The Ruan–Wang K-equivalence conjecture — advanced by Yongbin Ruan and Chin-Lung Wang and recorded as Conjecture 0.4 of Lee–Lin–Wang's Annals paper — predicts that K-equivalent smooth projective varieties, i.e. varieties related by a birational map that identifies their canonical classes (flops being the elementary examples), have canonically isomorphic quantum cohomology rings over the extended Kähler moduli spaces, the identification being induced by the flop correspondence after analytic continuation in the quantum parameters; its natural extension predicts canonical identification of the full Gromov–Witten theories in all genera. The open core that remains after Lee–Lin–Wang (2010) proved genus-zero invariance of quantum rings for simple ordinary flops is, as of August 2026: (i) canonical identification of the all-genus ancestor GW potentials for ordinary flops with arbitrary smooth base — the all-genus statement is known only for simple flops over a point (Iwao–Lee–Lin–Wang 2012), while genus zero is complete for all ordinary flops (Lee–Lin–Wang 2016, I–III); (ii) the conjecture for general K-equivalences, for which a decomposition of a general K-equivalence into elementary ordinary/Mukai-type flops between smooth varieties is the classical sufficient route — Kawamata's flop-connectivity theorem connects minimal models by general (possibly singular, non-elementary) flops in every dimension, but this does not supply the elementary decomposition that route would use, and no symplectic-deformation version exists in dimension at least four; and (iii) global convergence and chamber-wise analytic continuation of the quantum potential on the extended Kähler moduli for general targets. Note that the source paper's Theorem 6.3 already proves all-genus invariance for (twisted, global) Mukai flops, so that item is not an open GW problem.

Background

Gromov–Witten (GW) invariants count stable maps from marked nodal curves into a smooth projective variety XX and are assembled into generating functions: the descendant potential, whose coefficients are invariants with descendant (ψ\psi-class, i.e. cotangent-line) insertions, and the ancestor potential, built from ancestor (ψˉ\bar\psi-class) insertions obtained by pulling the cotangent-line classes back through the map that forgets the map to XX and stabilizes the curve. The genus-zero, primary-insertion part of these potentials encodes the quantum cohomology ring QH∗(X)QH^*(X), a deformation of the classical cohomology ring whose product is defined by 3-point GW invariants. The curve classes β\beta are bookkept by Novikov variables qβq^\beta (one formal variable per effective curve class, weighted by symplectic area), introduced precisely to sidestep convergence questions; a basic analytic issue is whether the resulting series converge and extend holomorphically over the extended Kähler moduli, the analytic space of complexified Kähler parameters obtained by exponentiating the Kähler cone.

Two smooth projective varieties related by a birational map f:X⇢X′f:X\dashrightarrow X' that is an isomorphism in codimension one and identifies KXK_X with f∗KX′f^*K_{X'} under this isomorphism are called KK-equivalent; flops — the KK-trivial birational operations produced by the minimal model program — are the elementary examples, and different crepant resolutions of the same singularity are KK-equivalent to one another, so the situation below contains the Crepant Resolution Conjecture (CRC), the prediction that the GW theories of crepant resolutions of the same target are identified after analytic continuation and a linear change of variables, as a special case. Two local models matter. An ordinary (or Pr\mathbb P^r-) flop is the birational map arising from a flopping contraction ψ:X→Xˉ\psi:X\to\bar X whose exceptional locus ZZ is a Pr\mathbb P^r-bundle Z=PS(F)→SZ=\mathbb P_S(F)\to S over a smooth base SS, with normal bundle restricting on each fiber Zs≅PrZ_s\cong\mathbb P^r to OPr(−1)⊕(r+1)\mathcal O_{\mathbb P^r}(-1)^{\oplus(r+1)}; globally NZ/X≅OP(F)(−1)⊗ψˉ∗F′N_{Z/X}\cong\mathcal O_{\mathbb P(F)}(-1)\otimes\bar\psi^*F' for another rank-(r+1)(r+1) bundle F′F' on SS, and the flop replaces Z=PS(F)Z=\mathbb P_S(F) by Z′=PS(F′)Z'=\mathbb P_S(F') (blow up XX along ZZ and contract the exceptional divisor along the other fiber direction). The flop is called simple when SS is a point. A Mukai flop instead has NZ/X≅TZ/S∗⊗ψˉ∗LN_{Z/X}\cong T^*_{Z/S}\otimes\bar\psi^*\mathcal L, the relative cotangent bundle twisted by a line bundle from the base, so that the codimension of ZZ equals the fiber dimension and the flop replaces the fibers by their dual projective spaces. Because KK-equivalences preserve the canonical class, classical topological and Hodge-theoretic invariants largely agree across XX and X′X', which motivated Ruan's proposal that the quantum theories should agree as well.

An-Min Li and Yongbin Ruan (2001) proved the first case: for smooth Calabi–Yau 3-folds, GW invariants are invariant under flops, by symplectic surgery and degeneration. Their strategy had two ingredients — (1) decompose a general KK-equivalence, by symplectic deformation, into a composite of elementary ordinary P1\mathbb P^1 flops, and (2) prove invariance for each elementary flop. Ingredient (1) has never been extended beyond dimension three in a form usable for GW theory: Kawamata's flop-connectivity theorem (2008) does connect birationally equivalent minimal models by a sequence of flops in every dimension, but its flops are general crepant flips whose intermediate varieties may be singular and which are not the elementary ordinary/Mukai-type flops between smooth varieties that the local GW computations require. Ruan and Chin-Lung Wang then advanced the general conjecture, recorded as Conjecture 0.4 of Lee–Lin–Wang's Annals paper: K-equivalent smooth varieties have canonically isomorphic quantum cohomology rings over the extended Kähler moduli spaces — the identification induced by the correspondence defined by the closure of the graph of the birational map, acting on products of moduli spaces of stable maps, after analytic continuation in the quantum parameters. The conjecture is expected to extend to the full GW theory in all genera; the paper proves the genus-zero quantum-ring statement for simple ordinary flops and explicitly defers higher genus.

