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Bordism and resolution of singularities

Mohammed Abouzaid, Shaoyun Bai

TL;DR

This work develops a comprehensive, algorithmic framework to split and compare various bordism theories through resolution of singularities in normally complex orbifolds and derived orbifolds. By combining functorial embedded resolution, abelianization of stabilizers, and destackification (root stacks and divisorialification), the authors construct natural splittings of complex bordism maps to equivariant and derived theories, and transfer these splittings to orbifold bordism. Central to the approach are Fukaya–Ono–Parker perturbations adapted to the normally complex setting, and a perturbation–resolution pipeline that produces stably complex manifolds from derived orbifolds, thereby relating $dΩ^{U}$ to $Ω^{U}$ and to equivariant $Ω^{U,Γ}$. These results yield complex cobordism-valued Gromov–Witten invariants for general symplectic manifolds and constrain the topology of Hamiltonian-fibration spaces, with broad implications for moduli spaces of holomorphic curves and Floer-theoretic frameworks. The methods bridge algebraic resolution techniques with modern symplectic and homotopy-theoretic constructions, hinting at deeper global-homotopy interpretations and future refinements to $E_ fty$-structures in derived orbifold bordism.

Abstract

We adapt algorithms for resolving the singularities of complex algebraic varieties to prove that the natural map of homology theories from complex bordism to the bordism theory of complex derived orbifolds splits. In equivariant stable homotopy theory, our techniques yield a splitting of homology theories for the map from bordism to the equivariant bordism theory of a finite group $Γ$, given by assigning to a manifold its product with $Γ$. In symplectic topology, and using recent work of Abouzaid-McLean-Smith and Hirschi-Swaminathan, we conclude that one can define complex cobordism-valued Gromov-Witten invariant for arbitrary (closed) symplectic manifolds. We apply our results to constrain the topology of the space of Hamiltonian fibrations over $S^2$. The methods we develop apply to normally complex orbifolds, and will hence lead to applications in symplectic topology that rely on moduli spaces of holomorphic curves with Lagrangian boundary conditions.

Bordism and resolution of singularities

TL;DR

This work develops a comprehensive, algorithmic framework to split and compare various bordism theories through resolution of singularities in normally complex orbifolds and derived orbifolds. By combining functorial embedded resolution, abelianization of stabilizers, and destackification (root stacks and divisorialification), the authors construct natural splittings of complex bordism maps to equivariant and derived theories, and transfer these splittings to orbifold bordism. Central to the approach are Fukaya–Ono–Parker perturbations adapted to the normally complex setting, and a perturbation–resolution pipeline that produces stably complex manifolds from derived orbifolds, thereby relating to and to equivariant . These results yield complex cobordism-valued Gromov–Witten invariants for general symplectic manifolds and constrain the topology of Hamiltonian-fibration spaces, with broad implications for moduli spaces of holomorphic curves and Floer-theoretic frameworks. The methods bridge algebraic resolution techniques with modern symplectic and homotopy-theoretic constructions, hinting at deeper global-homotopy interpretations and future refinements to -structures in derived orbifold bordism.

Abstract

We adapt algorithms for resolving the singularities of complex algebraic varieties to prove that the natural map of homology theories from complex bordism to the bordism theory of complex derived orbifolds splits. In equivariant stable homotopy theory, our techniques yield a splitting of homology theories for the map from bordism to the equivariant bordism theory of a finite group , given by assigning to a manifold its product with . In symplectic topology, and using recent work of Abouzaid-McLean-Smith and Hirschi-Swaminathan, we conclude that one can define complex cobordism-valued Gromov-Witten invariant for arbitrary (closed) symplectic manifolds. We apply our results to constrain the topology of the space of Hamiltonian fibrations over . The methods we develop apply to normally complex orbifolds, and will hence lead to applications in symplectic topology that rely on moduli spaces of holomorphic curves with Lagrangian boundary conditions.

Paper Structure

This paper contains 50 sections, 79 theorems, 240 equations.

Table of Contents

  1. Introduction
  2. Applications: equivariant stable homotopy theory
  3. Applications: enumerative geometry and symplectic topology
  4. Results: bordism for orbispaces
  5. Methods of proof
  6. Outlook and further context
  7. Outline of the paper
  8. Normally complex orbifolds
  9. Orbifolds and vector bundles
  10. Differential geometry on orbifolds: metrics and normal complex structures
  11. Fukaya--Ono--Parker sections
  12. Abelianizing normally complex orbifolds
  13. Blow-ups
  14. The abelianization algorithm
  15. Discussions
  16. ...and 35 more sections

Key Result

Theorem 1.3

For each finite group $\Gamma$, the map of homology theories $\Omega^{U} \to \Omega^{U,\Gamma}$ splits as a map of $\Omega^{U}_*$-modules.

Theorems & Definitions (212)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 202 more