Cohomological Hall algebra of Higgs sheaves on a curve

TL;DR

This paper constructs and analyzes the cohomological Hall algebra of Higgs sheaves on a smooth projective curve, revealing its structure as a torsion-free module over the cohomology ring of coherent sheaves, with explicit generators.

Contribution

It defines the cohomological Hall algebra for Higgs sheaves on a curve and proves its torsion-free property as a module over the cohomology ring, providing explicit generators.

Findings

${AHA}_{Higgs(X)}$ is a torsion-free module over $ ext{H}$.

Explicit generators are given by fundamental classes of zero-sections.

The algebra structure is established in the context of an arbitrary oriented Borel-Moore homology theory.

Abstract

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