Invariance and decomposition of quantum cohomology under flips and birational maps between minimal models

The minimal model program (MMP) relates smooth projective varieties by birational transformations — flops and flips — and a two-decade program has established that quantum cohomology behaves well under the K-trivial ones: big quantum cohomology rings are invariant under ordinary flops (Lee–Lin–Wang, completed with Qu in 2016), and the small quantum cohomology algebras of any two birational projective Calabi–Yau manifolds are isomorphic after a change of Novikov rings (McLean, 2020). The open problem is the remaining, general part of the question 'to what extent is quantum cohomology invariant under the birational transformations of the MMP?': (i) for flips, where the two sides are not K-equivalent, the expected phenomenon is a decomposition of the quantum D-module into that of the flipped variety plus copies of that of the center — conjectured by Gu–Yu–Yu (2025) for all standard flips but proved only for local models and VGIT-realizable flips; (ii) globalization of Lee–Lin–Wang's F-embedding of big quantum cohomology from local models of simple flips to global flips; and (iii) Ruan's 1999 conjecture that any two birational Calabi–Yau manifolds have isomorphic big quantum cohomology rings up to analytic continuation. As of August 2026 no resolution of any of these is known.

Background

The minimal model program (MMP) classifies smooth projective varieties up to birational equivalence by surgically modifying them along rational curves: a sequence of birational operations — divisorial contractions, flips, and flops — is expected to terminate at a minimal model (or a Mori fiber space). A flip contracts curves spanning an extremal ray of the Mori cone on which the canonical class is negative and replaces them by other curves; a flop is the K-trivial cousin, in which the canonical class vanishes on the contracted curves. Since each MMP step replaces a variety by a birationally equivalent one, one can ask which invariants of smooth projective varieties survive the whole program.

Quantum cohomology is a deformation of ordinary cohomology built from Gromov–Witten invariants — integrals over moduli spaces of stable maps, which count rational curves in a fixed homology class numerically. The small quantum cohomology $QH^*(X)$ of a smooth projective variety $X$ is a commutative algebra over the Novikov ring $\Lambda$ (a formal completion of the group ring of $H_2(X;\mathbb{Z})$ whose monomials $Q^\beta$ record curve classes $\beta$ and whose valuation encodes growth against a Kähler class), deforming the cup product by rational-curve contributions in each class $\beta$. The big quantum cohomology is the family of products obtained when the deformation parameter is allowed to be an arbitrary class $\tau \in H^*(X)$; it forms a germ of a Frobenius (in particular F-)manifold structure, whose potential satisfies the associativity equations of enumerative geometry. The quantum D-module $QDM(X)$ repackages the same data as a flat connection (the quantum connection) on the trivial bundle with fiber $H^*(X)$, and is the object through which mirror symmetry and the Riemann–Hilbert correspondence enter. Quantum cohomology is a central structure of enumerative geometry, and its behavior under birational maps is governed by the crepant transformation conjecture (Ruan 1999 and successors), which predicts that for crepant (K-preserving) birational maps the two sides' quantum theories agree after analytic continuation in the complexified Kähler moduli space — the space in which the Novikov variables $Q$ are complexified and continued along paths.

For flops this prediction is now largely a theorem. Li–Ruan (2001) proved that birational Calabi–Yau threefolds related by a flop have identical small quantum cohomology rings. Lee–Lin–Wang ('Flops, motives and invariance of quantum rings', Annals 2010; series completed with Qu, 2016) proved that the correspondence defined by the closure of the graph of an ordinary flop — a flop whose two exceptional loci are projective bundles $\mathbb{P}(V_\pm) \to S$ of equal rank over a common smooth base $S$ — identifies the big quantum cohomology rings after analytic continuation, over an arbitrary smooth base. Beyond ordinary flops the results are case-by-case: stratified Mukai flops — flops whose exceptional loci carry Grassmannian stratifications — (Fu–Wang 2008), Grassmannian flops (2024), and non-smooth toric flops of the simplest type (2026). In the Calabi–Yau setting, McLean (Annals 2020) proved, via symplectic cohomology (a Hamiltonian-Floer-theoretic invariant of open symplectic manifolds), that any two birational projective Calabi–Yau manifolds — connected by arbitrary sequences of flops of unrestricted type, by Kawamata's theorem — have isomorphic small quantum cohomology algebras after a change of Novikov rings; he states explicitly that this concerns only the small algebras, with non-explicit isomorphisms.

