The quantum flips program of Lee–Lin–Wang studies how genus-zero Gromov–Witten theory transforms under simple (r,r′) flips, the non-K-equivalent elementary links of the minimal model program. Its first instalment proved, for projective local models only, that after analytic continuation in the extremal-ray quantum variable the Dubrovin connection of X splits into regular and irregular eigenbundles, and that the big quantum cohomology QH(X′) embeds non-linearly as an F-manifold into the regular integrable loci of QH(X) near the extremal degeneration. The paper's explicitly stated next step, which remains open, is to construct the eigenbundle decomposition and the F-embedding Ψ̂ for a general global simple flip between smooth projective varieties, identifying the analytic hypotheses (convergence, treatment of essential singularities) under which it holds. The core remains open: Gu–Yu–Yu (2025) prove the corresponding quantum-D-module decomposition for simple VGIT wall-crossings and for local models of standard flips but state the global standard-flip case as an open conjecture; Shen–Shoemaker (2025) prove the extremal quantum spectrum and Gamma-class decomposition for global standard flips.

Background

For a smooth complex projective variety X, genus-zero Gromov–Witten (GW) invariants — counts of rational stable maps into X — assemble into the big quantum cohomology QH(X), a deformation of the cup product on the cohomology H*(X) depending on a parameter t ∈ H*(X) and on Novikov variables q^β (one formal variable per effective curve class β, recording the degrees of the curves being counted). QH(X) underlies a Frobenius manifold, and its flat Dubrovin connection ∇ = d − (1/z)(·⋆) on the trivial H*(X)-bundle encodes the quantum product ⋆. If one keeps only the commutative, associative multiplication of tangent vectors with unit and Euler field and drops the metric compatibility, one obtains the weaker notion of an F-manifold; quantum cohomology is always an F-manifold, and this is the structure that survives the construction below.

The minimal model program (MMP) relates birational models by elementary links. A link f: X ⇢ X′ with exceptional loci Z ⊂ X, Z′ ⊂ X′ is a flop when it is K-equivalent (the canonical classes agree under the map) and a flip when it is not. For flops, Lee–Lin–Wang proved that the big quantum cohomology rings are isomorphic after analytic continuation in the Novikov variables (Ann. of Math. 2010, with sequels); this is part of the broader Crepant Transformation Conjecture. Flips are the genuinely non-crepant case, first studied in the program's instalment 'Quantum flips I: local model' (arXiv:1912.03012; published in Proc. Sympos. Pure Math. 103.2). A simple (r,r′) flip, with r > r′ > 0, has exceptional loci Z ≅ P^r ⊂ X and Z′ ≅ P^{r′} ⊂ X′ with normal bundles O(−1)^{⊕(r′+1)} and O(−1)^{⊕(r+1)}. Classically the cohomology rings are not isomorphic: the graph-correspondence Ψ = [Γ̄_f]^* embeds H(X′), and even the Chow motive of X′, into that of X preserving the Poincaré pairing, with orthogonal complement K (the 'vanishing cycles') of dimension r − r′. The question is therefore not invariance but embedding: whether QH(X′) sits inside QH(X).

Instalment I answers this for the projective local models X = P_{P^r}(O(−1)^{⊕(r′+1)} ⊕ O) and X′ = P_{P^{r′}}(O(−1)^{⊕(r+1)} ⊕ O). Analytically continuing the Dubrovin connection along the Kähler moduli in the extremal-ray variable q^ℓ (the Novikov variable of the curve class ℓ of a line in Z) produces a connection Φ∇ with an irregular singularity at q^ℓ = ∞ exactly along the directions of K. The paper proves an eigenbundle decomposition of the tangent bundle over the localized Novikov ring R′[1/q^{ℓ′}] (here R′ = C[[NE(X′)]] is the Novikov ring of X′ and q^{ℓ′} = 1/q^ℓ is the X′-side extremal variable, so this localization is a punctured neighborhood of q^ℓ = ∞) into an irregular eigenbundle extending K and its orthogonal complement, the regular eigenbundle, both integrable; and (Theorem 0.1) it constructs an R′-point σ0(q′) and a generally non-linear embedding Ψ̂(q′,s): QH(X′) → QH(X) of F-manifolds onto the regular integrable loci near q^ℓ = ∞, deforming Ψ. The proof relies on tools special to the local models: Picard–Fuchs differential equations (the differential equations satisfied by period integrals) coming from C^×-localization, explicit Birkhoff factorization (a standard normalization of flat sections of a meromorphic connection), generalized mirror transforms (GMT, the coordinate change between A-model quantum parameters and B-model period coordinates), and semisimplicity of the local models, which are Fano (their anti-canonical class is ample). The authors emphasize that the analytically continued connection develops essential singularities in naive coordinates — they circumvent this by working in small quantum cohomology and by cyclic extensions, noting but not developing an alternative 'adic' topology — and they describe the paper as the 'first instalment' of a project aimed at global flips.

