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arXiv · 2009.05542

Counting sheaves on Calabi-Yau 4-folds, I

Jeongseok Oh·Richard P. Thomas
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Abstract

Borisov-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 localisation of Edidin-Graham's square root Euler class for $SO(r,\mathbb C)$ bundles to the zero locus of an isotropic section, or to the support of an isotropic cone. We prove a torus localisation 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 localised versions. In a sequel we prove our invariants reproduce those of Borisov-Joyce.

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Jeongseok Oh, Richard P. Thomas. 2020-09-11. Counting sheaves on Calabi-Yau 4-folds, I. https://doi.org/10.1215/00127094-2022-0059

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