arXiv subjects

Travis Schedler

Publications and source records attributed to Travis Schedler.

At least 19 recordsLinked to original sources

arXiv2026

New symplectic singularities from $SU(2)$ gauge theories

Families of $\mathrm{Sp}(1)\simeq\mathrm{SU}(2)$ gauge theories with eight supercharges are found to have a Higgs branch which is an isolated symplectic singularity. These are, in some sense, the most ``minimal'' gauge theories as they only Higgs to a trivial theory. The matter content is $N$ fundamental half-hypermultiplets and one half-hypermultiplet in the $\mathrm{Sym}^k$ representation where $k=1,3,5,7$. The cases of $k=1,3$ reproduce the known Kraft--Procesi construction for minimal nilpotent orbit closures of $\mathrm{SO}(N+1)$ and the $g\mathrm{SO}(N)$ singularities of \cite{Bourget:2025wsp}, respectively. The cases $k=5,7$ are new isolated symplectic singularities which are termed $h\mathrm{SO}(N)$ and $i\mathrm{SO}(N)$, respectively. The classification of these isolated symplectic singularities is argued for through the Higgs mechanism, with Hilbert series and highest weight generating (HWG) functions computed for some cases. Each of the $g\mathrm{SO}(N)$, $h\mathrm{SO}(N)$, and $i\mathrm{SO}(N)$ families has a (quaternionic) one-dimensional member; these are the Klein $A_3$, $E_6$, and $E_8$ singularities, respectively. Our construction hence provides realisations of these Klein singularities as Higgs branches (hyper-K\"ahler quotients) of $\mathrm{Sp}(1)$ gauge theories, complementary to Kronheimer's construction \cite{Kronheimer:1989zs} using the $\widehat A_3$, $\widehat E_6$, and $\widehat E_8$ affine quivers. The Klein $E_7$ singularity is also realised as a Higgs branch (hyper-K\"ahler quotient) of an $\mathrm{Sp}(1)\times\mathrm{O}(1)$ gauge theory.

hep-th↗

arXiv2025

Singularities in Calogero--Moser Varieties

In this article we describe completely the singularities appearing in Calogero--Moser varieties associated (at any parameter) to the wreath product symplectic reflection groups. We do so by parameterizing the symplectic leaves in the variety, describing combinatorially the resulting closure relation and computing a transverse slice to each leaf. We also show that the normalization of the closure of each symplectic leaf is isomorphic to a Calogero--Moser variety for an associated (explicit) subquotient of the symplectic reflection group. This confirms a conjecture of Bonnaf\'e for these groups. We use the fact that the Calogero--Moser varieties associated to wreath products can be identified with certain Nakajima quiver varieties. In particular, our result identifying the normalization of the closure of each symplectic leaf with another quiver variety holds for arbitrary quiver varieties.

math.AG↗

arXiv2025

Nonprojective crepant resolutions of quiver varieties

In this paper, we construct a large class of examples of proper, nonprojective crepant resolutions of singularities for Nakajima quiver varieties. These include four and six dimensional examples and examples with $Q$ containing only three vertices. There are two main techniques: by taking a locally projective resolution of a projective partial resolution as in our previous work arXiv:2311.07539, and more generally by taking quotients of open subsets of representation space which are not stable loci, related to Arzhantsev--Derental--Hausen--Laface's construction in the setting of Cox rings. By the latter method we exhibit a proper crepant resolution that does not factor through a projective partial resolution. Most of our quiver settings involve one-dimensional vector spaces, hence the resolutions are toric hyperk\"ahler, which were studied from a different point of view in Arbo and Proudfoot arXiv:1511.09138. This builds on the classification of projective crepant resolutions of a large class of quiver varieties in arXiv:2212.09623 and the classification of proper crepant resolutions for the hyperpolygon quiver varieties in arXiv:2406.04117.

math.AG↗

arXiv2024

Creating quantum projective spaces by deforming q-symmetric algebras

We construct a large collection of "quantum projective spaces", in the form of Koszul, Calabi-Yau algebras with the Hilbert series of a polynomial ring. We do so by starting with the toric ones (the q-symmetric algebras), and then deforming their relations using a diagrammatic calculus, proving unobstructedness of such deformations under suitable nondegeneracy conditions. We then prove that these algebras are identified with the canonical quantizations of corresponding families of quadratic Poisson structures, in the sense of Kontsevich. In this way, we obtain the first broad class of quadratic Poisson structures for which his quantization can be computed explicitly, and shown to converge, as he conjectured in 2001.

