Refinements of Kool-Thomas Invariants via Equivariant $K$-theoretic invariants

Abstract

In this article we are defining a refinement of Kool-Thomas invariants of local surfaces via the equivariant $K$-theoretic invariants proposed by Nekrasov and Okounkov. Kool and Thomas defined the reduced obstruction theory for the moduli of stable pairs ${\mathcal{P}}_{\chi }(X,i_{\ast }\beta )$ as the degree of the virtual class ${\left[{\mathcal{P}}_{\chi }(S,\beta )\right]}^{red}$ after we apply $\tau ([pt]{)}^m\in H^{\ast }({\mathcal{P}}_{\chi }(X,i_{\ast }\beta ),\mathbb{Z})$. $\tau ([pt])$ contain the information of the incidence of a point and a curve supporting a $(\mathcal{F},s)$. $\text{Keywords: }$Kool-Thomas invariants, $K$-theoretic invariants, Göttsche Shende invariants

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