Imperial College London


Faculty of Natural SciencesDepartment of Mathematics

Reader in Pure Mathematics



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Publication Type

17 results found

Bellamy G, Schedler T, 2021, Symplectic resolutions of character varieties, Geometry and Topology, ISSN: 1364-0380

In this article we consider the connected component of the identity ofG-character varieties of compact Riemann surfaces of genusg>0 , for connected complex reductive groups G of type A (e.g., SLn and GLn). We show that these varieties are Q-factorial symplectic singularities and classify which admit symplectic resolutions. The classification reduces to the semi-simple case, where we show that a resolution exists if and only if either g=1 and G is a product of special linear groups of any rank and copies of the group PGL2, or ifg=2 and G=(SL2)m for some m.

Journal article

Bellamy G, Schedler T, 2021, Symplectic resolutions of quiver varieties and character varieties

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 $\theta$-polystable points, generalizing a result of Le Bruyn; we study their \'etale local structure, find their symplectic leaves, and we describe the Namikawa Weyl group. An interesting consequence of our results is that not all symplectic resolutions of quiver varieties appear to come from variation of GIT. We apply this to the $G$-character variety of a compact Riemann surface of genus $g > 0$, when $G$ is $\mathrm{SL}(n,\mathbb{C})$ or $\mathrm{GL}(n,\mathbb{C})$. We show that these varieties are symplectic singularities and classify when they admit symplectic resolutions: they do when $g = 1$ or $(g,n)=(2,2)$ (assuming $n \geq 2$). This is analogous to the case of a quiver with one vertex, $g$ arrows, and dimension vector $(n)$.

Working paper

Bellamy G, Schedler T, 2020, Symplectic resolutions of quiver varieties, Selecta Mathematica, ISSN: 1022-1824

In this article, we consider Nakajima quiver varieties from the point of view of symplec-tic 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 canonicallyθ-polystable points, generalizing a result of Le Bruyn; we study their ́etale 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.

Journal article

Negron C, Schedler T, 2020, The Hochschild cohomology ring of a global quotient orbifold, Advances in Mathematics, Vol: 364, Pages: 1-49, ISSN: 0001-8708

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 literature

Journal article

Bellamy G, Schedler T, 2019, On symplectic resolutions and factoriality of Hamiltonian reductions, Mathematische Annalen, Vol: 375, Pages: 165-176, ISSN: 0025-5831

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.

Journal article

Bitoun T, Schedler TJ, 2018, On D-modules related to the b-function and Hamiltonian flow, Compositio Mathematica, Vol: 154, Pages: 2426-2440, ISSN: 0010-437X

Let f be a quasi-homogeneous polynomial with an isolated singularity in Cn . We compute the length of the D -modules Df𝜆/Df𝜆+1 generated by complex powers of f in terms of the Hodge filtration on the top cohomology of the Milnor fiber. When 𝜆=−1 we obtain one more than the reduced genus of the singularity ( dimHn−2(Z,OZ) for Z the exceptional fiber of a resolution of singularities). We conjecture that this holds without the quasi-homogeneous assumption. We also deduce that the quotient Df𝜆/Df𝜆+1 is nonzero when 𝜆 is a root of the b -function of f (which Saito recently showed fails to hold in the inhomogeneous case). We obtain these results by comparing these D -modules to those defined by Etingof and the second author which represent invariants under Hamiltonian flow.

Journal article

Etingof P, Schedler TJ, 2018, Poisson traces, D-modules, and symplectic resolutions, Letters in Mathematical Physics, Vol: 108, Pages: 633-678, ISSN: 0377-9017

We survey the theory of Poisson traces (or zeroth Poisson homology) developed by the authors in a series of recent papers. The goal is to understand this subtle invariant of (singular) Poisson varieties, conditions for it to be finite-dimensional, its relationship to the geometry and topology of symplectic resolutions, and its applications to quantizations. The main technique is the study of a canonical D-module on the variety. In the case the variety has finitely many symplectic leaves (such as for symplectic singularities and Hamiltonian reductions of symplectic vector spaces by reductive groups), the D-module is holonomic, and hence, the space of Poisson traces is finite-dimensional. As an application, there are finitely many irreducible finite-dimensional representations of every quantization of the variety. Conjecturally, the D-module is the pushforward of the canonical D-module under every symplectic resolution of singularities, which implies that the space of Poisson traces is dual to the top cohomology of the resolution. We explain many examples where the conjecture is proved, such as symmetric powers of du Val singularities and symplectic surfaces and Slodowy slices in the nilpotent cone of a semisimple Lie algebra. We compute the D-module in the case of surfaces with isolated singularities and show it is not always semisimple. We also explain generalizations to arbitrary Lie algebras of vector fields, connections to the Bernstein–Sato polynomial, relations to two-variable special polynomials such as Kostka polynomials and Tutte polynomials, and a conjectural relationship with deformations of symplectic resolutions. In the appendix we give a brief recollection of the theory of D-modules on singular varieties that we require.

