arXiv subjects

Thomas Nevins

Publications and source records attributed to Thomas Nevins.

At least 19 recordsLinked to original sources

arXiv2024

Lagrangian Skeleta and Koszul Duality on Bionic Symplectic Varieties

We consider the category of modules over sheaves of Deformation-Quantization (DQ) algebras on bionic symplectic varieties. These spaces are equipped with both an elliptic $\mathbb{G}_m$-action and a Hamiltonian $\mathbb{G}_m$-action, with finitely many fixed points. On these spaces one can consider geometric category $\mathcal{O}$: the category of (holonomic) modules supported on the Lagrangian attracting set of the Hamiltonian action. We show that there exists a local generator in geometric category $\mathcal{O}$ whose dg endomorphism ring, cohomologically supported on the Lagrangian attracting set, is derived equivalent to the category of all DQ-modules. This is a version of Koszul duality generalizing the equivalence between D-modules on a smooth variety and dg-modules over the de Rham complex.

math.AG↗

arXiv2019

Counterexamples to hyperkahler Kirwan surjectivity

Suppose that M is a complete hyperkahler manifold with a compact Lie group K acting via hyperkahler isometries and with hyperkahler moment map $(μ_{\mathbb{C}}, μ_{\mathbb{R}}): M\rightarrow \mathfrak{k}^*\otimes\operatorname{Im}(\mathbb{H})$. It is a long-standing problem to determine when the hyperkahler Kirwan map $H^*_K(M,\mathbb{Q}) \longrightarrow H^*(M//K, \mathbb{Q})$ is surjective. We show that for each $n\geq 2$, the natural $U(n)$-action on $M = T^*(SL_n\times\mathbb{C}^n)$ admits a hyperkahler quotient for which the hyperkahler Kirwan map fails to be surjective. As a tool, we establish a ``Kahler $=$ GIT quotient'' assertion for products of cotangent bundles of reductive groups, equipped with the Kronheimer metric, and representations.

math.AG↗

arXiv2019

Springer theory for symplectic Galois groups

A classical and beautiful story in geometric representation theory is the construction by Springer of an action of the Weyl group on the cohomology of the fibres of the Springer resolution of the nilpotent cone. We establish a natural extension of Springer's theory to arbitrary symplectic resolutions of conical symplectic singularities. We analyse features of the action in the case of affine quiver varieties, constructing Weyl group actions on the cohomology of $ADE$ quiver varieties, and also consider "symplectically dual" examples arising from slices in the affine Grassmannian. Along the way, we document some basic features of the symplectic geometry of quiver varieties.

math.AG↗

arXiv2019

The pure cohomology of multiplicative quiver varieties

To a quiver $Q$ and choices of nonzero scalars $q_i$, non-negative integers $α_i$, and integers $θ_i$ labeling each vertex $i$, Crawley-Boevey--Shaw associate a "multiplicative quiver variety" $\mathcal{M}_θ^q(α)$, a trigonometric analogue of the Nakajima quiver variety associated to $Q$, $α$, and $θ$. We prove that the pure cohomology, in the Hodge-theoretic sense, of the stable locus $\mathcal{M}_θ^q(α)^s$ is generated as a $\mathbb{Q}$-algebra by the tautological characteristic classes. In particular, the pure cohomology of genus $g$ twisted character varieties of $GL_n$ is generated by tautological classes.

math.AG↗

arXiv2018

On the Kirwan map for moduli of Higgs bundles

Let $C$ be a smooth complex projective curve and $G$ a connected complex reductive group. We prove that if the center $Z(G)$ of $G$ is disconnected, then the Kirwan map $H^*\big(\operatorname{Bun}(G,C),\mathbb{Q}\big)\rightarrow H^*\big(\mathcal{M}_{\operatorname{Higgs}}^{\operatorname{ss}},\mathbb{Q}\big)$ from the cohomology of the moduli stack of $G$-bundles to the moduli stack of semistable $G$-Higgs bundles, fails to be surjective: more precisely, the "variant cohomology" (and variant intersection cohomology) of the stack $\mathcal{M}_{\operatorname{Higgs}}^{\operatorname{ss}}$ of semistable $G$-Higgs bundles, is always nontrivial. We also show that the image of the pullback map $H^*\big(M_{\operatorname{Higgs}}^{\operatorname{ss}},\mathbb{Q}\big)\rightarrow H^*\big(\mathcal{M}_{\operatorname{Higgs}}^{\operatorname{ss}},\mathbb{Q}\big)$, from the cohomology of the moduli space of semistable $G$-Higgs bundles to the stack of semistable $G$-Higgs bundles, cannot be contained in the image of the Kirwan map. The proof uses a Borel-Quillen--style localization result for equivariant cohomology of stacks to reduce to an explicit construction and calculation.

