Symplectic resolutions of quiver varieties

Abstract

In this article, we consider Nakajima quiver varieties from the point of view of symplectic algebraic geometry. We prove that they are all symplectic singularities in the sense of Beauville and completely classify which admit symplectic resolutions. Moreover we show that the smooth locus coincides with the locus of canonically $$\theta $$ θ -polystable points, generalizing a result of Le Bruyn; we study their étale local structure and find their symplectic leaves. An interesting consequence of our results is that not all symplectic resolutions of quiver varieties appear to come from variation of GIT.

BPS Lie algebras and the less perverse filtration on the preprojective CoHA

“…for |a| and |b| the Z-degrees of a and b respectively. This explains the introduction of the signs in (3). Via a similar argument to the previous subsection, we may show that the Borel-Moore homology of Higgs sst r,0 (C) is pure, so that the only terms that contribute on the left hand side of (2) have n = i.…”

“…Moreover, there is an isomorphism of algebras L ≤0 HA S ΠQ ∼ = U(g S ΠQ ) where g S ΠQ is isomorphic to the BPS Lie algebra [14] determined by a quiver Q, potential W and Serre subcategory S of the category of C Q-modules defined in §1. 3.…”

“…See[3] for a comprehensive treatment of when we may expect to find such a resolution 4. As a consequence of this difference, there is value in considering them both simultaneously; see §6.1.1 for example.…”

Quot Schemes for Kleinian Orbifolds

“…It remains to establish the properties of π I . Adapting the proof from [4, Lemma 2.4] shows that π I is a projective morphism, while [4,Theorem 1.15] gives that M θ (1, v) is non-singular as θ is generic. Our explicit description of π I shows that it factors via the morphism…”

Quiver gauge theories and symplectic singularities

“…This implies that M κ C (G, N) has symplectic singularities by [Bea00, Remark 1.2], and is terminal[Nam, Cor. 1].In either case, sinceπ κ : M κ C (G, N) → M C (G, N) is a proper birational Poisson map, we conclude that M C (G, N) has symplectic singularities by[BS, Lemma 6.12].…”