arXiv subjects

Olivier Schiffmann

Publications and source records attributed to Olivier Schiffmann.

At least 19 recordsLinked to original sources

arXiv2026

Hecke operators on symplectic surfaces and $χ$-independence

We prove Toda's chi-independence conjecture for the BPS cohomology of moduli spaces of one-dimensional sheaves on quasi-projective symplectic surfaces, relative to the Chow variety. We also identify the BPS Lie algebra associated with one-dimensional Mukai vectors with the subspace of tautological classes, giving an extension of Markman's tautological generation theorem from primitive to arbitrary Mukai vectors. The main structure input is a bialgebra structure on the cohomological Hall algebra of coherent sheaves on a quasi-projective symplectic variety S. The coproduct is obtained, by dimensional reduction, from a factorization coproduct for 3d cohomological Hall algebras, and gives rise to a global BPS Lie algebra attached to the stack of coherent sheaves on S. The link between this structure and the applications to chi-independence and tautological generation is provided by Hecke operators on BPS cohomology, which modify one-dimensional sheaves by zero-dimensional quotients. To make this construction work, we prove that there is an identification between the affinized BPS cohomology of the semistable locus and the primitive part of the coproduct on the entire moduli stack

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arXiv2026

Nilpotent cohomological Hall algebras of surfaces

This paper develops a framework for systematically studying cohomological "Hecke operators" associated with modifications of coherent sheaves on a smooth surface $X$ along a fixed proper curve $Z \subset X$ (possibly singular and reducible), using the theory of cohomological Hall algebras. More precisely, we construct a moduli stack of coherent sheaves $\mathbf{Coh}(\widehat{X}_Z)$ on $X$ with set-theoretic support $Z$ and we prove that its reduced is an Artin stack locally of finite type. This provides a vast generalization of the global nilpotent cone. Subsequently, we develop the needed background to define the (motivic, $T$-equivariant) cohomological Hall algebra $\mathbf{HA}^{T}_{X,Z}$ of the moduli stack of coherent sheaves on $X$ with set-theoretic support on $Z$, in the setting of a general motivic formalism $\mathbf{D}$ in the sense of Khan. The algebra $\mathbf{HA}^{\mathbf{D}, A}_{X,Z}$ is functorial with respect to closed immersions $Z' \subset Z$ and transformations of the motivic formalism $\mathbf{D}$, and only depends on the formal neighborhood $\widehat{X}_Z$ of $Z$ in $X$. In the companion paper arXiv:2603.03386, we use the nilpotent COHA $\mathbf{HA}^{T}_{X,Z}$ to answer a question previously raised in arXiv:2004.13685 about the precise relationship between the COHA of a minimal resolution of a Kleinian singularity and the corresponding preprojective COHA.

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arXiv2026

Cohomological Hall algebras of one-dimensional sheaves on surfaces and Yangians

This paper provides the first algebraic characterization of an algebra of cohomological Hecke operators associated with modifications of coherent sheaves on a smooth surface $X$ along a fixed proper curve $Z \subset X$ (possibly singular and reducible), establishing a direct connection with Yangians. It is based on the theory of equivariant nilpotent cohomological Hall algebras $\mathbf{HA}^T_{X,Z}$, developed by the same authors. More precisely, let $X$ be a resolution of a Kleinian singularity (for example, $X = T^\ast\mathbb{P}^1$) and let $Z$ be the exceptional divisor. One of the main results of this paper is an explicit isomorphism $\mathbf{HA}^T_{X,Z} \simeq \mathbb{Y}^+_\infty$, where $\mathbb{Y}^+_\infty$ is a completed, nonstandard, positive half of the affine Yangian $\mathbb{Y}(\mathfrak{g})$ of the corresponding affine ADE Lie algebra $\mathfrak{g}$. Furthermore, the generators of $\mathbf{HA}^T_{X,Z}$--given by fundamental classes of substacks of zero-dimensional sheaves and of pushforwards of line bundles on $Z$--are expressed explicitly in terms of Yangian generators. Our main tools, which may be of independent interest, are: (i) a `continuity' theorem describing the behavior of cohomological Hall algebras of objects in the heart of $t$-structures $τ_n$ when the sequence $(τ_n)_n$ converges, in an appropriate sense, to a fixed $t$-structure $τ_\infty$; (ii) the definition of a multi-parameter Yangian $\mathbb{Y}_Q$ for an arbitrary quiver $Q$, given by generators and relations; (iii) a theorem relating the algebraic action of the braid group $B_Q$ on the Yangian $\mathbb{Y}_Q$ to the action of $B_Q$ on the equivariant 2-dimensional cohomological Hall algebra $\mathbf{HA}^T_Q$ of $Q$, where the latter can be described in terms of derived reflection functors of the bounded derived category of modules over the preprojective algebra of $Q$.

