ncts

Let G be a finite group acting on an abelian variety A. We say that G acts freely in codimension k if the fixed locus by every element of G has codimension at least k+1. In particular, if G acts freely in codimension 1 on A, then the quotient A/G is a potentially singular variety with trivial canonical class; the higher the k such that G acts freely in codimension 2, the less singular A/G is. In this talk, I explain how to produce examples and show non-existence results for "particularly nice" desingularizations for such quotients A/G. Here, "particularly nice" ranges from resolutions that still have trivial canonical class (the so-called crepant resolutions) to resolutions that are Calabi-Yau manifolds. This can be thought of as a nice way to produce new Calabi-Yau varieties whose geometry may reflect that of the initial abelian variety A. I will spend some time at first explaining local phenomena when investigating crepant resolutions of a given quotient singularity, recalling notably some early successes of the McKay correspondence due to Y. Ito and M. Reid. I will then describing results of K. Oguiso for the 3-dimensional case, and present some recent results in the 4- and higher dimensional cases.