arXiv subjects

Lucien Hennecart

Publications and source records attributed to Lucien Hennecart.

17 recordsLinked to original sources

arXiv2026

Commutativity of nilpotent cohomological Hall algebras of $\mathbf{A}^2$

In this paper, we prove that both the seminilpotent and the fully nilpotent CoHAs of $\mathbf{A}^2$ are commutative. This result is in strong contrast with the CoHA of $\mathbf{A}^2$ without nilpotency conditions, previously studied by Davison, which is related to the Lie algebra $W_{1+\infty}$ of differential operators on $\mathbf{C}^*$. The latter is highly noncommutative. Our proof combines two constraints on the Lie bracket on the affinized BPS Lie algebra: it is filtered with respect to the perverse filtration and it is graded with respect to the cohomological degree. In the case of the Jordan quiver and nilpotent CoHAs, these constraints force the Lie bracket to vanish. We also describe the equivariant nilpotent CoHAs in the presence of the action of a one-dimensional torus rescaling the first coordinate of $\mathbf{A}^2$ with weight $1$ and the second with weight $-1$. In this case, one obtains enveloping algebras of Rees Lie algebras associated with the nilpotent and the seminilpotent filtrations on the Lie algebra $W_{1+\infty}^+$, reminiscent of the description of the equivariant non-nilpotent CoHA given by Davison.

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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

The Deligne-Simpson problem via 2-Calabi-Yau categories

We provide a short proof of the necessity of Crawley-Boevey's condition in his solution to the Deligne-Simpson problem. The proof relies on the local neighbourhood theorem for $2$-Calabi-Yau categories due to Davison together with Crawley-Boevey's sufficient condition for the existence of local systems with prescribed conjugacy classes of monodromy around the punctures.

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arXiv2026

Degenerations of CoHAs of 2-Calabi-Yau categories

By work of Davison and Meinhardt, the cohomological Hall algebra of a symmetric quiver with potential admits a geometrically defined filtration (the perverse filtration) whose associated graded is a supercommutative algebra. In the case of the triple quiver of a quiver with the canonical cubic potential, which corresponds to the preprojective algebra of the quiver via dimensional reduction, there is an additional filtration (the less perverse filtration), which is defined more generally for cohomological Hall algebras of suitably geometric $2$-Calabi-Yau categories in work of Davison. In this paper, we show that the degenerations of the cohomological Hall algebras of preprojective algebras and more generally $2$-Calabi-Yau categories with respect to the less perverse filtration is isomorphic to the enveloping algebra of the current Lie algebra of the BPS Lie algebra. This result applies in particular to CoHAs of local systems on Riemann surfaces and Higgs bundles on smooth projective curves. We extend this description to deformations of the cohomological Hall algebra obtained via torus actions on the arrows of the quiver and deformed canonical cubic potentials via the deformed dimensional reduction of Davison-P\u adurariu. We prove all our results at the level of sheafified CoHAs, which allows us to deduce similar statements for all versions of nilpotent CoHAs. Last, we use our results to compare the less perverse filtration on CoHAs of preprojective algebras with the order filtration on the Maulik-Okounkov Yangian, via the comparison isomorphism of Botta-Davison and Schiffmann-Vasserot.

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arXiv2025

The BPS decomposition theorem

We prove the BPS decomposition theorem (a.k.a. cohomological integrality theorem) decomposing the cohomology of smooth symmetric stacks into the Weyl-invariant part of the cohomological Hall induction of the intersection cohomology of good moduli spaces. As a consequence, we establish the BPS decomposition theorem for the Borel--Moore homology of $0$-shifted symplectic stacks and for the critical cohomology of symmetric $(-1)$-shifted symplectic stacks, thereby generalizing the main result of Bu--Davison--Ibáñez Nuñez--Kinjo--Pădurariu to the non-orthogonal setting. We will present three applications of our main result. First, we confirm Halpern-Leistner's conjecture on the purity of the Borel--Moore homology of $0$-shifted symplectic stacks admitting proper good moduli spaces, extending Davison's work on the moduli stack of objects in $2$-Calabi--Yau categories. Second, we prove versions of Kirwan surjectivity for the critical cohomology of symmetric $(-1)$-shifted symplectic stacks and for the Borel--Moore homology of $0$-shifted symplectic stacks. Finally, by applying our main result to the character stacks associated with compact oriented $3$-manifolds, we reduce the quantum geometric Langlands duality conjecture for $3$-manifolds, as formulated by Safronov, from an isomorphism between infinite-dimensional critical cohomologies to a comparison of finite-dimensional BPS cohomologies.