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arXiv2026

Coherent sheaves on surfaces, COHAs and deformed $W_{1+\infty}$-algebras

We compute the cohomological Hall algebra of zero-dimensional sheaves on an arbitrary smooth quasi-projective surface $S$ with pure cohomology, deriving an explicit presentation by generators and relations. When $S$ has trivial canonical bundle, this COHA is isomorphic to the enveloping algebra of deformed trigonometric $W_{1+\infty}$-algebra associated to the ring $H^*(S,\mathbb{Q})$. We also define a double of this COHA, show that it acts on the homology of various moduli stacks of sheaves on $S$ and explicitly describe this action on the products of tautological classes. Examples include Hilbert schemes of points on surfaces, the moduli stack of Higgs bundles on a smooth projective curve and the moduli stack of $1$-dimensional sheaves on a $K3$ surface in an ample class. The double COHA is shown to contain Nakajima's Heisenberg algebra, as well as a copy of the Virasoro algebra.

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arXiv2026

KLR-Schur algebra of coherent sheaves on the projective line: Tilting and PBW bases

We begin the study of Khovanov-Lauda-Rouquier type algebras associated to moduli stacks of coherent sheaves on smooth projective curves. We consider the case of $\mathbb{P}^1$ and define, for any pair $(r,d)$ of a rank and a degree, the KLR and Schur algebras $A_{r,d}, \mathcal{R}_{r,d}$ as suitable convolution algebras in the Borel-Moore homology of an analog of the Steinberg stack built from the stacks $Coh_{r,d}(\mathbb{P}^1)$. We use the tilting equivalence and Bridgeland stability conditions to construct an interpolation between the KLR or Schur algebras of the categories of coherent sheaves on $\mathbb{P}^1$ and the KLR or Schur algebras of the categories of representations of the Kronecker quiver. We also introduce a stratification of the Steinberg stacks into cohomologically pure pieces and use this to construct a PBW basis of the corresponding algebra.

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arXiv2025

Kleinian orbifolds, Cohomological Hall Algebras, and Yangians

We establish, for each orbifold crepantly resolving a Kleinian singularity, the existence of the cohomological Hall algebra (COHA) of coherent sheaves supported on the exceptional locus and explicitly compute this COHA as a completion of some positive half of the associated affine Yangian. Tracking these categories under derived autoequivalences and the McKay correspondence, we show that (1) every point in Bridgeland's space of stability conditions on the resolution arises from a Kleinian orbifold, and (2) every positive half of the affine Yangian can be recovered from the COHA associated to some such stability condition. This provides the first example of a family of (pointwise) COHAs defined over the space of stability conditions.

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arXiv2025

$P=W$ via $\mathcal{H}_2$

Let $\mathcal{H}_2$ be the Lie algebra of polynomial Hamiltonian vector fields on the symplectic plane. Let $X$ be the moduli space of stable Higgs bundles of fixed relatively prime rank and degree, or more generally the moduli space of stable parabolic Higgs bundles of arbitrary rank and degree for a generic stability condition. Let $H^*(X)$ be the cohomology with rational coefficients. Using the operations of cup-product by tautological classes and Hecke correspondences we construct an action of $\mathcal{H}_2$ on $H^*(X)[x,y]$, where $x$ and $y$ are formal variables. We show that the perverse filtration on $H^*(X)$ coincides with the filtration canonically associated to $\mathfrak{sl}_2\subset \mathcal{H}_2$ and deduce the $P=W$ conjecture of de Cataldo-Hausel-Migliorini.

