Related papers: Uniqueness of crepant resolutions and symplectic s…

We give some necessary conditions for the existence of a symplectic resolution for quotient singularities. The McKay correspondence is also worked out for these resolutions.

We prove that any symplectic resolution of the closure of a nilpotent orbit in a semi-simple complex Lie algebra is isomorphic to the collapsing of the cotangent bundle of a projective homogenous variety. Then we give a complete…

We describe an explicit symplectic resolution for the quotient singularity arising from the four-dimensional symplectic represenation of the binary tetrahedral group.

This short note is a supplement to the previous article with the same title. Here we treat a conical symplectic variety obtained as a finite covering of a (not necessarily normal) nilpotent orbit closure of a complex semisimple Lie algebra.

We study the existence of symplectic resolutions of quotient singularities V/G where V is a symplectic vector space and G acts symplectically. Namely, we classify the symplectically irreducible and imprimitive groups, excluding those of the…

Let G be a finite group acting on a symplectic complex vector space V. Assume that the quotient V/G has a holomorphic symplectic resolution. We prove that G is generated by "symplectic reflectionsd"', i.e. symplectomorphisms with fixed…

We introduce the notion of a ``non-commutative crepant'' resolution of a singularity and show that it exists in certain cases. We also give some evidence for an extension of a conjecture by Bondal and Orlov, stating that different crepant…

We prove that a conical symplectic variety with maximal weight 1 is isomorphic to one of the following: (i) an affine space with the standard symplectic form (ii) a nilpotent orbit closure of a complex semisimple Lie algebra with the…

In this paper we generalize standard results about non-commutative resolutions of quotient singularities for finite groups to arbitrary reductive groups. We show in particular that quotient singularities for reductive groups always have…

We give a criterion for the existence of non-commutative crepant resolutions (NCCR's) for certain toric singularities. In particular we recover Broomhead's result that a 3-dimensional toric Gorenstein singularity has a NCCR. Our result also…

Over the past two decades, there has been much progress on the classification of symplectic linear quotient singularities V/G admitting a symplectic (equivalently, crepant) resolution of singularities. The classification is almost complete…

We prove existence of non-commutative crepant resolutions (in the sense of van den Bergh) of quotient singularities by finite and linearly reductive group schemes in positive characteristic. In dimension two, we relate these to resolutions…

We introduce the notion of a conical symplectic variety, and show that any symplectic resolution of such a variety is isomorphic to the Springer resolution of a nilpotent orbit in a semisimple Lie algebra, composed with a linear projection.

We study symplectic deformation types of minimal symplectic fillings of links of quotient surface singularities. In particular, there are only finitely many symplectic deformation types for each quotient surface singularity.

In this paper we shall prove that the singular locus of a symplectic singularity has no codimension 3 irreducible components. As a corollary, a symplectic singularity is terminal if and only if its singular locus has codimension $\geq 4$.…

Some Poisson structures do admit resolutions by symplectic manifolds of the same dimension. We give examples and simple conditions under which such resolutions can not exist.

In this paper, we show a condition for two-parameter Gorenstein cyclic quotient singularities to have a crepant resolution by using the remainder polynomial in any dimension.

We introduce special classes of non-commutative crepant resolutions (= NCCR) which we call steady and splitting. We show that a singularity has a steady splitting NCCR if and only if it is a quotient singularity by a finite abelian group.…

We provide an infinite family of diffeomorphic symplectic forms on ruled surfaces, which are pairwise non-isotopic. This answers a uniqueness question regarding symplectic structures up to isotopy on closed symplectic four-manifolds.

In this paper we present a method to obtain resolutions of symplectic orbifolds arising from symplectic reduction of a Hamiltonian S^1-manifold at a regular value. As an application, we show that all isolated cyclic singularities of a…