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arXiv2025

BPS Lie algebras for totally negative 2-Calabi-Yau categories and nonabelian Hodge theory for stacks

We define and study a sheaf-theoretic cohomological Hall algebra for suitably geometric Abelian categories $\mathcal{A}$ of homological dimension at most two, and a sheaf-theoretic BPS algebra under the conditions that $\mathcal{A}$ is 2-Calabi-Yau and has a good moduli space. We show that the BPS algebra for the preprojective algebra $Π_Q$ of a totally negative quiver is the free algebra generated by the intersection cohomology of the closure of the locus parameterising simple $Π_Q$-modules in the coarse moduli space. We define and study the BPS Lie algebra of arbitrary 2-Calabi-Yau categories $\mathcal{A}$ for which the Euler form is negative on all pairs of non-zero objects, which recovers the BPS algebra as its universal enveloping algebra for such "totally negative" 2CY categories. We show that for totally negative 2CY categories the BPS algebra is freely generated by intersection complexes of certain coarse moduli spaces as above, and the Borel-Moore homology of the stack of objects in such $\mathcal{A}$ satisfies a Yangian-type PBW theorem for the BPS Lie algebra. In this way we prove the cohomological integrality theorem for these categories. We use our results to prove that for $C$ a smooth projective curve, and for $r$ and $d$ not necessarily coprime, there is a nonabelian Hodge isomorphism between the Borel-Moore homologies of the stack of rank $r$ and degree $d$ Higgs bundles, and the appropriate stack of twisted representations of the fundamental group of $C$. In addition we prove the Bozec-Schiffmann positivity conjecture for totally negative quivers; we prove that their polynomials counting cuspidal functions in the constructible Hall algebra for $Q$ have positive coefficients, strengthening the positivity theorem for the Kac polynomials of such quivers.

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arXiv2025

BPS algebras and generalised Kac-Moody algebras from 2-Calabi-Yau categories

We determine the structure of the BPS algebra of 2-Calabi-Yau Abelian categories for which the stack of objects admits a good moduli space. We prove that this algebra is isomorphic to the positive part of the enveloping algebra of a generalised Kac-Moody Lie algebra generated by the intersection cohomology of certain connected components (corresponding to roots) of the good moduli space. Some major examples include the BPS algebras of (1) the category of semistable coherent sheaves of given slope on a K3 surface or, more generally, quasiprojective symplectic surface, (2) semistable Higgs bundles on a smooth projective curve, (3) preprojective algebras of quivers, (4) multiplicative preprojective algebras and (5) fundamental groups of (quiver) Riemann surfaces. We define the BPS Lie algebras of 2-Calabi-Yau categories and prove that they coincide with the ones obtained by dimensional reduction from the critical cohomological Hall algebra in the case in which the 2-Calabi-Yau category is the category of representations of a preprojective algebra. Consequences include (1) A proof in full generality of the Bozec-Schiffmann positivity conjecture for absolutely cuspidal polynomials, a strengthening of the Kac positivity conjecture (2) A proof of the cohomological integrality conjecture for the category of semistable coherent sheaves on local K3 surfaces (3) A description of the cohomology (in all degrees) of Nakajima quiver varieties as direct sums of irreducible lowest weight representations over the BPS Lie algebra.

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arXiv2025

Cohomological integrality for weakly symmetric representations of reductive groups

In this paper, we prove the integrality conjecture for quotient stacks arising from weakly symmetric representations of reductive groups. Our main result is a decomposition of the cohomology of the stack into finite-dimensional components indexed by some equivalence classes of cocharacters of a maximal torus. This decomposition enables the definition of new enumerative invariants associated with the stack, which we begin to explore.

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arXiv2025

Cohomological Mackey formula for quotient stacks

In this paper, we construct a restriction morphism on the critical cohomology of an equivariant Landau-Ginzburg model associated to a representation of a reductive group equipped with an invariant function. We show a compatibility formula between the restriction and induction maps as a Mackey-type formula, thereby giving the critical cohomology the structure of a localized induction-restriction system.

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arXiv2025

Nonabelian Hodge isomorphisms for stacks and cohomological Hall algebras

In this paper, we complete the nonabelian Hodge theory (NAHT) triangle of isomorphisms for stacks between the Borel-Moore homologies of the Dolbeault, Betti, and de Rham moduli stacks. We first explain how to realise the category of connections on a smooth projective curve as a subcategory of a $2$-Calabi-Yau dg-category satisfying some appropriate geometric conditions. Then, we define a cohomological Hall algebra (CoHA) product on the Borel-Moore homology of the stack of connections on a smooth projective curve. This allows us to not only compare the Borel-Moore homologies of the stacks at the relative and absolute levels for the three sides of NAHT, but also to compare their CoHA structures: they all coincide. To compare the Dolbeault and de Rham sides, we define a relative CoHA for the Hodge-Deligne moduli space parametrising $λ$-connections. The Betti and de Rham sides are compared using the (derived) Riemann-Hilbert correspondence. The comparison of the Borel-Moore homologies of the Dolbeault and Betti moduli stacks was previously considered by the author with Davison and Schlegel Mejia (without taking the cohomological Hall algebra structures into account). This paper completes this study and provides a CoHA enhancement of the classical NAHT for curves.