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arXiv2024

The Langlands formula and perverse sheaves

For a complex reductive Lie algebra $\mathfrak{g}$ with Cartan subalgebra $\mathfrak{h}$ and Weyl group $W$ we consider the category $\text{Perv}(W \backslash \mathfrak{h})$ of perverse sheaves on $W \backslash \mathfrak{h}$ smooth w.r.t. the natural stratification. We construct a category $\boldsymbol{\mathcal{C}}$ such that $\text{Perv}(W\backslash \mathfrak{h})$ is identified with the category of functors from $\boldsymbol{\mathcal{C}}$ to vector spaces. Objects of $\boldsymbol{\mathcal{C}}$ are labelled by standard parabolic subalgebras in $\mathfrak{g}$. It has morphisms analogous to the operations of parabolic induction (Eisenstein series) and restriction (constant term) of automorphic forms. In particular, the Langlands formula for the constant term of an Eisenstein series has a counterpart in the form of an identity in $\boldsymbol{\mathcal{C}}$. We define $\boldsymbol{\mathcal{C}}$ as the category of $W$-invariants (in an appropriate sense) in the category $Q$ describing perverse sheaves on $\mathfrak{h}$ smooth w.r.t. the root arrangement. This matches, in an interesting way, the definition of $W \backslash \mathfrak{h}$ itself as the spectrum of the algebra of $W$-invariants.

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arXiv2024

Shuffle algebras for quivers as quantum groups

We define a quantum loop group $\mathbf{U}^+_Q$ associated to an arbitrary quiver $Q=(I,E)$ and maximal set of deformation parameters, with generators indexed by $I \times \mathbb{Z}$ and some explicit quadratic and cubic relations. We prove that $\mathbf{U}^+_Q$ is isomorphic to the (generic, small) shuffle algebra associated to the quiver $Q$ and hence, by [Neg21a], to the localized K-theoretic Hall algebra of $Q$. For the quiver with one vertex and $g$ loops, this yields a presentation of the spherical Hall algebra of a (generic) smooth projective curve of genus $g$ (invoking the results of [SV12]). We extend the above results to the case of non-generic parameters satisfying a certain natural metric condition. As an application, we obtain a description by generators and relations of the subalgebra generated by absolutely cuspidal eigenforms of the Hall algebra of an arbitrary smooth projective curve (invoking the results of [KSV17]).

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arXiv2024

Cohomological Hall algebras of quivers and Yangians

We construct an isomorphism between the preprojective cohomological Hall algebra of an arbitrary quiver and a positive half of the corresponding Maulik-Okounkov Yangian, which intertwines the respective actions on the cohomology of the Nakajima quiver varieties. We use this to prove a conjecture of Okounkov relating the character of the Maulik-Okounkov Lie algebra to Kac polynomials.

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arXiv2022

On cohomological Hall algebras of quivers : generators

We study the cohomological Hall algebra Y of a lagrangian substack of the moduli stack of representations of the preprojective algebra of an arbitrary quiver Q, and their actions on the cohomology of Nakajima quiver varieties. We prove that Y is pure and we compute its Poincare polynomials in terms of (nilpotent) Kac polynomials. We also provide a family of algebra generators. We conjecture that Y is equal, after a suitable extension of scalars, to the Yangian introduced by Maulik and Okounkov. As a corollary, we prove a variant of Okounkov's conjecture, which is a generalization of the Kac conjecture relating the constant term of Kac polynomials to root multiplicities of Kac-Moody algebras.

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arXiv2021

Continuum Kac-Moody algebras

We introduce a new class of infinite-dimensional Lie algebras, which we refer to as continuum Kac-Moody algebras. Their construction is closely related to that of usual Kac-Moody algebras, but they feature a continuum root system with no simple roots. Their Cartan datum encodes the topology of a one-dimensional real space and can be thought of as a generalization of a quiver, where vertices are replaced by connected intervals. For these Lie algebras, we prove an analogue of the Gabber-Kac-Serre theorem, providing a complete set of defining relations featuring only quadratic Serre relations. Moreover, we provide an alternative realization as continuum colimits of symmetric Borcherds-Kac-Moody algebras with at most isotropic simple roots. The approach we follow deeply relies on the more general notion of a semigroup Lie algebra and its structural properties.

