Gong's two-point Gromov-Witten cup-length conjecture for degenerate symplectic fixed points (Conjecture 1.7 of arXiv:2507.04191)

Conjecture 1.7 of Wenmin Gong's preprint "Degenerate symplectic fixed points and Gromov-Witten invariants" (arXiv:2507.04191) predicts a uniform cup-length lower bound for fixed points of Hamiltonian diffeomorphisms that is powered by enumerative geometry: on a closed rational symplectic manifold with minimal symplectic period $p$ that carries a nonzero genus-zero Gromov-Witten invariant with two point insertions, every Hamiltonian diffeomorphism $\varphi$ should have at least $\lceil p\cdot\operatorname{cuplength}(M;\mathbb{C})/\Theta(M,\omega)\rceil$ fixed points, where $\Theta(M,\omega)$ is the infimum of the symplectic areas $\langle[\omega],A\rangle$ over classes $A$ with such a nonzero invariant. In particular, the degenerate Arnol'd conjecture over the complex numbers would follow for every such manifold, a class expected to include all symplectic rationally connected manifolds. The claim is stated as an open conjecture in the latest revision of the preprint.

Background

The Arnol'd conjecture is a foundational problem of symplectic topology. A Hamiltonian diffeomorphism $\varphi$ of a closed symplectic manifold $(M,\omega)$ — the time-one map of the flow of a time-dependent Hamiltonian function, the natural class of transformations of classical mechanics that preserve $\omega$ — is predicted to have at least as many fixed points as a Morse-theoretic lower bound for $M$. When all fixed points are nondegenerate (the linearization of $\varphi$ at each fixed point has no eigenvalue 1) the expected bound is the sum of the Betti numbers; in the general degenerate case the expected bound is the cuplength $\operatorname{cuplength}(M;\mathbb{C})$, defined as the maximal number $k+1$ such that there exist $k$ cohomology classes $a_1,\dots,a_k\in H^*(M;\mathbb{C})$ of positive degree, $\deg(a_i)>0$, satisfying $a_1\cup\cdots\cup a_k\neq 0$ (with this convention $\operatorname{cuplength}(\mathbb{CP}^n;\mathbb{C})=n+1$). The degenerate form is proved only in special classes: the standard torus $T^{2n}$ (Conley–Zehnder), manifolds with $\pi_2(M)=0$ (Floer, Hofer), complex projective spaces (Fortune's theorem), negatively monotone manifolds whose minimal Chern number is at least the complex dimension (Lê and Ono), and closed symplectically aspherical manifolds — those on which $[\omega]$ and the first Chern class $c_1$ vanish on $\pi_2(M)$ — for which Rudyak and Oprea obtained the most general answer; per Gong's own account, $(\mathbb{CP}^n,\omega_{FS})$ is the only monotone symplectic manifold for which the degenerate homological Arnol'd conjecture has been proven. Schwarz also proved the conjecture over $\mathbb{Z}/2$ for Hamiltonian diffeomorphisms of Hofer norm (the $L^{\infty}$-type energy of the generating Hamiltonian) smaller than the minimal symplectic period. For general closed symplectic manifolds the degenerate conjecture remains open.

Gong's approach converts Gromov-Witten invariants into fixed-point bounds. A genus-zero Gromov-Witten invariant $GW_{0,k+3,A}([pt],a_0,[pt],a_1,\dots,a_k)$ virtually counts rational (genus-zero) holomorphic stable curves in a homology class $A\in H_2(M;\mathbb{Z})/\text{torsion}$ that pass through two generic points and through cycles Poincaré-dual to the classes $a_0,\dots,a_k$. The manifold is assumed rational, meaning $\langle[\omega],\pi_2(M)\rangle=p\mathbb{Z}$ for a positive generator $p$, the minimal symplectic period (the smallest positive symplectic area of a sphere class); any nonzero invariant with a point insertion forces $\langle[\omega],A\rangle>0$, so the quantity $\Theta(M,\omega):=\inf\{\langle[\omega],A\rangle : GW_{0,k+3,A}([pt],a_0,[pt],a_1,\dots,a_k)\neq 0\}$ satisfies $\Theta\ge p>0$.

The machinery of the source paper extends Schwarz's quantum cup-length estimate through big quantum homology — the quantum product on $H^*(M;\Lambda)$ deformed by a bulk class $\zeta$ in the even-degree part of the Novikov ring $\Lambda_0$ of formal power series tracking symplectic areas — and $\zeta$-deformed spectral invariants of Hamiltonian Floer theory (minimax values of the Floer action functional, deformed by the same bulk class, in the spirit of Usher's deformed Hamiltonian Floer theory), connected to ordinary homology by deformed PSS isomorphisms (the standard chain maps between the Morse complex of a function and the Floer complex of its Hamiltonian). The paper's Theorems 1.1-1.6 turn nonvanishing of Gromov-Witten data into lower bounds for $\#\operatorname{Fix}(\varphi)$, but each falls short of the plain cup-length bound under the bare two-point hypothesis: Theorem 1.1 requires a minimizing-area condition (the relevant class $A$ realizes the infimum among classes in related mixed invariants), which the author calls technical and expects to be removable; Theorem 1.6 requires the manifold to be point invertible of order $g$ (in the small quantum homology ring with product $\ast$, there is a class $\beta$ of degree below the top with $[pt]\ast\beta=T^{kp}[M]$ for some $k\in\mathbb{N}$, where $T$ is the Novikov variable and $[M]$ the fundamental class — a condition enjoyed by $\mathbb{CP}^n$ and the quadrics and stable under products) together with minimal Chern number $N\ge n+1$ ($N$ being the positive generator of $\langle c_1,\pi_2(M)\rangle$ for a monotone manifold), and yields $\#\operatorname{Fix}(\varphi)\ge\lceil\operatorname{cuplength}(M;\mathbb{C})/g\rceil$.

