The degenerate Arnold conjecture on $\mathbb T^{2m}\times\mathbb{CP}^n$: the cup-length bound $2m+n+1$ for arbitrary Hamiltonian diffeomorphisms

The degenerate Arnold conjecture predicts that a Hamiltonian diffeomorphism of a closed symplectic manifold has at least as many contractible fixed points as a smooth function has critical points; its most widely studied form, the cup-length version, demands at least $\mathrm{CL}(M)+1$ fixed points. For the product $\mathbb T^{2m}\times\mathbb{CP}^n$ ($m,n\ge 1$) with the standard symplectic form, the expected bound is $2m+n+1$, but before 2024 only $n+1$ (the $\mathbb{CP}^n$ factor's bound, due to Oh) was known. Asselle and Starostka (arXiv 2024, journal version in Topological Methods in Nonlinear Analysis) proved the full bound $2m+n+1$ only under a smallness hypothesis, $\max\{2\|H_0\|_\infty,\|\nabla_{\mathbb{CP}^n}H_0\|_\infty\}<1/2$, via a Conley-index argument for a modified Hamiltonian action functional whose coercivity rests on a non-resonance Condition (A). The open problem is to decide, without any smallness hypothesis, whether every Hamiltonian diffeomorphism of $\mathbb T^{2m}\times\mathbb{CP}^n$ has at least $2m+n+1$ contractible fixed points; a secondary question posed in that paper is the sharp threshold for Condition (A), known to lie in $[\sqrt{\pi}/2,\pi]$. As of August 2026, no resolution, refutation, or weakening of the smallness hypothesis is known.

Background

Let $(M,\omega)$ be a closed symplectic manifold. A smooth one-periodic Hamiltonian $H:\mathbb T\times M\to\mathbb R$, $\mathbb T=\mathbb R/\mathbb Z$, generates a flow $\varphi_t$ through $\iota_{X_H}\omega=-dH$, where $X_H$ is the Hamiltonian vector field; a diffeomorphism is Hamiltonian if it equals the time-one map $\varphi_1$ of some such $H$. A fixed point $x$ of $\varphi=\varphi_1$ is called contractible if the loop $t\mapsto\varphi_t(x)$ is contractible in $M$. In the 1960s Arnold conjectured that $\varphi$ has at least as many contractible fixed points as a smooth function on $M$ has critical points. By Lusternik–Schnirelmann theory the number of critical points of any smooth function is bounded below by $\mathrm{LS}(M)\ge\mathrm{CL}(M)+1$, where $\mathrm{CL}(M)$ is the cup-length of $M$ — the maximal number of positive-degree cohomology classes whose cup product is nonzero — so the conjecture is commonly studied in the weaker form $|\mathrm{Fix}_{\mathrm{contr}}(\varphi)|\ge\mathrm{CL}(M)+1$. For nondegenerate fixed points the corresponding homological version (bound by the sum of Betti numbers) was settled in the 1990s by Hamiltonian Floer theory (Floer homology, the $S^1$-equivariant homology of the Hamiltonian action functional), but the degenerate version — no nondegeneracy assumed on the fixed points — remains open in general.

The degenerate conjecture is known for symplectically aspherical manifolds (those with $\omega$ vanishing on $\pi_2(M)$, so that there are no symplectic spheres and Floer-theoretic bubbling is absent): Conley and Zehnder proved it for the torus $\mathbb T^{2m}$ (1983), Fortune for $\mathbb{CP}^n$ with the Fubini–Study form (1985), and Rudyak and Oprea for all closed symplectically aspherical manifolds (1999). Beyond the aspherical world almost nothing is known. The simplest product mixing an aspherical factor with one carrying symplectic spheres is $\mathbb T^{2m}\times\mathbb{CP}^n$ with $\omega_{\mathrm{std}}\oplus\omega_{FS}$, whose cup-length is $2m+n$; both factors are individually settled (the torus case was reproved by Starostka and Waterstraat via an $E$-cohomological Conley index, and the $\mathbb{CP}^n$ case was reproved by Asselle, Izydorek and Starostka via the Conley index), yet for the product the best previously available bound, due to Oh (1990), was only $n+1=\mathrm{CL}(\mathbb{CP}^n)+1$ as recalled in that paper — the torus factor contributed nothing. Oh also raised doubts about the validity of the degenerate Arnold conjecture, in any of its forms, in full generality.

