A new linear quotient of C⁴ admitting a symplectic resolution

We show that the quotient C^4/G admits a symplectic resolution for G = (Q_8 x D_8)/(Z/2) < Sp(4,C). Here Q_8 is the quaternionic group of order eight and D_8 is the dihedral group of order eight, and G is the quotient of their direct product which identifies the nontrivial central elements -1 of each. It is equipped with the tensor product of the defining two-dimensional representations of Q_8 and D_8. This group is also naturally a subgroup of the wreath product group of Q_8 by S_2. We compute the singular locus of the family of commutative spherical symplectic reflection algebras deforming C^4/G. We also discuss preliminary investigations on the more general question of classifying linear quotients V / G admitting symplectic resolutions

Poisson traces for symmetric powers of symplectic varieties

We compute the space of Poisson traces on symmetric powers of affine symplectic varieties. In the case of symplectic vector spaces, we also consider the quotient by the diagonal translation action, which includes the quotient singularities C^{2n-2}/S_n associated to the type A Weyl group S_n and its reflection representation C^{n-1}. We also compute the full structure of the natural D-module, previously defined by the authors, whose solution space over algebraic distributions identifies with the space of Poisson traces. As a consequence, we deduce bounds on the numbers of finite-dimensional irreducible representations and prime ideals of quantizations of these varieties. Finally, motivated by these results, we pose conjectures on symplectic resolutions, and give related examples of the natural D-module. In an appendix, the second author computes the Poisson traces and associated D-module for the quotients C^{2n}/D_n associated to type D Weyl groups. In a second appendix, the same author provides a direct proof of one of the main theorems.Comment: 28 page

Symplectic resolutions for multiplicative quiver varieties and character varieties for punctured surfaces

We study the algebraic symplectic geometry of multiplicative quiver varieties, which are moduli spaces of representations of certain quiver algebras, introduced by Crawley-Boevey and Shaw, called multiplicative preprojective algebras. They are multiplicative analogues of Nakajima quiver varieties. They include character varieties of (open) Riemann surfaces fixing conjugacy class closures of the monodromies around punctures, when the quiver is "crab-shaped". We prove that, under suitable hypotheses on the dimension vector of the representations, or the conjugacy classes of monodromies in the character variety case, the normalisations of such moduli spaces are symplectic singularities and that the existence of a symplectic resolution depends on a combinatorial condition on the quiver and the dimension vector. These results are analogous to those obtained by Bellamy and the first author in the ordinary quiver variety case, and for character varieties of closed Riemann surfaces. At the end of the paper, we outline some conjectural generalisations to moduli spaces of objects in 2-Calabi--Yau categories

The Hochschild cohomology ring of a global quotient orbifold

We study the cup product on the Hochschild cohomology of the stack quotient [X/G] of a smooth quasi-projective variety X by a finite group G. More specifically, we construct a G-equivariant sheaf of graded algebras on X whose G-invariant global sections recover the associated graded algebra of the Hochschild cohomology of [X/G], under a natural filtration. This sheaf is an algebra over the polyvector fields T^{poly}_X on X, and is generated as a T^{poly}_X-algebra by the sum of the determinants det(N_{X^g}) of the normal bundles of the fixed loci in X. We employ our understanding of Hochschild cohomology to conclude that the analog of Kontsevich's formality theorem, for the cup product, does not hold for Deligne--Mumford stacks in general. We discuss relationships with orbifold cohomology, extending Ruan's cohomological conjectures. This employs a trivialization of the determinants in the case of a symplectic group action on a symplectic variety X, which requires (for the cup product) a nontrivial normalization missing in previous literature

On symplectic resolutions and factoriality of Hamiltonian reductions

Recently, Herbig–Schwarz–Seaton have shown that 3-large representations of a reductive group G give rise to a large class of symplectic singularities via Hamiltonian reduction. We show that these singularities are always terminal. We show that they are Q -factorial if and only if G has finite abelianization. When G is connected and semi-simple, we show they are actually locally factorial. As a consequence, the symplectic singularities do not admit symplectic resolutions when G is semi-simple. We end with some open questions

The Hochschild cohomology ring of a global quotient orbifold

We study the cup product on the Hochschild cohomology of the stack quotient of a smooth quasi-projective variety X by a finite group G. More specifically, we construct a G-equivariant sheaf of graded algebras on X whose G-invariant global sections recover the associated graded algebra of the Hochschild cohomology of , under a natural filtration. This sheaf is an algebra over the polyvector fields on X, and is generated as a -algebra by the sum of the determinants of the normal bundles of the fixed loci in X. We employ our understanding of Hochschild cohomology to conclude that the analog of Kontsevich's formality theorem, for the cup product, does not hold for Deligne–Mumford stacks in general. We discuss, in the case of a symplectic group action on a symplectic variety X, relationships with orbifold cohomology and Ruan's cohomological conjectures. In describing the Hochschild cohomology in the symplectic situation, we employ compatible trivializations of the determinants , which requires (for the cup product) a nontrivial normalization missing in previous literatur

Liposomal amphotericin B twice weekly as antifungal prophylaxis in paediatric haematological malignancy patients

AbstractData on antifungal prophylaxis in paediatric cancer patients at high risk for invasive fungal disease (IFD) are scant. Intermittent administration of liposomal amphotericin B (LAMB) has been shown to be safe and effective in adult patients with haematological malignancies. We prospectively evaluated the safety and efficacy of prophylactic LAMB at a dosage of 2.5 mg/kg twice weekly in children at high risk for IFD. Efficacy was compared with that in a historical control group of patients with similar demographic characteristics not receiving LAMB prophylaxis. A total of 46 high-risk patients (24 boys; mean age, 7.7 years) with 187 episodes of antifungal prophylaxis were analysed. The median duration of neutropenia (<500/µL) was 10 days. LAMB was discontinued in four patients because of acute allergic reactions. Median values for creatinine and liver enzymes at end of treatment did not differ significantly from those at baseline. Hypokalaemia (<3.0 mmol/L) occurred with 13.5% of the prophylactic episodes, but was usually mild and always reversible. No proven/probable IFD occurred in patients receiving LAMB prophylaxis. In comparison, five proven and two probable IFDs were observed in 45 historical controls not receiving LAMB prophylaxis (p 0.01). LAMB prophylaxis had no impact on the use of empirical antifungal therapy. Systemic antifungal prophylaxis with LAMB 2.5 mg/kg twice weekly is feasible and safe, and seems to be an effective approach for antifungal prophylaxis in high-risk paediatric cancer patients