Flips are the unexplored side, and there invariance cannot hold: the two sides are not K-equivalent (their canonical classes differ on the contracted curves), so even the classical cohomology rings need not agree. The expected phenomenon, modeled on Iritani's blowup theorem — the quantum D-module of a blowup of $X$ along a smooth center $Z$ decomposes into $QDM(X)$ plus copies of $QDM(Z)$ — is decomposition rather than isomorphism. Gu–Yu–Yu (2025) proved such a decomposition for every simple variation-of-GIT (VGIT) wall-crossing — pairs of GIT quotients of a common variety separated by one wall, with the wall's quotient $S$ contributing residue factors — and, as an application, for the local models of standard flips. A standard flip of type $(r_+-1, r_- - 1)$ is one whose exceptional loci are projective bundles $\mathbb{P}(V_\pm) \to S$ of ranks $r_\pm$ over a common smooth center $S$ with $r_+ \le r_-$; the case $r_+ = r_-$ is an ordinary flop, and $r_+ < r_-$ is a genuine flip, where the predicted wall multiplicity is $r_- - r_+$ copies of the wall's quantum D-module. The globalization of this statement — from local models (and from VGIT-realizable flips) to all standard flips, and the analogous globalization of Lee–Lin–Wang's F-embedding for big quantum cohomology — is the precise open core, together with Ruan's big-quantum-ring conjecture for birational Calabi–Yau manifolds. A survey of the whole program is Lee–Lin–Wang, 'Quantum Cohomology under Birational Maps and Transitions' (2017).

Problem Statement

The overarching question is: to what extent is quantum cohomology — small algebra, big ring, or quantum D-module — invariant, or decomposable, under the birational transformations of the minimal model program, beyond the settled case of ordinary flops? The common objective uniting the parts below is to complete the flop-invariance picture for all MMP birational links between minimal models: flops preserve (the settled template), flips must decompose, and arbitrary birational maps between Calabi–Yau minimal models should factor through these two behaviors.

The three parts below are independent, each a self-contained prove-or-refute question.

(1) Global flip decomposition. Let $f: X^- \dashrightarrow X^+$ be a standard flip between smooth projective varieties of type $(r_+ - 1, r_- - 1)$ with $r_+ < r_-$, i.e. the exceptional loci are projective bundles $\mathbb{P}(V_\pm) \to S$ of ranks $r_\pm$ over a common smooth center $S$. Prove or refute Gu–Yu–Yu's Conjecture 1.8 (arXiv:2508.15770): after formal changes of variables $\tau_{X^+}$ and $\zeta_j$ and base change to the ring $\mathbb{C}[z]\big(\big(Q^{a/2(r_- - r_+)}\big)\big)\llbracket Q^{X^-}, \tau_{X^-}\rrbracket$ — where $z$ is the quantum D-module parameter and $Q^a$ is the Novikov variable for the fiber curve class in the exceptional locus of $f$ in $X^-$ — there is an isomorphism of quantum D-modules

$QDM(X^-)^{\mathrm{La,red}} \;\longrightarrow\; (\tau_{X^+})^* QDM(X^+)^{\mathrm{La,red}} \;\oplus\; \bigoplus_{j=0}^{r_- - r_+ - 1} (\zeta_j)^* QDM(S)^{\mathrm{La,red}},$

compatible with the quantum connections and the Poincaré pairings (the superscript $\mathrm{La,red}$ denotes this base change, and the precise decorated statement of the modules and the base-change ring is Conjecture 1.8 of the cited preprint), inducing the corresponding decomposition of the big quantum cohomology ring of $X^-$ into that of $X^+$ and $r_- - r_+$ factors of $S$ over the same base-change ring. (Currently proved only for the projective local models of such flips and for flips satisfying the VGIT-realizability condition of Corollary 6.12 of the cited preprint.)

(2) Globalization of quantum flips. Lee–Lin–Wang constructed, for local models of simple flips, a deformation of the inverse graph correspondence inducing a nonlinear F-embedding of big quantum cohomology $QH(X') \hookrightarrow QH(X)$ near $q^\ell = \infty$ ('Quantum flips I', PSPM 2021). Establish or refute the corresponding statement for global (projective) standard flips (the simple case $S = \mathrm{pt}$ included).

(3) Big quantum cohomology of birational Calabi–Yau manifolds (Ruan's conjecture, 1999). Prove or refute: any two birational projective Calabi–Yau manifolds have isomorphic big quantum cohomology rings up to analytic continuation over the Kähler moduli space. (Known when the connecting map is a sequence of ordinary flops; McLean's Annals 2020 theorem covers only the small quantum algebras, non-explicitly.)

References

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