Borel-Moore Homology Research Articles

Pseudocycles for Borel-Moore Homology

Pseudocycles for Borel-Moore Homology

Affine Springer Fibers and Generalized Haiman Ideals (with an Appendix by Eugene Gorsky and Joshua P. Turner)

Abstract We compute the Borel–Moore homology of unramified affine Springer fibers for $\textrm{Gr}_n$ under the assumption that they are equivariantly formal and relate them to certain ideals discussed by Haiman. For $n=3$, we give an explicit description of these ideals, compute their Hilbert series, generators, and relations, and compare them to generalized $(q,t)$-Catalan numbers. We also compare the homology to the Khovanov–Rozansky homology of the associated link, and prove a version of a conjecture of Oblomkov, Rasmussen, and Shende in this case.

A type C study of Braverman–Gaitsgory–Ginzburg’s construction of sln representations

In [Z. Fan, H. Ma and H. Xiao, Equivariant K-theory approach to [Formula: see text]-quantum groups, Publ. Res. Inst. Math. Sci. 58 (2022), no.3, 635–668], it is proved that the convolution algebra of the top Borel–Moore homology on the Steinberg variety of type [Formula: see text] realizes [Formula: see text], where [Formula: see text] is the fixed point subalgebra of involution on [Formula: see text]. So, the top Borel–Moore homology of the partial Springer fibers gives the representations of [Formula: see text]. In this paper, we study these representations using Schur–Weyl duality and Springer theory.

The integrality conjecture and the cohomology of preprojective stacks

Abstract We study the Borel–Moore homology of stacks of representations of preprojective algebras Π Q \Pi_{Q} , via the study of the DT theory of the undeformed 3-Calabi–Yau completion Π Q ⁢ [ x ] \Pi_{Q}[x] . Via a result on the supports of the BPS sheaves for Π Q ⁢ [ x ] ⁢ -mod \Pi_{Q}[x]\textup{-mod} , we prove purity of the BPS cohomology for the stack of Π Q ⁢ [ x ] \Pi_{Q}[x] -modules and define BPS sheaves for stacks of Π Q \Pi_{Q} -modules. These are mixed Hodge modules on the coarse moduli space of Π Q \Pi_{Q} -modules that control the Borel–Moore homology and geometric representation theory associated to these stacks. We show that the hypercohomology of these objects is pure and thus that the Borel–Moore homology of stacks of Π Q \Pi_{Q} -modules is also pure. We transport the cohomological wall-crossing and integrality theorems from DT theory to the category of Π Q \Pi_{Q} -modules. We use our results to prove positivity of a number of “restricted” Kac polynomials, determine the critical cohomology of Hilb n ⁡ ( A 3 ) \operatorname{Hilb}_{n}(\mathbb{A}^{3}) , and the Borel–Moore homology of genus one character stacks, as well as providing various applications to the cohomological Hall algebras associated to Borel–Moore homology of stacks of modules over preprojective algebras, including the PBW theorem, and torsion-freeness.

Borel–Moore homology of determinantal varieties

Borel–Moore homology of determinantal varieties

A sheaf-theoretic approach to tropical homology

We introduce a sheaf-theoretic approach to tropical homology, especially for tropical homology with potentially non-compact supports. Our setup is suited to study the functorial properties of tropical homology, and we show that it behaves analogously to classical Borel-Moore homology in the sense that there are proper push-forwards, cross products, and cup products with tropical cohomology classes, and that it satisfies identities like the projection formula and the Künneth theorem. Our framework allows for a natural definition of the tropical cycle class map, which we show to be a natural transformation. Finally, we characterize the rational polyhedral spaces that satisfy Poincaré-Verdier duality as those that are smooth.

Tropical Poincaré duality spaces

Abstract The tropical fundamental class of a rational balanced polyhedral fan induces cap products between tropical cohomology and tropical Borel–Moore homology. When all these cap products are isomorphisms, the fan is said to be a tropical Poincaré duality space. If all the stars of faces also are such spaces, such as for fans of matroids, the fan is called a local tropical Poincaré duality space. In this article, we first give some necessary conditions for fans to be tropical Poincaré duality spaces and a classification in dimension one. Next, we prove that tropical Poincaré duality for the stars of all faces of dimension greater than zero and a vanishing condition implies tropical Poincaré duality of the fan. This leads to necessary and sufficient conditions for a fan to be a local tropical Poincaré duality space. Finally, we use such fans to show that certain abstract balanced polyhedral spaces satisfy tropical Poincaré duality.