math.AG↗

arXiv2017

Kirwan surjectivity for quiver varieties

For algebraic varieties defined by hyperkahler or, more generally, algebraic symplectic reduction, it is a long-standing question whether the "hyperkahler Kirwan map" on cohomology is surjective. We resolve this question in the affirmative for Nakajima quiver varieties. We also establish similar results for other cohomology theories and for the derived category. Our proofs use only classical topological and geometric arguments.

math.AG↗

arXiv2014

Compatibility of t-structures for quantum symplectic resolutions

Let W be a smooth complex quasiprojective variety with the action of a connected reductive group G. Adapting the stratification approach of Teleman to a microlocal context, we prove a vanishing theorem for the functor of G-invariant sections---i.e., of quantum Hamiltonian reduction---for G-equivariant twisted D-modules on W. As a consequence, when W is affine we establish an effective combinatorial criterion for exactness of the global sections functors of microlocalization theory. When combined with our earlier derived equivalence results, this gives precise criteria for "microlocalization of representation categories."

math.AG↗

arXiv2014

Morse decomposition for D-module categories on stacks

Let Y be a smooth algebraic stack exhausted by quotient stacks. Given a Kirwan-Ness stratification of the cotangent stack T^*Y, we establish a recollement package for twisted D-modules on Y, gluing the category from subquotients described via modules microsupported on the Kirwan-Ness strata of T^*Y. The package includes unusual existence and "preservation-of-finiteness'' properties for functors of the full category of twisted D-modules, extending the standard functorialities for holonomic modules. In the case that Y = X/G is a quotient stack, our results provide a higher categorical analogue of the Atiyah-Bott--Kirwan--Ness "equivariant perfection of Morse theory'' for the norm-squared of a real moment map. As a consequence, we deduce a modified form of Kirwan surjectivity for the cohomology of hyperkaehler/algebraic symplectic quotients of cotangent bundles.

math.AG↗

arXiv2013

Categorical Cell Decomposition of Quantized Symplectic Algebraic Varieties

We prove a new symplectic analogue of Kashiwara's Equivalence from D-module theory. As a consequence, we establish a structure theory for module categories over deformation quantizations that mirrors, at a higher categorical level, the Bialynicki-Birula stratification of a variety with an action of the multiplicative group. The resulting categorical cell decomposition provides an algebro-geometric parallel to the structure of Fukaya categories of Weinstein manifolds. From it, we derive concrete consequences for invariants such as K-theory and Hochschild homology of module categories of interest in geometric representation theory.

math.AG↗

arXiv2013

Derived equivalence for quantum symplectic resolutions

Using techniques from the homotopy theory of derived categories and noncommutative algebraic geometry, we establish a general theory of derived microlocalization for quantum symplectic resolutions. In particular, our results yield a new proof of derived Beilinson-Bernstein localization and a derived version of the more recent microlocalization theorems of Gordon-Stafford and Kashiwara-Rouquier as special cases. We also deduce a new derived microlocalization result linking cyclotomic rational Cherednik algebras with quantized Hilbert schemes of points on minimal resolutions of cyclic quotient singularities.

math.AG↗

arXiv2009

Mirabolic Langlands duality and the quantum Calogero-Moser system

We give a generic spectral decomposition of the derived category of twisted D-modules on a moduli stack of mirabolic vector bundles on a curve X in characteristic p: that is, we construct an equivalence with the derived category of quasi-coherent sheaves on a moduli stack of mirabolic local systems on X. This equivalence may be understood as a tamely ramified form of the geometric Langlands equivalence. When X has genus 1, this equivalence generically solves (in the sense of noncommutative geometry) the quantum Calogero-Moser system.

math.AG↗

arXiv2008

W-Symmetry of the Adelic Grassmannian

We give a geometric construction of the W_{1+infty} vertex algebra as the infinitesimal form of a factorization structure on an adelic Grassmannian. This gives a concise interpretation of the higher symmetries and Backlund-Darboux transformations for the KP hierarchy and its multicomponent extensions in terms of a version of "W_{1+infty}-geometry": the geometry of D-bundles on smooth curves, or equivalently torsion-free sheaves on cuspidal curves.

math.RT↗

arXiv2008

Descent of coherent sheaves and complexes to geometric invariant theory quotients