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arXiv2025

Cohomological integrality for symmetric quotient stacks

In this paper, we establish the sheafified version of the cohomological integrality conjecture for stacks obtained as a quotient of a smooth affine symmetric algebraic variety by a reductive algebraic group equipped with an invariant function. A crucial step is the definition of the BPS sheaf as a complex of monodromic mixed Hodge modules. We prove the purity of the BPS sheaf when the situation arises from a smooth affine weakly symplectic algebraic variety with a weak moment map. This situation gives local models for 1-Artin derived stacks with self-dual cotangent complex. We then apply these results to prove a conjecture of Halpern-Leistner predicting the purity of the Borel-Moore homology of $0$-shifted symplectic stacks (or more generally, derived stacks with self-dual cotangent complex) having a proper good moduli space. One striking application is the purity of the Borel--Moore homology of the moduli stack of principal Higgs bundles over a smooth projective curve for a reductive group.

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arXiv2024

Geometric realisations of the unipotent enveloping algebra of a quiver

We compare and generalise the various geometric constructions (due to Ringel, Lusztig, Schofield, Bozec, Davison...) of the unipotent generalised Kac-Moody algebra associated with an arbitrary quiver. These constructions are interconnected through several geometric operations, including the stalk Euler characteristic of constructible complexes, the characteristic cycle, the Euler obstruction map, and the intersection multiplicities of Lagrangian subvarieties. We provide a proof that these geometric realisations hold for the integral form of the Lie algebra. Furthermore, by modifying the generators of the enveloping algebra, we ensure compatibility with the natural coproducts that can be defined in terms of restriction diagrams. As a result, we establish that the top cohomological Hall algebra of the strictly seminilpotent stack is isomorphic to the positive part of the enveloping algebra of the generalised Kac-Moody algebra associated with the quiver. This appears to be one of the cornerstones needed to describe the BPS algebra of very general $2$-Calabi-Yau categories, which is the subject of the author's work on BPS Lie algebras for $2$-Calabi-Yau categories with Davison and Schlegel Mejia.

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arXiv2022

A Quiver Analogue of Higman's Conjecture

An unresolved conjecture by Graham Higman states that for all $n\geq 1$ the number of conjugacy classes of the group of $n \times n$ unitriangular matrices with entries in the finite field $\mathbb{F}_q$ is a polynomial in $q$. In this paper we introduce a new quiver generalization of the conjecture. Motivated by this generalization, we prove that certain operations on quivers leave the relevant counts unchanged. Based on these invariance properties, we solve the introduced conjecture for quivers containing no path of length exceeding two, providing explicit formulas.

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arXiv2021

Perverse sheaves with nilpotent singular support on the stack of coherent sheaves on an elliptic curve

We define a stratification of the moduli stack of coherent sheaves on an elliptic curve which allows us (1) to give an explicit description of the irreducible components of the global nilpotent cone of elliptic curves, (2) to establish an explicit bijection between the simple objects of the category of perverse sheaves defined by Schiffmann to categorify the elliptic Hall algebra (the so-called spherical Eisenstein sheaves) and the irreducible components of the global nilpotent cone and (3) to give an explicit description and parametrization of the perverse sheaves on the moduli stack of coherent sheaves on an elliptic curve having nilpotent singular support. Along the way, we find a combinatorial parametrization of the irreducible components of the semistable locus of the elliptic global nilpotent cone.

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arXiv2021

Asymptotic behaviour of Kac polynomials

We conjecture a formula supported by computations for the valuation of Kac polynomials of a quiver, which only depends on the number of loops at each vertex. We prove a convergence property of renormalized Kac polynomials of quivers when increasing the number of arrows: they converge in the ring of power series, with a linear rate of convergence. Then, we propose a conjecture concerning the global behaviour of the coefficients of Kac polynomials. All computations were made using SageMath.

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arXiv2020

Microlocal characterization of Lusztig sheaves for affine quivers and $g$-loops quivers

We prove that for extended Dynkin quivers, simple perverse sheaves in Lusztig category are characterized by the nilpotency of their singular support. This proves a conjecture of Lusztig in the case of affine quivers. For cyclic quivers, we prove a similar result for a larger nilpotent variety and a larger class of perverse sheaves. We formulate conjectures for similar results for quivers with loops, for which we have to use the appropriate notion of nilpotent variety, due to Bozec, Schiffmann and Vasserot. We prove our conjecture for $g$-loops quivers ($g\geq 2$).

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arXiv2019

Isotropic cuspidal functions in the Hall algebra of a quiver

From the structure of the category of representations of an affine cycle-free quiver, we determine an explicit linear form on the space of regular cuspidal functions over a finite field: its kernel is exactly the space of cuspidal functions. Moreover, we show that any isotropic cuspidal dimension has an affine support. Brought together, this two results give an explicit description of isotropic cuspidal functions of any quiver. The main theorem together with an appropriate action of some permutation group on the Hall algebra provides a new elementary proof of two conjectures of Berenstein and Greenstein previously proved by Deng and Ruan. We also prove a statement giving non-obvious constraints on the support of the comultiplication of a cuspidal regular function allowing us to connect both mentioned conjectures of Berenstein and Greenstein. Our results imply the positivity conjecture of Bozec and Schiffmann concerning absolutely cuspidal polynomials in isotropic dimensions.

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