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arXiv2020

The circle quantum group and the infinite root stack of a curve (with an appendix by Tatsuki Kuwagaki)

In the present paper, we give a definition of the quantum group $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1))$ of the circle $S^1\colon =\mathbb{R}/\mathbb{Z}$, and its fundamental representation. Such a definition is motivated by a realization of a quantum group $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1_\mathbb{Q}))$ associated to the rational circle $S^1_\mathbb{Q}\colon= \mathbb{Q}/\mathbb{Z}$ as a direct limit of $\mathbf{U}_\upsilon(\widehat{\mathfrak{sl}}(n))$'s, where the order is given by divisibility of positive integers. The quantum group $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1_\mathbb{Q}))$ arises as a subalgebra of the Hall algebra of coherent sheaves on the infinite root stack $X_\infty$ over a fixed smooth projective curve $X$ defined over a finite field. Via this Hall algebra approach, we are able to realize geometrically the fundamental and the tensor representations, and a family of symmetric tensor representations, depending on the genus $g_X$, of $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1_\mathbb{Q}))$. Moreover, we show that $\mathbf{U}_\upsilon(\widehat{\mathfrak{sl}}(+\infty))$ and $\mathbf{U}_\upsilon(\widehat{\mathfrak{sl}}(\infty))$ are subalgebras of $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1_\mathbb{Q}))$. As proved by T. Kuwagaki in the appendix, the quantum group $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1))$ naturally arises as well in the mirror dual picture, as a Hall algebra of constructible sheaves on the circle $S^1$.

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arXiv2020

Fock space representation of the circle quantum group

In [arXiv:1711.07391] we have defined quantum groups $\mathbf{U}_\upsilon(\mathfrak{sl}(\mathbb{R}))$ and $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1))$, which can be interpreted as continuous generalizations of the quantum groups of the Kac-Moody Lie algebras of finite, respectively affine type $A$. In the present paper, we define the Fock space representation $\mathcal{F}_{\mathbb{R}}$ of the quantum group $\mathbf{U}_\upsilon(\mathfrak{sl}(\mathbb{R}))$ as the vector space generated by real pyramids (a continuous generalization of the notion of partition). In addition, by using a variant of the "folding procedure" of Hayashi-Misra-Miwa, we define an action of $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1))$ on $\mathcal{F}_{\mathbb{R}}$.

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arXiv2020

Cohomological Hall algebra of Higgs sheaves on a curve

We define the cohomological Hall algebra ${AHA}_{Higgs(X)}$ of the ($2$-dimensional) Calabi-Yau category of Higgs sheaves on a smooth projective curve $X$, as well as its nilpotent and semistable variants, in the context of an arbitrary oriented Borel-Moore homology theory. In the case of usual Borel-Moore homology, ${AHA}_{Higgs(X)}$ is a module over the (universal) cohomology ring $\mathbb{H}$ of the stacks of coherent sheaves on $X$ . We show that it is a torsion-free $\mathbb{H}$-module, and we provide an explicit collection of generators (the collection of fundamental classes $[Coh_{r,d}]$ of the zero-sections of the map $Higgs_{r,d} \to Coh_{r,d}$, for $r \geq 0, d \in Z$).

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arXiv2018

Kac polynomials and Lie algebras associated to quivers and curves

A survey of the theory of Kac polynomials for quivers and for curves. In particular, we describe the representation-theoretic meaning of Kac polynomials in terms of Hall algebras, and the geometric meaning of Kac polynomials in relation to the geometry of moduli spaces of representations of quivers or vector bundles on smooth projective curves. We end with some heuristics concerning a family of infinite-dimensional $\mathbb{Z}^2$-graded Lie algebras attached to curves of a fixed genus (over a finite field), whose 'Cartan datum' encodes the dimension of the spaces of absolutely cuspidal functions.

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arXiv2017

On cohomological Hall algebras of quivers : Yangians

We consider the cohomological Hall algebra Y of a Lagrangian substack of the moduli stack of representations of the preprojective algebra of an arbitrary quiver Q, and its actions on the cohomology of quiver varieties. We conjecture that Y is equal, after a suitable extension of scalars, to the Yangian introduced by Maulik and Okounkov, and we construct an embedding of Y in the Yangian, intertwining the respective actions of both algebras on the cohomology of quiver varieties.

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arXiv2017

Counting Higgs bundles and type A quiver bundles

We prove a closed formula counting semistable twisted (or meromorphic) Higgs bundles of fixed rank and degree over a smooth projective curve of genus g, defined over a finite field, when the degree of the twisting line bundle is at least 2g-2; this includes the case of usual Higgs bundles. We obtain at the same time a closed expression for the Donaldson-Thomas invariants of the moduli spaces of twisted Higgs bundles. We similarly deal with twisted quiver bundles of type A (affine or finite), obtaining in particular a Harder-Narasimhan-type formula counting semistable U(p,q)-bundles over any smooth projective curve over a finite field.

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