Towards the classification of symplectic linear quotient singularities admitting a symplectic resolution

Over the past 2 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 but there is an infinite series of groups in dimension 4—the symplectically primitive but complex imprimitive groups—and 10 exceptional groups up to dimension 10, for which it is still open. In this paper, we treat the remaining infinite series and prove that for all but possibly 39 cases there is no symplectic resolution. We thereby reduce the classification problem to finitely many open cases. We furthermore prove non-existence of a symplectic resolution for one exceptional group, leaving $$39+9=48$$ 39 + 9 = 48 open cases in total. We do not expect any of the remaining cases to admit a symplectic resolution.

The Kodaira Problem for Kähler Spaces with Vanishing First Chern Class

Let X be a normal compact Kähler space with klt singularities and torsion canonical bundle. We show that X admits arbitrarily small deformations that are projective varieties if its locally trivial deformation space is smooth. We then prove that this unobstructedness assumption holds in at least three cases: if X has toroidal singularities, if X has finite quotient singularities and if the cohomology group ${\mathrm {H}^{2} \!\left ( X, {\mathscr {T}_{X}} \right )}$ vanishes.

Nearly Gorenstein cyclic quotient singularities

We investigate the nearly Gorenstein property among d-dimensional cyclic quotient singularities $$\Bbbk \llbracket x_1,\dots ,x_d\rrbracket ^G$$ k 〚 x 1 , ⋯ , x d 〛 G , where $$\Bbbk $$ k is an algebraically closed field and $$G\subseteq {\text {GL}}(d,\Bbbk )$$ G ⊆ GL ( d , k ) is a finite small cyclic group whose order is invertible in $$\Bbbk $$ k . We prove a necessary and sufficient condition to be nearly Gorenstein that also allows us to find several new classes of such rings.

Elliptic classes, McKay correspondence and theta identities

We revisit the construction of elliptic class given by Borisov and Libgober for singular algebraic varieties. Assuming torus action we adjust the theory to the equivariant local situation. We study theta function identities having a geometric origin. In the case of quotient singularities $${\mathbb {C}}^n/G$$ C n / G , where G is a finite group the theta identities arise from McKay correspondence. The symplectic singularities are of special interest. The Du Val surface singularity $$A_n$$ A n leads to a remarkable formula.

Birational geometry of symplectic quotient singularities

For a finite subgroup $$\Gamma \subset \mathrm {SL}(2,\mathbb {C})$$ Γ ⊂ SL ( 2 , C ) and for $$n\ge 1$$ n ≥ 1 , we use variation of GIT quotient for Nakajima quiver varieties to study the birational geometry of the Hilbert scheme of n points on the minimal resolution S of the Kleinian singularity $$\mathbb {C}^2/\Gamma $$ C 2 / Γ . It is well known that $$X:={{\,\mathrm{{\mathrm {Hilb}}}\,}}^{[n]}(S)$$ X : = Hilb [ n ] ( S ) is a projective, crepant resolution of the symplectic singularity $$\mathbb {C}^{2n}/\Gamma _n$$ C 2 n / Γ n , where $$\Gamma _n=\Gamma \wr \mathfrak {S}_n$$ Γ n = Γ ≀ S n is the wreath product. We prove that every projective, crepant resolution of $$\mathbb {C}^{2n}/\Gamma _n$$ C 2 n / Γ n can be realised as the fine moduli space of $$\theta $$ θ -stable $$\Pi $$ Π -modules for a fixed dimension vector, where $$\Pi $$ Π is the framed preprojective algebra of $$\Gamma $$ Γ and $$\theta $$ θ is a choice of generic stability condition. Our approach uses the linearisation map from GIT to relate wall crossing in the space of $$\theta $$ θ -stability conditions to birational transformations of X over $$\mathbb {C}^{2n}/\Gamma _n$$ C 2 n / Γ n . As a corollary, we describe completely the ample and movable cones of X over $$\mathbb {C}^{2n}/\Gamma _n$$ C 2 n / Γ n , and show that the Mori chamber decomposition of the movable cone is determined by an extended Catalan hyperplane arrangement of the ADE root system associated to $$\Gamma $$ Γ by the McKay correspondence. In the appendix, we show that morphisms of quiver varieties induced by variation of GIT quotient are semismall, generalising a result of Nakajima in the case where the quiver variety is smooth.