Yuan-Pin Lee, Hui-Wen Lin, and Chin-Lung Wang (2010) proved that genus-zero statement by degenerating to the local model W=P(NZ/X⊕OZ)W=\mathbb P(N_{Z/X}\oplus\mathcal O_Z), computing the extremal quantum corrections, and matching the two sides after explicit analytic continuation; the same paper (Theorem 6.3) proved all-genus invariance for twisted global Mukai flops, where the quantum corrections vanish identically. Iwao–Lee–Lin–Wang (2012) then established invariance in all genera for simple flops (S=ptS=\mathrm{pt}) at the level of ancestor potentials, again after analytic continuation. Lee–Lin–Wang (2016), Parts I–III with Feng Qu, completed genus zero for ordinary flops with arbitrary base via quantum corrections and reduction to local models, a quantum Leray–Hirsch theorem (expressing the quantum cohomology of the total space through a basis lifted from the base), and a quantum splitting principle. The surrounding program remains active only in genus zero: quantum flips — the analogue for flipping contractions, which change the canonical class — beyond KK-equivalence, for the local model (Lee–Lin–Wang, PSPM 2021); non-smooth toric flops of the simplest orbifold type (Chen–Chong–Lin 2026); and descendant and Fourier–Mukai equivalences (i.e. equivalences of derived categories induced by a kernel correspondence) for simple flops (Chen–Tseng 2026). On the analytic side, convergence of GW potentials is known for targets whose quantum cohomology is analytic and generically semisimple — including toric varieties and complete flag varieties (Coates–Iritani 2015) — and, conditionally on base convergence, for toric bundles (Koto 2022); no general smooth projective result exists. What no later work supplies is the all-genus statement beyond simple flops, any decomposition of a general KK-equivalence into elementary flops in dimension at least four, or the general convergence of the potential over the extended Kähler moduli.

Problem Statement

Prove or refute the K-equivalence conjecture for Gromov–Witten theory — advanced by Yongbin Ruan and Chin-Lung Wang and recorded as Conjecture 0.4 of Lee–Lin–Wang (2010), together with its natural all-genus extension. Let f:X⇢X′f:X\dashrightarrow X' be a KK-equivalence between smooth projective varieties, i.e. a birational map that is an isomorphism in codimension one and identifies KXK_X with f∗KX′f^*K_{X'} under this isomorphism (compositions of flops being the typical case). Conjecture 0.4 asserts that such XX and X′X' have canonically isomorphic quantum cohomology rings over the extended Kähler moduli spaces, the identification being induced by the correspondence defined by the closure of the graph of ff on products of moduli spaces of stable maps and achieved after analytic continuation in the quantum parameters; the all-genus extension predicts the same canonical identification for the full GW theory in every genus. Concretely, the open core as positioned by the 2010–2026 literature consists of three linked faces of this single objective:

(i) All genera, general base: establish the canonical identification of the all-genus ancestor potentials of the two sides of an ordinary flop with arbitrary smooth base SS and bundle data, after chamber-wise analytic continuation — extending Iwao–Lee–Lin–Wang's all-genus theorem for simple flops (S=ptS=\mathrm{pt}) to the generality where genus zero is already settled by Lee–Lin–Wang I–III.

(ii) General KK-equivalences: establish the conjecture for arbitrary smooth projective KK-equivalent varieties. A decomposition of a general KK-equivalence into a composition of elementary ordinary- or Mukai-type flops between smooth varieties — the higher-dimensional analogue of ingredient (1) of the Li–Ruan (2001) strategy, called desirable in Remark 0.9 of the source paper — would be a sufficient route to this face, but is not logically necessary: a direct proof, or an explicit counterexample, settles the face without any decomposition theorem. Kawamata's flop-connectivity theorem already connects birationally equivalent minimal models by a sequence of flops in every dimension, but its flops may be singular and non-elementary, so it does not by itself supply such a decomposition.

(iii) Global analytic structure: for general smooth projective XX, prove that the genus-zero three-point series Φijk\Phi_{ijk} — and, in concert with (i), the full potential in all genera — converges for B∈KXB\in\mathcal K_X, the Kähler cone of XX (at least for BB sufficiently large) and analytically continues between the Kähler chambers of KK-equivalent models (the Kähler cones of the various models form disjoint chambers in the space of complexified Kähler parameters) over the entire extended Kähler moduli, as conjectured in Remark 3.3 of the source paper; this includes a systematic treatment of the term-by-term analytic continuations of the Novikov-variable transformation in general relative splittings.

A refutation of any face — e.g. an explicit KK-equivalent pair whose genus-gg invariants provably cannot be matched by any analytic continuation of the kind prescribed by Conjecture 0.4, i.e. one induced by the closure-of-graph correspondence, continued chamber-wise over the extended Kähler moduli with functorial Novikov-variable transformations — equally settles the corresponding part of the problem. Invariance for Mukai flops is not part of the open problem: Theorem 6.3 of the source paper already proves all-genus invariance for (twisted, global) Mukai flops.

References

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