Abstract
Borisov and Joyce constructed a real virtual cycle on compact moduli spaces of stable sheaves on Calabi–Yau 4-folds, using derived differential geometry. We construct an algebraic virtual cycle. A key step is a localization of Edidin and Graham’s square root Euler class for -bundles to the zero locus of an isotropic section, or to the support of an isotropic cone. We prove a torus localization formula, making the invariants computable and extending them to the noncompact case when the fixed locus is compact. We give a K-theoretic refinement by defining K-theoretic square root Euler classes and their localized versions. In a sequel, we prove that our invariants reproduce those of Borisov and Joyce.
Citation
Jeongseok Oh. Richard P. Thomas. "Counting sheaves on Calabi–Yau 4-folds, I." Duke Math. J. 172 (7) 1333 - 1409, 15 May 2023. https://doi.org/10.1215/00127094-2022-0059
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