Directed double-point degeneration: an algebraic-cobordism route to quantum-ring invariance under ordinary flops (Question 8.3.1 of Lee–Lin–Qu–Wang)

Lee, Lin, Qu and Wang proved that for a general ordinary $\mathbb{P}^r$-flop $f: X \dashrightarrow X'$ the correspondence $\mathscr{F} = [\bar{\Gamma}_f]_*$ induced by the closure of the flop graph identifies the big quantum cohomology rings of $X$ and $X'$ after analytic continuation in the extremal Novikov variables (genus zero, primary insertions). The proof reduces arbitrary bundle data to split bundles through a long chain of blow-ups and deformation arguments. In Section 8.3 the authors observe that a far shorter 'dream proof' would follow from an algebraic-cobordism classification: Lee–Pandharipande proved that, up to double-point degeneration, any list of vector bundles is equivalent to a rational combination of split bundles on products of projective spaces, whose Gromov–Witten theory is toric and semisimple. The obstacle is that flop-invariance of Gromov–Witten generating functions is only known to be preserved by forward degenerations and by backward deformation to the normal cone, not by inversion of a general double-point degeneration. The open question — Question 8.3.1, unanswered in the literature as of August 2026 — asks whether the cobordism classification can nevertheless be obtained from the directed double-point degeneration relation generated by those two moves. A positive answer would simplify the proof of flop invariance and open a route to all genera; a negative answer consists of a proof that some triple cannot be reduced to split toric models under the directed moves. Note also that naive descendant invariance is refuted (counterexamples appear in the authors' Annals of Mathematics 2010 paper), so the descendant question is context rather than an open extension.

Background

Two smooth projective varieties connected by flops have the same Betti numbers but in general different cup-product structures on cohomology; since the minimal model program produces such pairs abundantly, one wants to know which enumerative invariants are nonetheless preserved. An ordinary $\mathbb{P}^r$-flop is a birational map $f: X \dashrightarrow X'$ between smooth projective varieties whose exceptional loci are projective bundles $Z = \mathbb{P}_S(F)$ and $Z' = \mathbb{P}_S(F')$ over a common smooth base variety $S$, where $F$ and $F'$ are vector bundles of rank $r+1$ on $S$ with $\mathbb{P}_S(F) \cong \mathbb{P}_S(F')$; the flop is the blow-up of $X$ along $Z$ followed by contraction of the exceptional divisor along the other ruling. The K-equivalence conjecture predicts that Gromov–Witten (GW) theory — the genus-graded system of invariants counting stable maps — is invariant under such crepant birational maps (birational maps under which the canonical class pulls back trivially), in a suitable sense. Let $\mathscr{F} = [\bar{\Gamma}_f]_*$ be the correspondence on cohomology induced by the closure of the graph of $f$, extended to quantum variables by $\mathscr{F}(q^\beta) = q^{\mathscr{F}(\beta)}$, where $q^\beta$ ranges over Novikov variables (one formal variable per effective curve class, ensuring convergence of GW generating functions). Then $\mathscr{F}$ preserves the Poincaré pairing but not the cup product, and it maps the class of a flopping curve to a non-effective class; both defects are cured simultaneously by quantum corrections together with analytic continuation in the Novikov variable $q^\ell$ of the extremal ray. In the series 'Invariance of quantum rings under ordinary flops I–III' (with the 2010 Annals of Mathematics paper as forerunner), Lee–Lin–(Qu)–Wang proved Theorem 0.1.1: for every general ordinary flop, $\mathscr{F}$ induces an isomorphism of big quantum cohomology rings (cohomology rings deformed by genus-zero three-point GW invariants) after analytic continuation in the extremal Novikov variables, and similarly for relative primary invariants and ancestors. Here 'ancestors' denote insertions that behave well under the degeneration formula, unlike 'descendants' (insertions of Chern classes of the tautological cotangent line bundles at the marked points), for which the corresponding statement is false. The proof reduces to local models built functorially from the triple $(S, F, F')$ — toric double projective bundles over $S$ — then applies a quantum Leray–Hirsch theorem for split bundles (Part II) and a quantum splitting principle removing the splitting hypothesis (Part III), via a lengthy sequence of blow-ups of the base and deformation arguments.