Conjecture 1.7 asserts that the two-point-insertion hypothesis alone suffices, with no minimizing-area, semipositivity, or point-invertibility condition. Its premise is itself a major question: a folklore conjecture asserts that every rationally connected projective manifold (a complex variety in which any two general points can be joined by a rational curve) carries a nonzero genus-zero Gromov-Witten invariant with two point insertions; Zhiyu Tian proved that rationally connected projective threefolds that are Fano or have second Betti number 2 are symplectically rationally connected and constructed further symplectic rationally connected fourfolds (including all Fano fourfolds of pseudo-index at least 2), and Pieloch proved that enumeratively rationally connected closed symplectic manifolds — those carrying a nonzero two-point-insertion genus-zero Gromov-Witten invariant — have finite fundamental group. The conjecture is stated conditionally, so it can be pursued on any class of manifolds where the premise is verified.

Problem Statement

Let $(M^{2n},\omega)$ be a closed rational symplectic manifold with minimal symplectic period $p>0$, i.e. $\langle[\omega],\pi_2(M)\rangle=p\mathbb{Z}$. Assume that $M$ admits a nonzero genus-zero Gromov-Witten invariant with two point insertions,

$$GW_{0,k+3,A}\big([pt],a_0,[pt],a_1,\dots,a_k\big)\neq 0,$$

for some $A\in H_2(M;\mathbb{Z})/\text{torsion}$ and rational homology classes $a_0,\dots,a_k$ of even degree, and define $\Theta(M,\omega):=\inf\{\langle[\omega],A\rangle : GW_{0,k+3,A}([pt],a_0,[pt],a_1,\dots,a_k)\neq 0\}$, where the infimum runs over all $k\ge 0$, all even-degree rational homology classes $a_0,\dots,a_k$, and all $A\in H_2(M;\mathbb{Z})/\text{torsion}$ for which such an invariant is nonzero, so that $\Theta(M,\omega)\ge p>0$. Prove or refute: every Hamiltonian diffeomorphism $\varphi\in\operatorname{Ham}(M,\omega)$ satisfies

$$\#\operatorname{Fix}(\varphi)\ \ge\ \left\lceil\frac{p\cdot\operatorname{cuplength}(M;\mathbb{C})}{\Theta(M,\omega)}\right\rceil.$$

In particular, when $\Theta(M,\omega)=p$, every $\varphi\in\operatorname{Ham}(M,\omega)$ should have at least $\operatorname{cuplength}(M;\mathbb{C})$ fixed points, which would establish the degenerate Arnol'd conjecture over $\mathbb{C}$ for every manifold carrying a nonzero two-point-insertion genus-zero Gromov-Witten invariant — a class expected to include all symplectic rationally connected manifolds. This is Conjecture 1.7 of Gong (arXiv:2507.04191, v5). The statement must be proved under the bare two-point hypothesis: no minimizing-area condition, semipositivity, monotonicity, or point-invertibility hypothesis may be added, and a single explicit counterexample (a manifold and Hamiltonian diffeomorphism as above violating the displayed bound) refutes it.

References

1. Wenmin Gong, "Degenerate symplectic fixed points and Gromov-Witten invariants", arXiv:2507.04191 (2025), DOI 10.48550/arxiv.2507.04191, https://arxiv.org/abs/2507.04191
2. Hao Jiao, "Estimate of Periodic Orbits of Degenerate Hamiltonians", arXiv:2606.15103 (2026), https://arxiv.org/abs/2606.15103
3. L. Asselle and M. Starostka, "On the degenerate Arnold conjecture on T^{2m} x CP^n", arXiv:2411.19636 (2024), https://arxiv.org/abs/2411.19636
4. Zhiyu Tian, "Symplectic geometry of rationally connected threefolds", arXiv:1006.2486 (2010), https://arxiv.org/abs/1006.2486
5. Zhiyu Tian, "Some examples of symplectic rationally connected 4-folds", arXiv:1112.1369 (2011), https://arxiv.org/abs/1112.1369
6. Alex Pieloch, "Fundamental groups of rationally connected symplectic manifolds", arXiv:2212.07882 (2022), https://arxiv.org/abs/2212.07882
7. Matthias Schwarz, "A quantum cup-length estimate for symplectic fixed points", Inventiones Mathematicae 133 (1998), no. 2, 353–397, DOI 10.1007/s002220050248, https://doi.org/10.1007/s002220050248
8. Michael Usher, "Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms", Geometry & Topology 15 (2011), no. 3, 1313–1417, DOI 10.2140/gt.2011.15.1313, https://doi.org/10.2140/gt.2011.15.1313
9. Yuli B. Rudyak and John Oprea, "On the Lusternik–Schnirelmann category of symplectic manifolds and the Arnold conjecture", Mathematische Zeitschrift 230 (1999), no. 4, 673–678, DOI 10.1007/pl00004709, https://doi.org/10.1007/pl00004709