2. M. Andreatta, J. Wiśniewski, On the Kummer construction, Rev. Mat. Complut. 23, 2010, no. 1, 191{215.
  3. G. Bellamy, T. Schedler, A new linear quotient of C4 admitting a symplectic resolution , Math. Z. 273, 2013, no. 3-4, 753-769.
Print version