{"paper":{"title":"Enumerative invariants and wall-crossing formulae in abelian categories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Dominic Joyce","submitted_at":"2021-11-08T18:17:15Z","abstract_excerpt":"Enumerative invariants in Algebraic Geometry 'count' $\\tau$-(semi)stable objects $E$ with fixed topological invariants $[E]=a$ in some geometric problem, using a virtual class $[{\\cal M}_a^{\\rm ss}(\\tau)]_{\\rm virt}$ in homology, for the moduli spaces ${\\cal M}_a^{\\rm st}(\\tau)\\subseteq{\\cal M}_a^{\\rm ss}(\\tau)$ of $\\tau$-(semi)stable objects. We get numbers by taking integrals $\\int_{[{\\cal M}_a^{\\rm ss}(\\tau)]_{\\rm virt}}\\Upsilon$ for cohomology classes $\\Upsilon$.\n  Let $\\cal A$ be a $\\mathbb C$-linear abelian category in Algebraic Geometry. There are two moduli stacks of objects in $\\cal A"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2111.04694","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/2111.04694/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"}