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

Alexandre Minets

• 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 A\mathbf{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 A\mathbf{Ha}_C^0-action. These triples can be interpreted as certain sheaves on \mathbb {P}_C(\omega _C\oplus \mathcal {O}_C). In particular, we obtain an action of A\mathbf{Ha}_C^0 on the cohomology of Hilbert schemes of points on T^*C.