Algebraic cobordism suggests a potential shortcut. Levine–Pandharipande showed that the algebraic cobordism ring — the universal oriented cohomology theory on smooth projective varieties, generalizing the Chow ring — is governed by double-point degenerations: one-parameter families whose general fiber is a smooth projective variety $S$ and whose special fiber is a normal-crossing union $S_0 \cup_D S_1$ of smooth varieties glued along a smooth divisor $D$. Lee–Pandharipande extended this to bundles in 'Algebraic cobordism of bundles on varieties': Theorem 1 there states that, modulo double-point degeneration relations, any list of vector bundles on a smooth projective variety is equivalent to a rational combination of split bundles (direct sums of line bundles) over products of projective spaces. Such split models are toric; their genus-zero GW theory is explicitly computable, their quantum cohomology is generically semisimple (the quantum product diagonalizes over a Zariski-open set of parameters), and semisimple genus-zero data determine the entire higher-genus theory by Givental's quantization formalism.

Section 8.3 of Part III observes that a 'dream proof' of Theorem 0.1.1 would combine this classification with flop-invariance bookkeeping: reduce an arbitrary triple $(S, F, F')$ to split toric triples, where everything is computable. The obstacle is that invariance of GW generating functions is only known to be preserved under 'forward' degenerations (passing from a variety to the components of a double-point fiber; Section 7.1) and under inversion of deformations to the normal cone (Section 7.4): the DNC degeneration replaces $S$ by $S \cup_D \mathbb{P}_D(N_{D/S} \oplus \mathcal{O})$, where $\mathbb{P}_D(N_{D/S} \oplus \mathcal{O})$ is the projectivization of the normal bundle of a smooth divisor $D \subset S$ together with a trivial summand (a $\mathbb{P}^1$-bundle over $D$), and this particular degeneration can be run backward. Inversion for a general double-point degeneration is not known. This motivates the directed (the authors say non-reflexive) double-point degeneration relation on triples: the relation generated only by forward degenerations and backward DNC, each move used as a one-way substitution rather than as an equation.

Problem Statement

Decide Question 8.3.1 of Lee–Lin–Qu–Wang, 'Invariance of quantum rings under ordinary flops III' (arXiv:1401.7097; Cambridge J. Math. 4 (2016) 333–401). Consider triples $(S, F, F')$ consisting of a smooth projective variety $S$ over $\mathbb{C}$ and two vector bundles $F, F'$ of equal rank on $S$, together with formal $\mathbb{Q}$-linear combinations of such triples. Define the directed double-point degeneration relation to be the directed reachability relation on these formal combinations generated by the following two moves, used only as substitutions in the stated directions:

(i) Forward double-point degeneration — the Lee–Pandharipande double-point relation (Definition 3 of 'Algebraic cobordism of bundles on varieties', arXiv:1002.1500) used in the degeneration direction. Given a projective morphism $\pi: Y \to \mathbb{P}^1$ with $Y$ smooth of pure dimension, special fiber $\pi^{-1}(0) = A \cup B$ a transverse union of smooth codimension-one components with double point locus $D = A \cap B$, and smooth general fiber $Y_\zeta = \pi^{-1}(\zeta)$, replace $[Y_\zeta, F|_{Y_\zeta}, F'|_{Y_\zeta}] \;\longmapsto\; [A, F|_A, F'|_A] + [B, F|_B, F'|_B] - [P(\pi), F|_{P(\pi)}, F'|_{P(\pi)}],$ where $P(\pi) = \mathbb{P}(\mathcal{O}_D \oplus N_{A/D}) \cong \mathbb{P}(\mathcal{O}_D \oplus N_{B/D}) \to D$ is the $\mathbb{P}^1$-bundle over the double point locus, the bundle data on $A$ and $B$ are obtained by restriction, and those on $P(\pi)$ by pullback along $P(\pi) \to D \subset Y$. The correction term $-[P(\pi), \ldots]$ over the double point locus is part of every forward move.