math.QA↗

arXiv2024

The semi-invariant ring as the Cox ring of a GIT quotient

We study GIT quotients $X_\theta=V\!/\!\!/\!_\theta G$ whose linearisation map defines an isomorphism between the group of characters of $G$ and the Picard group of $X_\theta$ modulo torsion. Our main result establishes that the Cox ring of $X_\theta$ is isomorphic to the semi-invariant ring of the $\theta$-stable locus in $V$. This applies to quiver flag varieties, Nakajima quiver varieties, hypertoric varieties, and crepant resolutions of threefold Gorenstein quotient singularities with fibre dimension at most one. As an application, we present a simple, explicit calculation of the Cox ring of the Hilbert scheme of $n$-points in the affine plane.

math.AG↗

arXiv2023

Crepant resolutions of stratified varieties via gluing

Let $X$ be a variety with a stratification $\mathcal{S}$ into smooth locally closed subvarieties such that $X$ is locally a product along each stratum (e.g., a symplectic singularity). We prove that assigning to each open subset $U \subset X$ the set of isomorphism classes of locally projective crepant resolutions of $U$ defines an $\mathcal{S}$-constructible sheaf of sets. Thus, for each stratum $S$ and basepoint $s \in S$, the fundamental group acts on the set of germs of projective crepant resolutions at $s$, leaving invariant the germs extending to the entire stratum. Global locally projective crepant resolutions correspond to compatible such choices for all strata. For example, if the local projective crepant resolutions are unique, they automatically glue uniquely. We give criteria for a locally projective crepant resolution $\rho: \tilde X \to X$ to be globally projective. We show that the sheafification of the presheaf $U \mapsto \text{Pic}(\rho^{-1}(U)/U)$ of relative Picard classes is also constructible. The resolution is globally projective only if there exist local relatively ample bundles whose classes glue to a global section of this sheaf. The obstruction to lifting this section to a global ample line bundle is encoded by a gerbe on the singularity $X$. We show the gerbes are automatically trivial if $X$ is a symplectic quotient singularity. Our main results hold in the more general setting of partial crepant resolutions, that need not have smooth source. We apply the theory to symmetric powers and Hilbert schemes of surfaces with du Val singularities, finite quotients of tori, multiplicative and Nakajima quiver varieties, as well as to canonical threefold singularities.

math.AG↗

arXiv2022

Holonomic Poisson geometry of Hilbert schemes

We undertake a detailed study of the geometry of Bottacin's Poisson structures on Hilbert schemes of points in Poisson surfaces, i.e. smooth complex surfaces equipped with an effective anticanonical divisor. We focus on three themes that, while logically independent, are linked by the interplay between (characteristic) symplectic leaves and deformation theory. Firstly, we construct the symplectic groupoids of the Hilbert schemes and develop the classification of their symplectic leaves, using the methods of derived symplectic geometry. Secondly, we establish local normal forms for the Poisson brackets, and combine them with a toric degeneration argument to verify that Hilbert schemes satisfy our recent conjecture characterizing holonomic Poisson manifolds in terms of the geometry of the modular vector field. Finally, using constructible sheaf methods, we compute the space of first-order Poisson deformations when the anti-canonical divisor is reduced and has only quasi-homogeneous singularities. (The latter is automatic if the surface is projective.) Along the way, we find a tight connection between the Poisson geometry of the Hilbert schemes and the finite-dimensional Lie algebras of affine transformations, which is mediated by syzygies. In particular, we find that the Hilbert scheme has a natural subvariety that serves as a global counterpart of the nilpotent cone, and we prove that the Lie algebras of affine transformations have holonomic dual spaces -- the first such series of Lie algebras to be discovered.

math.AG↗

arXiv2022

Birational geometry of quiver varieties and other GIT quotients

We prove that all projective crepant resolutions of Nakajima quiver varieties satisfying natural conditions are also Nakajima quiver varieties. More generally, we classify the small birational models of many Geometric Invariant Theory (GIT) quotients by introducing a sufficient condition for the GIT quotient of an affine variety $V$ by the action of a reductive group $G$ to be a relative Mori Dream Space. Two surprising examples illustrate that our new condition is optimal. When the condition holds, we show that the linearisation map identifies a region of the GIT fan with the Mori chamber decomposition of the relative movable cone of $V /\!/_{\theta} G$. If $V/\!/_{\theta} G$ is a crepant resolution of $Y\!\!:= V/\!/_{0} G$, then every projective crepant resolution of $Y$ is obtained by varying $\theta$. Under suitable conditions, we show that this is the case for quiver varieties and hypertoric varieties. Similarly, for any finite subgroup $\Gamma\subset \mathrm{SL}(3,\mathbb{C})$ whose nontrivial conjugacy classes are all junior, we obtain a simple geometric proof of the fact that every projective crepant resolution of $\mathbb{C}^3/\Gamma$ is a fine moduli space of $\theta$-stable $\Gamma$-constellations.

math.AG↗

arXiv2022

An investigation into Lie algebra representations obtained from regular holonomic D-modules

