Vector Bundles on Curves – Quivers and Sheaves

For a finite subgroup $G$ of $\mathrm{SL}(3,\mathbb{C})$, the $G$-Hilbert scheme is a crepant resolution of the quotient singularity $\mathbb{C}^3/G$, and the universal family determines a derived equivalence between the derived category of $G$-equivariant coherent sheaves on $\mathbb{C}^3$ on one hand, and the derived category of coherent sheaves on $G$-Hilb on the other. In their 2009 paper, Cautis and Logvinenko conjectured that this derived equivalence sends the vertex simple $G$-sheaves, one for each nontrivial irreducible representation of $G$, to pure sheaves on $G$-Hilb. I will explain the importance of this statement to the McKay correspondence, and I will report on joint work with Ryo Yamagishi towards a proof of this conjecture.

What is the "maximal" amount of algebraic/representation theoretic structure one expects to exist on CoHAs? In the critical case, we will explain the answer to this question --- constructing a quantum group structure with a coproduct for each cohomology generator, generalising Drinfeld's meromorphic coproduct and abelian R-matrix on the Yangian. (Joint with S. Kaubrys and S. Jindal) We will also discuss how to bosonise/double in this setting, and if time allows, future work on obtaining these structures geometrically via factorisation spaces.

In this work we introduce a notion of tensor product of (twisted) quiver representations with relations in the category of O_X-modules. As a first application of our notion, we see that tensor products of polystable quiver bundles are polystable and later we use this to both deduce a quiver version of the Segre embedding and to identify distinguished closed subschemes of $\mathrm{GL}(n, \mathbb{C})$-character varieties of free abelian groups.

In this talk, I describe the cohomological Hall algebra of torsion sheaves on a weighted projective line with weights $(2,\ldots,2)$ in terms of generators and relations. This extends work of Franzen--Reineke on the CoHA of semistable representations of the Kronecker quiver.

Kronecker moduli are algebraic varieties parametrizing linear algebra data up to base change. They are special cases of quiver moduli spaces, with close connections and similarities to moduli spaces of vector bundles on curves. We consider generating series of motives of these spaces, and discuss recent results characterizing these series by explicit functional equations.

We define a version of the cohomological Hall algebra of Higgs sheaves which lives over the moduli space of smooth genus $g$ curves, and show that it forms a local system. We deduce from this the existence of a 'generic' version of the BPS Lie algebra, which is a Lie algebra object in the category of $\mathrm{GSp}(2g,\mathbb{C})$-modules and use this to prove some strong positivity property of Kac polynomials counting indecomposable vector bundles over curves defined over finite fields.

Kac polynomials are counts of quiver representations over finite fields, whose coefficients are non-negative and encode the graded dimensions of the so-called BPS Lie algebra. This Lie algebra sits inside the cohomological Hall algebra built from the cohomology of certain quiver moduli.

In this work, I study a generalisation of Kac polynomials to rings of truncated power series over finite fields, for certain dimension vectors. The main result is that these polynomials have non-negative coefficients and encode the graded dimensions of a certain subspace in the cohomology of jet spaces over the aforementioned quiver moduli. This behaviour is quite similar to that of the usual Kac polynomials.