(ii) Backward deformation to the normal cone (backward DNC) — the exact reverse of move (i) along a deformation to the normal cone $\mathrm{Bl}_{Z \times \{0\}}(S \times \mathbb{A}^1) \to \mathbb{A}^1$ with smooth center $Z \subset S$, whose singular fiber is a union $\tilde{S} \cup_E \mathbb{P}_Z(N_{Z/S} \oplus \mathcal{O})$ of two smooth components, where $\tilde{S} = \mathrm{Bl}_Z S$ and $E = \mathbb{P}_Z(N_{Z/S})$ is their common divisor: the singular-fiber data are replaced by the general-fiber data, retaining the double-point correction term of (i). Given bundles $(F, F')$ on $S$ and the natural morphisms $\phi: \tilde{S} \to S, \qquad p: \mathbb{P}_Z(N_{Z/S} \oplus \mathcal{O}) \to Z \subset S, \qquad r: P(\pi) \to E \subset S,$ replace $[\tilde{S}, \phi^*F, \phi^*F'] + [\mathbb{P}_Z(N_{Z/S} \oplus \mathcal{O}), p^*F, p^*F'] - [P(\pi), r^*F, r^*F'] \;\longmapsto\; [S, F, F'],$ where the bundle data on $\tilde{S}$ and on $\mathbb{P}_Z(N_{Z/S} \oplus \mathcal{O})$ are obtained by pullback along $\phi$ and $p$, and $P(\pi) = \mathbb{P}(\mathcal{O}_E \oplus N_{\tilde{S}/E}) \to E$ is the same $\mathbb{P}^1$-bundle correction term over the double point locus $E$ that occurs in (i) (bundle data on $P(\pi)$ by pullback along $r$). DNC degenerations are special instances of double-point degenerations; this is the inversion justified for Gromov–Witten generating functions in Sections 7.1 and 7.4 of Part III.

The directed semantics is essential: the moves are one-way substitutions inside formal $\mathbb{Q}$-combinations, not a symmetric closure and not equations in a group quotient. If the moves were instead imposed as equations, the resulting quotient would coincide with the symmetric bundle-cobordism group $\omega_{n,r} \otimes \mathbb{Q}$ of Lee–Pandharipande, and the classification below would follow formally from their Theorem 1; the open content of Question 8.3.1 is the directed reading.

Question: can one prove an analogue of Theorem 1 of Lee–Pandharipande ('Algebraic cobordism of bundles on varieties', JEMS 14 (2012)) with the usual symmetric double-point degeneration relations replaced by this directed relation? Concretely: is every triple $(S, F, F')$ reachable, via finitely many directed moves, as a rational combination of split triples $(\prod_{i=1}^{m} \mathbb{P}^{n_i}, \bigoplus_j \mathcal{O}(k_{1j}, \ldots, k_{mj}), \bigoplus_j \mathcal{O}(k'_{1j}, \ldots, k'_{mj}))$ — where $\mathcal{O}(k_{1j}, \ldots, k_{mj})$ denotes the line bundle of multidegree $(k_{1j}, \ldots, k_{mj})$ on the product — consisting of direct sums of line bundles over products of projective spaces? A positive answer supplies the algebraic-cobordism 'dream proof' of flop invariance (Theorem 0.1.1 of Part III) and, through semisimplicity of the toric split models and Givental's quantization formalism, opens a route to $\mathscr{F}$-invariance of Gromov–Witten theory under general ordinary flops in all genera. A negative answer consists of a proof, by whatever rigorous means, that some triple $(S, F, F')$ reaches no $\mathbb{Q}$-linear combination of split toric triples under the directed moves. The problem is answered by establishing the directed classification theorem or by proving such a non-reachability result; a restatement or reproof of the genus-zero flop theorem alone, without the directed degeneration theory, does not answer it.

References

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