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A new transversality condition on orbifolds and integer-valued Gromov-Witten type invariants

Shaoyun Bai, Guangbo Xu

TL;DR

The paper develops a new transversality framework, the FOP transversality condition, for sections of orbifold vector bundles endowed with normally complex structures, enabling integral virtual cycles and integer-valued Gromov–Witten type invariants. It builds a rigorous finite-dimensional setting via Whitney stratifications of a universal zero locus Z and a robust collection of extension/stabilization properties, yielding strongly controlled perturbations whose zero loci carry integral Euler-type cycles. These perturbations underpin two key applications: (i) a general construction of integer-valued Gromov–Witten type invariants for all compact symplectic manifolds, and (ii) an alternative cohomological splitting proof for Hamiltonian fibrations over S^2 with integer coefficients. The framework connects to broader themes in symplectic topology and algebraic geometry, and leverages a deep interplay between normally complex structures, derived orbifolds, and canonical Whitney stratifications to produce open, dense, and perturbation-stable perturbations with stratified regularity. The approach promises further refinements in curve-counting invariants and potential links to Kontsevich–Manin-type axioms in this integral setting.

Abstract

Following a proposal of Fukaya-Ono and the exploration by B. Parker, we introduce a new transversality condition, the FOP transversality condition, for sections of orbifold vector bundles $\mathcal{E} \rightarrow \mathcal{U}$ when both $\mathcal{E}$ and $\mathcal{U}$ have "normal complex structures." This notion allows one to define various integral virtual cycles on moduli spaces of pseudoholomorphic curves. Two immediate applications in symplectic topology are the definition of integer-valued Gromov-Witten type invariants in all genera for general compact symplectic manifolds using the global Kuranishi chart constructed by Abouzaid-McLean-Smith and Hirschi-Swaminathan, and an alternative proof of the cohomological splitting theorem for Hamiltonian fibrations over $S^2$ with integer coefficients by Abouzaid-McLean-Smith.

A new transversality condition on orbifolds and integer-valued Gromov-Witten type invariants

TL;DR

The paper develops a new transversality framework, the FOP transversality condition, for sections of orbifold vector bundles endowed with normally complex structures, enabling integral virtual cycles and integer-valued Gromov–Witten type invariants. It builds a rigorous finite-dimensional setting via Whitney stratifications of a universal zero locus Z and a robust collection of extension/stabilization properties, yielding strongly controlled perturbations whose zero loci carry integral Euler-type cycles. These perturbations underpin two key applications: (i) a general construction of integer-valued Gromov–Witten type invariants for all compact symplectic manifolds, and (ii) an alternative cohomological splitting proof for Hamiltonian fibrations over S^2 with integer coefficients. The framework connects to broader themes in symplectic topology and algebraic geometry, and leverages a deep interplay between normally complex structures, derived orbifolds, and canonical Whitney stratifications to produce open, dense, and perturbation-stable perturbations with stratified regularity. The approach promises further refinements in curve-counting invariants and potential links to Kontsevich–Manin-type axioms in this integral setting.

Abstract

Following a proposal of Fukaya-Ono and the exploration by B. Parker, we introduce a new transversality condition, the FOP transversality condition, for sections of orbifold vector bundles when both and have "normal complex structures." This notion allows one to define various integral virtual cycles on moduli spaces of pseudoholomorphic curves. Two immediate applications in symplectic topology are the definition of integer-valued Gromov-Witten type invariants in all genera for general compact symplectic manifolds using the global Kuranishi chart constructed by Abouzaid-McLean-Smith and Hirschi-Swaminathan, and an alternative proof of the cohomological splitting theorem for Hamiltonian fibrations over with integer coefficients by Abouzaid-McLean-Smith.
Paper Structure (99 sections, 108 theorems, 420 equations, 4 figures)

This paper contains 99 sections, 108 theorems, 420 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Statement of the main technical result
  3. The FOP transversality condition
  4. FOP pseudocycles
  5. Intuitions about the FOP transversality
  6. Applications in symplectic topology
  7. Integral Gromov--Witten type invariants
  8. Homological splitting for Hamiltonian loops
  9. Relations with other work
  10. $\mathbb{Z}$-valued enumerative invariants in symplectic topology
  11. $\mathbb{Z}$-valued invariants in algebraic geometry
  12. Wasserman's theorem and stable homotopy theory
  13. Plan of the paper
  14. Acknowledgements
  15. Normally Complex Orbifolds and Bundles
  16. ...and 84 more sections

Key Result

Theorem 1.1

Suppose ${\mathcal{U}}$ is a normally complex orbifold without boundary and ${\mathcal{E}} \to {\mathcal{U}}$ is a normally complex vector bundle. Let $\Gamma({\mathcal{U}}, {\mathcal{E}})$ be the space of smooth sections. Then there is a $C^0$-denseIn general it is not $C^1$-dense. See Remark rem_d In addition, the structure of the zero loci of FOP transverse sections can be described as follows.

Figures (4)

  • Figure 1: Smooth perturbations and FOP transverse perturbations.
  • Figure 2: The fibration $P \to M$.
  • Figure 3: Comparing Whitney stratifications
  • Figure 4: The fibers of $\varphi_1$ and $\varphi_0$ over the same point are contained in the same preimage $M_p = f^{-1}(p)$. Their tangents at the zero section are the same so their distance is small compared to the distance from the zero section. One interpolates between them along the shortest geodesics $\gamma\subseteq M_p$ in a region near $D$.

Theorems & Definitions (284)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 274 more