{"paper":{"title":"Refinements of Kool-Thomas Invariants via Equivariant $K$-theoretic invariants","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Rizal Afgani","submitted_at":"2020-12-09T19:31:05Z","abstract_excerpt":"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 $\\mathcal{P}_{\\chi}(X,i_{*}\\beta)$ as the degree of the virtual class $\\left[\\mathcal{P}_{\\chi}(S,\\beta)\\right]^{red}$ after we apply $\\tau([pt])^{m}\\in H^{*}(\\mathcal{P}_{\\chi}(X,i_{*}\\beta),\\mathbb{Z})$. $\\tau([pt])$ contain the information of the incidence of a point and a curve supporting a $(\\mathcal{F},s)$.\n  $\\textbf{Keywords: }$Kool-Th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2012.05278","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2012.05278/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}