Refinements of Kool-Thomas Invariants via Equivariant $K$-theoretic invariants
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 $\mathcal{P}_χ(X,i_{*}β)$ as the degree of the virtual class $\left[\mathcal{P}_χ(S,β)\right]^{red}$ after we apply $τ([pt])^{m}\in H^{*}(\mathcal{P}_χ(X,i_{*}β),\mathbb{Z})$. $τ([pt])$ contain the information of the incidence of a point and a curve supporting a $(\mathcal{F},s)$. $\textbf{Keywords: }$Kool-Thomas invariants, $K$-theoretic invariants, Göttsche Shende invariants
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