Nonabelian Hodge theory for stacks and a stacky P=W conjecture

We introduce a version of the P=W conjecture relating the Borel–Moore homology of the stack of representations of the fundamental group of a genus g Riemann surface with the Borel–Moore homology of the stack of degree zero semistable Higgs bundles on a smooth projective complex curve of genus g. In order to state the conjecture we propose a construction of a canonical isomorphism between these Borel–Moore homology groups. We relate the stacky P=W conjecture with the original P=W conjecture concerning the cohomology of smooth moduli spaces, and the PI=WI conjecture concerning the intersection cohomology groups of singular moduli spaces. In genus zero and one, we prove the conjectures that we introduce in this paper.

Enhanced bivariant homology theory attached to six functor formalism

Bivariant theory is a unified framework for cohomology and Borel–Moore homology theories. In this paper, we extract an ∞ $\infty$ -enhanced bivariant homology theory from Gaitsgory–Rozenblyum's six functor formalism.

Oriented Borel–Moore homologies of toric varieties

We generalize the well known Künneth formula for Chow groups to an arbitrary oriented Borel–Moore homology theory satisfying localization and descent (e.g. algebraic bordism) when taking a product with a toric variety. As a corollary we obtain a universal coefficient theorem for the operational cohomology rings. We also give a new, homological, description for the homology groups of smooth toric varieties, which allows us to compute the algebraic bordism groups of some singular toric varieties.

A boson-fermion correspondence in cohomological Donaldson–Thomas theory

AbstractWe introduce and study a fermionisation procedure for the cohomological Hall algebra $\mathcal{H}_{\Pi_Q}$ of representations of a preprojective algebra, that selectively switches the cohomological parity of the BPS Lie algebra from even to odd. We do so by determining the cohomological Donaldson–Thomas invariants of central extensions of preprojective algebras studied in the work of Etingof and Rains, via deformed dimensional reduction. Via the same techniques, we determine the Borel–Moore homology of the stack of representations of the $\unicode{x03BC}$ -deformed preprojective algebra introduced by Crawley–Boevey and Holland, for all dimension vectors. This provides a common generalisation of the results of Crawley-Boevey and Van den Bergh on the cohomology of smooth moduli schemes of representations of deformed preprojective algebras and my earlier results on the Borel–Moore homology of the stack of representations of the undeformed preprojective algebra.

Dimensional reduction in cohomological Donaldson–Thomas theory

For oriented $-1$-shifted symplectic derived Artin stacks, Ben-Bassat, Brav, Bussi and Joyce introduced certain perverse sheaves on them which can be regarded as sheaf-theoretic categorifications of the Donaldson–Thomas invariants. In this paper, we prove that the hypercohomology of the above perverse sheaf on the $-1$-shifted cotangent stack over a quasi-smooth derived Artin stack is isomorphic to the Borel–Moore homology of the base stack up to a certain shift of degree. This is a global version of the dimensional reduction theorem due to Davison. We give two applications of our main theorem. Firstly, we apply it to the study of the cohomological Donaldson–Thomas invariants for local surfaces. Secondly, regarding our main theorem as a version of the Thom isomorphism theorem for dual obstruction cones, we propose a sheaf-theoretic construction of the virtual fundamental classes for quasi-smooth derived Artin stacks.

Hilbert schemes of nonreduced divisors in Calabi–Yau threefolds and W-algebras

A W-algebra action is constructed on the equivariant Borel-Moore homology of the Hilbert scheme of points on a nonreduced plane in three dimensional affine space, identifying it to the vacuum W-module. This is based on a generalization of the ADHM construction as well as the W-action on the equivariant Borel-Moore homology of the moduli space of instantons constructed by Schiffmann and Vasserot.