Fix a scheme $X$ over a field of characteristic zero that is equipped with an action of a reductive algebraic group $G$. We give necessary and sufficient conditions for a $G$-equivariant coherent sheaf on $X$ or a bounded-above complex of $G$-equivariant coherent sheaves on $X$ to descend to a good quotient $X//G$. This gives a description of the coherent derived category of $X//G$ as an admissible subcategory of the equivariant derived category of $X$.

math.AG↗

arXiv2007

Perverse Bundles and Calogero-Moser Spaces

We present a simple description of moduli spaces of torsion-free D-modules (``D-bundles'') on general smooth complex curves X, generalizing the identification of the space of ideals in the Weyl algebra with Calogero-Moser quiver varieties. Namely, we show that the moduli of D-bundles form twisted cotangent bundles to stacks of torsion sheaves on X, answering a question of Ginzburg. The corresponding (untwisted) cotangent bundles are identified with moduli of ``perverse vector bundles'' on T^*X, which contain as open subsets the moduli of framed torsion-free sheaves (the Hilbert schemes (T^*X)^[n] in the rank one case). The proof is based on the description of the derived category of D-modules on X by a noncommutative version of the Beilinson transform on the projective line.

math.AG↗

arXiv2006

D-Bundles and Integrable Hierarchies

We study the geometry of D-bundles--locally projective D-modules--on algebraic curves, and apply them to the study of integrable hierarchies, specifically the multicomponent Kadomtsev-Petviashvili (KP) and spin Calogero-Moser (CM) hierarchies. We show that KP hierarchies have a geometric description as flows on moduli spaces of D-bundles; in particular, we prove that the local structure of D-bundles is captured by the full Sato Grassmannian. The rational, trigonometric, and elliptic solutions of KP are therefore captured by D-bundles on cubic curves E, that is, irreducible (smooth, nodal, or cuspidal) curves of arithmetic genus 1. We develop a Fourier-Mukai transform describing D-modules on cubic curves E in terms of (complexes of) sheaves on a twisted cotangent bundle over E. We then apply this transform to classify D-bundles on cubic curves, identifying their moduli spaces with phase spaces of general CM particle systems (realized through the geometry of spectral curves in our twisted cotangent bundle). Moreover, it is immediate from the geometric construction that the flows of the KP and CM hierarchies are thereby identified and that the poles of the KP solutions are identified with the positions of the CM particles. This provides a geometric explanation of a much-explored, puzzling phenomenon of the theory of integrable systems: the poles of meromorphic solutions to KP soliton equations move according to CM particle systems.

math.AG↗

arXiv2006

Flows of Calogero-Moser Systems

The Calogero-Moser (or CM) particle system and its generalizations appear, in a variety of ways, in integrable systems, nonlinear PDE, representation theory, and string theory. Moreover, the partially completed CM systems--in which dynamics of particles are continued through collisions--have been identified as meromorphic Hitchin systems, giving natural ``geometric action-angle variables'' for the CM system. Motivated by relations of the CM system to nonlinear PDE, we introduce a new class of generalizations of the spin CM particle systems, the framed (rational, trigonometric and elliptic) CM systems. We give two algebro-geometric descriptions of these systems, via meromorphic Hitchin systems with decorations (framing data) on (cuspidal, nodal and smooth) cubic curves and via one-dimensional sheaves on corresponding ``twisted'' ruled surfaces. We also present a simple geometric formulation of the flows of all meromorphic GL_n Hitchin systems (with no regularity assumptions) as tweaking flows on spectral sheaves. Using this formulation, we show that all spin and framed CM systems are identified with hierarchies of tweaking flows on the corresponding spectral sheaves. This generalizes the well-known description of spinless CM systems in terms of tangential covers.

math.AG↗

arXiv2006

Stringy Chern classes of singular varieties

Motivic integration and MacPherson's transformation are combined in this paper to construct a theory of "stringy" Chern classes for singular varieties. These classes enjoy strong birational invariance properties, and their definition encodes data coming from resolution of singularities. The singularities allowed in the theory are those typical of the minimal model program; examples are given by quotients of manifolds by finite groups. For the latter an explicit formula is proven, assuming that the canonical line bundle of the manifold descends to the quotient. This gives an expression of the stringy Chern class of the quotient in terms of Chern-Schwartz-MacPherson classes of the fixed-point set data.

math.AG↗

arXiv2004

A Localization Principle for Orbifold Theories

In this article, written primarily for physicists and geometers, we survey several manifestations of a general localization principle for orbifold theories such as $K$-theory, index theory, motivic integration and elliptic genera.

hep-th↗