Journal article

Bellamy G, Schedler TJ, 2018, Filtrations on Springer fiber cohomology and Kostka polynomials, Letters in Mathematical Physics, Vol: 108, Pages: 679-698, ISSN: 0377-9017

We prove a conjecture which expresses the bigraded Poisson-de Rham homology of the nilpotent cone of a semisimple Lie algebra in terms of the generalized (one-variable) Kostka polynomials, via a formula suggested by Lusztig. This allows us to construct a canonical family of filtrations on the flag variety cohomology, and hence on irreducible representations of the Weyl group, whose Hilbert series are given by the generalized Kostka polynomials. We deduce consequences for the cohomology of all Springer fibers. In particular, this computes the grading on the zeroth Poisson homology of all classical finite W-algebras, as well as the filtration on the zeroth Hochschild homology of all quantum finite W-algebras, and we generalize to all homology degrees. As a consequence, we deduce a conjecture of Proudfoot on symplectic duality, relating in type A the Poisson homology of Slodowy slices to the intersection cohomology of nilpotent orbit closures. In the last section, we give an analogue of our main theorem in the setting of mirabolic D-modules.

Journal article

Pym B, Schedler TJ, 2017, Holonomic Poisson manifolds and deformations of elliptic algebras, Hitchin70, Publisher: Oxford University Press

Conference paper

Schedler T, Bellamy G, 2017, Hyperplane arrangements associated to symplectic quotient singularities, miniPAGES

Conference paper

Schedler T, 2017, Equivariant Slices for Symplectic Cones, International Mathematics Research Notices, Vol: 2017, Pages: 3801-3847, ISSN: 1073-7928

The Darboux–Weinstein decomposition is a central result in the theory of complex Poisson (degenerate symplectic) varieties, which gives a local decomposition at a point as a product of the formal neighborhood of the symplectic leaf through the point and a formal slice. Recently, conical symplectic resolutions, and more generally, Poisson cones, have been very actively studied in representation theory and algebraic geometry. This motivates asking for a C×C×-equivariant version of the Darboux–Weinstein decomposition. In this paper, we develop such a theory, prove basic results on their existence and uniqueness, and study examples (quotient singularities and hypertoric varieties) and applications to noncommutative algebra (their quantization). We also pose some natural questions on existence and quantization of C×C×-actions on slices to conical symplectic leaves.

Journal article

Schedler TJ, Etingof PI, 2017, Coinvariants of Lie algebras of vector fields on algebraic varieties, Asian Journal of Mathematics, Vol: 20, Pages: 795-868, ISSN: 1093-6106

We prove that the space of coinvariants of functions on an affine variety by a Lie algebraof vector fields whose flow generates finitely many leaves is finite-dimensional. Cases of the theoreminclude Poisson (or more generally Jacobi) varieties with finitely many symplectic leaves underHamiltonian flow, complete intersections in Calabi-Yau varieties with isolated singularities underthe flow of incompressible vector fields, quotients of Calabi-Yau varieties by finite volume-preservinggroups under the incompressible vector fields, and arbitrary varieties with isolated singularitiesunder the flow of all vector fields. We compute this quotient explicitly in many of these cases. Theproofs involve constructing a natural D-module representing the invariants under the flow of thevector fields, which we prove is holonomic if it has finitely many leaves (and whose holonomicitywe study in more detail). We give many counterexamples to naive generalizations of our results.These examples have been a source of motivation for us.

Journal article

Schedler TJ, Bellamy G, 2016, On the (non)existence of symplectic resolutions of linear quotients, Mathematical Research Letters, Vol: 23, Pages: 1537-1564, ISSN: 1073-2780

We study the existence of symplectic resolutions of quotient singularitiesV /G, where V is a symplectic vector space and G acts symplectically.Namely, we classify the symplectically irreducible andimprimitive groups, excluding those of the form K S2 where K <SL2(C), for which the corresponding quotient singularity admits aprojective symplectic resolution. As a consequence, for dim V = 4,we classify all symplectically irreducible quotient singularities V /Gadmitting a projective symplectic resolution, except for at mostfour explicit singularities, that occur in dimensions at most 10, forwhich the question of existence remains open.