Problem Statement

Let f: X ⇢ X′ be a simple (r,r′) flip between smooth complex projective varieties (r > r′ > 0): the exceptional loci are Z ≅ P^r ⊂ X and Z′ ≅ P^{r′} ⊂ X′ with normal bundles N_{Z/X} ≅ O_{P^r}(−1)^{⊕(r′+1)} and N_{Z′/X′} ≅ O_{P^{r′}}(−1)^{⊕(r+1)}, and f is no longer assumed to be the projective local model of the flip. Establish the globalization of Theorem 0.1 of Lee–Lin–Wang's 'Quantum flips I: local model': after analytic continuation of the Dubrovin connection of QH(X) in the extremal-ray Novikov variable q^ℓ to a punctured neighborhood of q^ℓ = ∞,

(i) prove the eigenbundle decomposition of the tangent bundle over the localized Novikov ring R′[1/q^{ℓ′}] (q^{ℓ′} = 1/q^ℓ) into the irregular eigenbundle extending the vanishing-cycle subspace K ≅ C^{r−r′} and the regular integrable eigenbundle K^⊥; and

(ii) construct an R′-point σ0(q′) and a (generally non-linear) embedding Ψ̂(q′,s): QH(X′) → QH(X) in the category of F-manifolds onto the regular integrable loci of QH(X) near q^ℓ = ∞, deforming the graph-correspondence (inverse correspondence) embedding Ψ = [Γ̄_f]^* and satisfying the tangent-space compatibility that quantum products on X′ map to the corresponding restricted products on X.

The globalization is sought together with the analytic hypotheses under which it holds — in particular the convergence or adic form of the big quantum product required for a non-Fano global flip, and a systematic treatment (or a characterization of removability) of the essential singularities that the analytically continued connection Φ∇ develops in naive coordinates, which in the local models were circumvented only through small-quantum-cohomology and cyclic-extension devices. Partial formulations in the literature — the quantum D-module decomposition (the quantum D-module being the system of flat differential equations encoding the Dubrovin connection) for standard flips conjectured by Gu–Yu–Yu (Conjecture 1.8) and the extremal quantum spectrum of Shen–Shoemaker — are strictly weaker than the target: the F-manifold embedding of the full big quantum cohomology for arbitrary smooth projective simple flips.

References

  1. Yuan-Pin Lee, Hui-Wen Lin, Chin-Lung Wang, 'Quantum flips I: local model', arXiv:1912.03012; in Integrability, quantization, and geometry II. Quantum theories and algebraic geometry (Dubrovin memorial volume), Proc. Sympos. Pure Math. 103.2, Amer. Math. Soc., Providence, RI, 2021, pp. 303–352. https://arxiv.org/abs/1912.03012
  2. Yuan-Pin Lee, Hui-Wen Lin, Chin-Lung Wang, 'Flops, motives and invariance of quantum rings', Annals of Mathematics (2) 172 (2010), no. 1, 243–290, doi:10.4007/annals.2010.172.243, arXiv:math/0608370. https://arxiv.org/abs/math/0608370 https://doi.org/10.4007/annals.2010.172.243
  3. Yuan-Pin Lee, Hui-Wen Lin, Chin-Lung Wang, 'Quantum cohomology under birational maps and transitions', arXiv:1705.04799; in Proc. Sympos. Pure Math. 96, Amer. Math. Soc., Providence, RI, 2017, pp. 149–168. https://arxiv.org/abs/1705.04799
  4. Hiroshi Iritani, 'Gamma classes and quantum cohomology', arXiv:2307.15938; to appear in the ICM 2022 proceedings. https://arxiv.org/abs/2307.15938
  5. Yefeng Shen, Mark Shoemaker, 'Quantum spectrum and Gamma structure for standard flips', arXiv:2502.08762. https://arxiv.org/abs/2502.08762
  6. Zhaoxing Gu, Song Yu, Tony Yue Yu, 'Quantum cohomology of variations of GIT quotients and flips', arXiv:2508.15770. https://arxiv.org/abs/2508.15770
  7. Tsung-Chen Chen, Hui-Wen Lin, Sz-Sheng Wang, 'Fiber products under toric flops and flips', arXiv:2411.12446; accepted at Mathematische Zeitschrift. https://arxiv.org/abs/2411.12446
  8. Tsung-Chen Chen, Jia-Hua Chong, Hui-Wen Lin, 'Quantum invariance under non-smooth toric flops of the simplest type', arXiv:2608.24066. https://arxiv.org/abs/2608.24066