Unramified affine Springer fibers and isospectral Hilbert schemes

For any connected reductive group G over \({{\mathbb {C}}}\), we revisit Goresky–Kottwitz–MacPherson’s description of the torus equivariant Borel–Moore homology of affine Springer fibers \({\mathrm {Sp}}_{\gamma }\subset {{\,\mathrm{Gr}\,}}_G\), where \(\gamma =zt^d\) and z is a regular semisimple element in the Lie algebra of G. In the case \(G=GL_n\), we relate the equivariant cohomology of \({\mathrm {Sp}}_\gamma \) to Haiman’s work on the isospectral Hilbert scheme of points on the plane. We also explain the connection to the HOMFLY homology of (n, dn)-torus links, and formulate a conjecture describing the homology of the Hilbert scheme of points on the curve \(\{x^n=y^{dn}\}\).

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 $r \geq 0, d \in Z$).

Cohomological Hall algebras for Higgs torsion sheaves, moduli of triples and sheaves on surfaces

For any free oriented Borel–Moore homology theory A, we construct an associative product on the A-theory of the stack of Higgs torsion sheaves over a projective curve C. We show that the resulting algebra Amathbf{Ha}_C^0 admits a natural shuffle presentation, and prove it is faithful when A is replaced with usual Borel–Moore homology groups. We also introduce moduli spaces of stable triples, heavily inspired by Nakajima quiver varieties, whose A-theory admits an Amathbf{Ha}_C^0-action. These triples can be interpreted as certain sheaves on mathbb {P}_C(omega _Coplus mathcal {O}_C). In particular, we obtain an action of Amathbf{Ha}_C^0 on the cohomology of Hilbert schemes of points on T^*C.

Cyclotomic double affine Hecke algebras

We show that the partially spherical cyclotomic rational Cherednik algebra (obtained from the full rational Cherednik algebra by averaging out the cyclotomic part of the underlying reflection group) has four other descriptions: (1) as a subalgebra of the degenerate DAHA of type A given by generators; (2) as an algebra given by generators and relations; (3) as an algebra of differential-reflection operators preserving some spaces of functions; (4) as equivariant Borel-Moore homology of a certain variety. Also, we define a new $q$-deformation of this algebra, which we call cyclotomic DAHA. Namely, we give a $q$-deformation of each of the above four descriptions of the partially spherical rational Cherednik algebra, replacing differential operators with difference operators, degenerate DAHA with DAHA, and homology with K-theory, and show that they give the same algebra. In addition, we show that spherical cyclotomic DAHA are quantizations of certain multiplicative quiver and bow varieties, which may be interpreted as K-theoretic Coulomb branches of a framed quiver gauge theory. Finally, we apply cyclotomic DAHA to prove new flatness results for various kinds of spaces of $q$-deformed quasiinvariants. In the appendix by H. Nakajima and D. Yamakawa (added in version 2), the authors explain the relations between multiplicative bow varieties and (various versions of) multiplicative quiver varieties for a cyclic quiver.

Characteristic cycles of highest weight Harish-Chandra modules and the Weyl group action on the conormal variety

We give an inductive algorithm that computes the action of simple reflections on a subset of basis-vectors of the Borel-Moore homology of the conormal variety associated to the symmetric pair $(\text{Sp}(2n), \text{GL}(n))$.

A FILTRATION ON EQUIVARIANT BOREL–MOORE HOMOLOGY

Let $G/H$ be a homogeneous variety and let $X$ be a $G$ -equivariant embedding of $G/H$ such that the number of $G$ -orbits in $X$ is finite. We show that the equivariant Borel–Moore homology of $X$ has a filtration with associated graded module the direct sum of the equivariant Borel–Moore homologies of the $G$ -orbits. If $T$ is a maximal torus of $G$ such that each $G$ -orbit has a $T$ -fixed point, then the equivariant filtration descends to give a filtration on the ordinary Borel–Moore homology of $X$ . We apply our findings to certain wonderful compactifications as well as to double flag varieties.

Suslin’s moving lemma with modulus

The moving lemma of Suslin states that a cycle on $X\times \mathbb{A} ^n$ meeting all faces properly can be moved so that it becomes equidimensional over $\mathbb{A}^n$. This leads to an isomorphism of motivic Borel-Moore homology and higher Chow groups. In this short paper we formulate and prove a variant of this. It leads to an isomorphism of Suslin homology with modulus and higher Chow groups with modulus, in an appropriate pro setting.