Journal article

Schedler T, 2016, Zeroth Hochschild homology of preprojective algebras over the integers, Advances in Mathematics, Vol: 299, Pages: 451-542, ISSN: 1090-2082

We determine the Z-module structure of the preprojective algebra and its zeroth Hochschild homology, for any non-Dynkin quiver (and hence the structure working over any base commutative ring, of any characteristic). This answers (and generalizes) a conjecture of Hesselholt and Rains, producing new p -torsion classes in degrees 2pℓ, ℓ≥1. We relate these classes by p-th power maps and interpret them in terms of the kernel of Verschiebung maps from noncommutative Witt theory. An important tool is a generalization of the Diamond Lemma to modules over commutative rings, which we give in the appendix.

Journal article

Schedler TJ, Proudfoot NJ, 2016, Poisson–de Rham homology of hypertoric varieties and nilpotent cones, Selecta Mathematica, Vol: 23, Pages: 179-202, ISSN: 1022-1824

We prove a conjecture of Etingof and the second author for hypertoric varieties that the Poisson–de Rham homology of a unimodular hypertoric cone is isomorphic to the de Rham cohomology of its hypertoric resolution. More generally, we prove that this conjecture holds for an arbitrary conical variety admitting a symplectic resolution if and only if it holds in degree zero for all normal slices to symplectic leaves. The Poisson–de Rham homology of a Poisson cone inherits a second grading. In the hypertoric case, we compute the resulting 2-variable Poisson–de Rham–Poincaré polynomial and prove that it is equal to a specialization of an enrichment of the Tutte polynomial of a matroid that was introduced by Denham (J Algebra 242(1):160–175, 2001). We also compute this polynomial for S3-varieties of type A in terms of Kostka polynomials, modulo a previous conjecture of the first author, and we give a conjectural answer for nilpotent cones in arbitrary type, which we prove in rank less than or equal to 2.

Journal article

Ginzburg V, Schedler TJ, 2016, A new construction of cyclic homology, Proceedings of the London Mathematical Society, Vol: 112, Pages: 549-587, ISSN: 0024-6115

Based on the ideas of Cuntz and Quillen, we give a simple construction of cyclic homology of unital algebras in terms of the noncommutative de Rham complex and a certain differential similar to the equivariant de Rham differential. We describe the Connes exact sequence in this setting. We define equivariant Deligne cohomology and construct, for each 𝑛⩾1 , a natural map from cyclic homology of an algebra to the GL𝑛 ‐equivariant Deligne cohomology of the variety of 𝑛 ‐dimensional representations of that algebra. The bridge between cyclic homology and equivariant Deligne cohomology is provided by extended cyclic homology, which we define and compute here, based on the extended noncommutative de Rham complex introduced previously by the authors.

Journal article

Etingof P, Schedler T, 2014, Invariants of Hamiltonian flow on locally complete intersections, Geometric and Functional Analysis, Vol: 24, Pages: 1885-1912, ISSN: 1016-443X

We consider the Hamiltonian flow on complex complete intersection surfaces with isolated singularities, equipped with the Jacobian Poisson structure. More generally we consider complete intersections of arbitrary dimension equipped with Hamiltonian flow with respect to the natural top polyvector field, which one should view as a degenerate Calabi–Yau structure. Our main result computes the coinvariants of functions under the Hamiltonian flow. In the surface case this is the zeroth Poisson homology, and our result generalizes those of Greuel, Alev and Lambre, and the authors in the quasihomogeneous and formal cases. Its dimension is the sum of the dimension of the top cohomology and the sum of the Milnor numbers of the singularities. In other words, this equals the dimension of the top cohomology of a smoothing of the variety. More generally, we compute the derived coinvariants, which replaces the top cohomology by all of the cohomology. Still more generally we compute the D-module which represents all invariants under Hamiltonian flow, which is a nontrivial extension (on both sides) of the intersection cohomology D-module, which is maximal on the bottom but not on the top. For cones over smooth curves of genus g, the extension on the top is the holomorphic half of the maximal extension.

Journal article

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