Beilinson--Bernstein localisation relates representations of a Lie algebra $\mathfrak{g}$ to certain $\mathcal{D}$-modules on the flag variety of $\mathfrak{g}$. In [arXiv:2002.01540], examples of $\mathfrak{sl}_2$-representations which correspond to $\mathcal{D}$-modules on $\mathbb{CP}^1$ were computed. In this expository article, we give a topological description of these and extended examples via the Riemann-Hilbert correspondence. We generalise this to a full characterisation of $\mathfrak{sl}_2$-representations which correspond to holonomic $\mathcal{D}$-modules on $\mathbb{CP}^1$ with at most 2 regular singularities. We construct further examples with more singularities and develop a computer program for the computation of this correspondence in more general cases.

math.AG↗

arXiv2021

All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces

We demonstrate that the linear quotient singularity for the exceptional subgroup G in Sp(4,C) of order 32 is isomorphic to an affine quiver variety for a 5-pointed star-shaped quiver. This allows us to construct uniformly all 81 projective crepant resolutions of the quotient singularity C4/G as hyperpolygon spaces by variation of GIT quotient, and we describe both the movable cone and the Namikawa Weyl group action via an explicit hyperplane arrangement. More generally, for the n-pointed star shaped quiver, we describe completely the birational geometry for the corresponding hyperpolygon spaces in dimension 2n - 6; for example, we show that there are 1684 projective crepant resolutions when n = 6. We also prove that the resulting affine cones are not quotient singularities for n >= 6.

math.AG↗

arXiv2021

A local Torelli theorem for log symplectic manifolds

We establish a local model for the moduli space of holomorphic symplectic structures with logarithmic poles, near the locus of structures whose polar divisor is normal crossings. In contrast to the case without poles, the moduli space is singular: when the cohomology class of a symplectic structure satisfies certain linear equations with integer coefficients, its polar divisor can be partially smoothed, yielding adjacent irreducible components of the moduli space that correspond to possibly non-normal crossings structures. These components are indexed by combinatorial data we call smoothing diagrams, and amenable to algorithmic classification. Applying the theory to four-dimensional projective space, we obtain a total of 40 irreducible components of the moduli space, most of which are new. Our main technique is a detailed analysis of the relevant deformation complex (the Poisson cohomology) as an object of the constructible derived category.

math.AG↗

arXiv2019

Deformations of algebras in noncommutative geometry

These are significantly expanded lecture notes for the author's minicourse at MSRI in June 2012, as published in the MSRI lecture note series, with some minor additional corrections. In these notes, we give an example-motivated review of the deformation theory of associative algebras in terms of the Hochschild cochain complex as well as quantization of Poisson structures, and Kontsevich's formality theorem in the smooth setting. We then discuss quantization and deformation via Calabi-Yau algebras and potentials. Examples discussed include Weyl algebras, enveloping algebras of Lie algebras, symplectic reflection algebras, quasihomogeneous isolated hypersurface singularities (including du Val singularities), and Calabi-Yau algebras.

math.RA↗

arXiv2019

Symplectic resolutions of character varieties

In this article we consider the connected component of the identity of $G$-character varieties of compact Riemann surfaces of genus $g > 0$, for connected complex reductive groups $G$ of type $A$ (e.g., $SL_n$ and $GL_n$). We show that these varieties are 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 $PGL_2$, or if $g=2$ and $G = (SL_2)^m$ for some $m$.

math.AG↗

arXiv2019

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.

math.AG↗

arXiv2019

Multiplicative preprojective algebras are 2-Calabi-Yau

We prove that multiplicative preprojective algebras, defined by Crawley-Boevey and Shaw, are 2-Calabi-Yau algebras, in the case of quivers containing unoriented cycles. If the quiver is not itself a cycle, we show that the center is trivial, and hence the Calabi-Yau structure is unique. If the quiver is a cycle, we show that the algebra is a non-commutative crepant resolution of its center, the ring of functions on the corresponding multiplicative quiver variety with a type A surface singularity. We also prove that the dg versions of these algebras (arising as certain Fukaya categories) are formal. We conjecture that the same properties hold for all non-Dynkin quivers, with respect to any extended Dynkin subquiver (note that the cycle is the type A case). Finally, we prove that multiplicative quiver varieties-for all quivers-are formally locally isomorphic to ordinary quiver varieties. In particular, they are all symplectic singularities (which implies they are normal and have rational Gorenstein singularities). This includes character varieties of Riemann surfaces with punctures and monodromy conditions. We deduce this from a more general statement about 2-Calabi--Yau algebras (following Bocklandt, Galluzzi, and Vaccarino).

math.RA↗

arXiv2019

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 $\mathbb{Q}$-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.

math.AG↗

arXiv2018

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.

math.AG↗

arXiv2018

Equivariant slices for symplectic cones

The Darboux-Weinstein decomposition is a central result in the theory of 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*-equivariant version of the Darboux-Weinstein decomposition. In this paper, we develop such a theory, prove basic results on their existence and uniqueness, 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*-actions on slices to conical symplectic leaves.

math.AG↗