Cohomological Hall algebra of Higgs sheaves on a curve

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 $[Co{h}_{r,d}]$ of the zero-sections of the map $Higg{s}_{r,d}\to Co{h}_{r,d}$, for $r\ge 0,d\in Z$).

Latex, 22 pages; v2: 28 pages, Final version published in Algebraic Geometry

References in corpus (3)

• The cohomological Hall algebra of a preprojective algebra
• Quivers with loops and generalized crystals
• Quivers with loops and perverse sheaves

Cited by in corpus (3)

• Quiver Yangian from Crystal Melting
• A slow review of the AGT correspondence
• McKay correspondence, cohomological Hall algebras and categorification