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Complex Kuranishi structures and counting sheaves on Calabi-Yau 4-folds, II

Jeongseok Oh, Richard P. Thomas

Abstract

We develop a theory of complex Kuranishi structures on projective schemes. These are sufficiently rigid to be equivalent to weak perfect obstruction theories, but sufficiently flexible to admit global complex Kuranishi charts. We apply the theory to projective moduli spaces M of stable sheaves on Calabi-Yau 4-folds. Using real derived differential geometry, Borisov-Joyce produced a virtual homology cycle on M. In the prequel to this paper we constructed an algebraic virtual cycle on M. We prove the cycles coincide in homology after inverting 2 in the coefficients. And when Borisov-Joyce's real virtual dimension is odd, their virtual cycle is 2-torsion.

Complex Kuranishi structures and counting sheaves on Calabi-Yau 4-folds, II

Abstract

We develop a theory of complex Kuranishi structures on projective schemes. These are sufficiently rigid to be equivalent to weak perfect obstruction theories, but sufficiently flexible to admit global complex Kuranishi charts. We apply the theory to projective moduli spaces M of stable sheaves on Calabi-Yau 4-folds. Using real derived differential geometry, Borisov-Joyce produced a virtual homology cycle on M. In the prequel to this paper we constructed an algebraic virtual cycle on M. We prove the cycles coincide in homology after inverting 2 in the coefficients. And when Borisov-Joyce's real virtual dimension is odd, their virtual cycle is 2-torsion.
Paper Structure (13 sections, 13 theorems, 81 equations)

This paper contains 13 sections, 13 theorems, 81 equations.

Table of Contents

  1. Some locally ringed spaces
  2. Smooth thickenings
  3. Schemes and $C^\infty$ functions
  4. $\mathcal{A}$-functions on smooth varieties
  5. $\mathcal{A}_2$-embeddings
  6. $\mathcal{A}_1$-bundles
  7. Complex Kuranishi neighbourhoods
  8. Perfect obstruction theories
  9. Complex Kuranishi spaces
  10. Equivalence with weak perfect obstruction theories
  11. Global complex Kuranishi charts
  12. The global smooth thickening $2M$.
  13. The $\mathcal{A}_2$-embedding

Key Result

Theorem 1

Complex Kuranishi structures on $M$ (Definition cK) are equivalent to weak perfect obstruction theories on $M$ (Definition wpot).

Theorems & Definitions (35)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Definition 1.3
  • Proposition 1.5
  • proof
  • Definition 1.8
  • Lemma 1.11
  • ...and 25 more