The paper of Asselle and Starostka attacks this product with Conley-index methods. The Conley index is a homotopy invariant of an isolated invariant set of a flow; for gradient flows of functionals on a Hilbert space that are compact perturbations of a fixed quadratic form (so-called LS-functionals), the index is defined through finite-dimensional projections, and its relative cup-length gives lower bounds on the number of critical points. Periodic orbits on $\mathbb T^{2m}\times\mathbb{CP}^n$ are lifted to $\mathbb R^{2m}\times\mathbb R^{2n+2}$ through the Hopf fibration $S^{2n+1}\to\mathbb{CP}^n$; since a lifted orbit need only return to the same Hopf fiber, the lifted problem acquires an extra real phase parameter $\lambda$, and the modified action functional $\mathbb A_H$ on $H^{1/2}(\mathbb T,\mathbb R^{2m}\times\mathbb R^{2n+2})\times\mathbb R$ has critical points corresponding exactly to contractible periodic orbits provided a coercivity hypothesis called Condition (A) holds near some $\lambda_0$: roughly, the $y$-component of the gradient of the unmodified action dominates $\|y\|$ uniformly for all phase parameters $\lambda$ in a neighborhood of $\lambda_0$. The paper shows that Condition (A) holds with $\lambda_0=\pm 1/2$ whenever $c:=\max\{2\|H_0\|_\infty,\|\nabla_{\mathbb{CP}^n}H_0\|_\infty\}<1/2$, and then a continuation of the Conley index to the zero Hamiltonian, whose flow splits as a product, yields $|\mathrm{Fix}(\varphi)|\ge(2m+1)+(n+1)-1=2m+n+1$. Remark 2 of the paper notes that a refined Fourier-splitting argument extends the sufficient range to $c<\sqrt{\pi}/2$, while Example (8), the Hamiltonian $H=(\pi/2)\sin(x)|y|^2$, which in the normalization above has $c=\pi$, shows that Condition (A) may genuinely fail, its lifted second equation having solutions for a continuum of $\lambda$ (a resonance phenomenon). Thus the unrestricted conjecture on $\mathbb T^{2m}\times\mathbb{CP}^n$, as well as the sharp coercivity threshold for the authors' method and the delineation of the resonant Hamiltonians for which the lifted variational problem loses coercivity, are left open by the source — and the smallness regime is not an artifact of the statement but the exact limit of the current technique.

Problem Statement

Decide the degenerate Arnold conjecture (cup-length version) on the product torus-times-projective-space: does every Hamiltonian diffeomorphism $\varphi$ of $(\mathbb T^{2m}\times\mathbb{CP}^n,\omega_{\mathrm{std}}\oplus\omega_{FS})$, $m,n\ge 1$, generated by an arbitrary smooth one-periodic Hamiltonian $H_0:\mathbb T\times\mathbb T^{2m}\times\mathbb{CP}^n\to\mathbb R$, have at least $\mathrm{CL}(\mathbb T^{2m}\times\mathbb{CP}^n)+1=2m+n+1$ contractible fixed points, with no $C^0$-smallness or $\mathbb{CP}^n$-directional $C^1$-smallness hypothesis (in particular without $\max\{2\|H_0\|_\infty,\|\nabla_{\mathbb{CP}^n}H_0\|_\infty\}<1/2$ and without any replacement coercivity hypothesis)?

References

1. Luca Asselle and Maciej Starostka, On the degenerate Arnold conjecture on $\mathbb T^{2m}\times\mathbb{CP}^n$, arXiv:2411.19636 (2024), DOI 10.48550/arXiv.2411.19636, https://arxiv.org/abs/2411.19636
2. Luca Asselle and Maciej Starostka, On the degenerate Arnold conjecture on $\mathbb T^{2m}\times\mathbb{CP}^n$, Topological Methods in Nonlinear Analysis 67(1), 19–34 (2026; online first June 2025), DOI 10.12775/tmna.2025.011, https://doi.org/10.12775/tmna.2025.011
3. Luca Asselle, Marek Izydorek, and Maciej Starostka, The Arnold conjecture in $\mathbb{CP}^n$ and the Conley index, Discrete and Continuous Dynamical Systems B 28(4), 2603–2620 (2023), DOI 10.3934/dcdsb.2022184, https://doi.org/10.3934/dcdsb.2022184
4. Yong-Geun Oh, A symplectic fixed point theorem on $\mathbb T^{2n}\times\mathbb{CP}^k$, Mathematische Zeitschrift 203(1), 535–552 (1990), DOI 10.1007/BF02570755, https://doi.org/10.1007/BF02570755
5. Barry Fortune, A symplectic fixed point theorem for $\mathbb{CP}^n$, Inventiones Mathematicae 81(1), 29–46 (1985), DOI 10.1007/BF01388770, https://doi.org/10.1007/BF01388770
6. Charles Conley and Eduard Zehnder, The Birkhoff–Lewis fixed point theorem and a conjecture of V.I. Arnold, Inventiones Mathematicae 73(1), 33–49 (1983), DOI 10.1007/BF01393824, https://doi.org/10.1007/BF01393824
7. Yuli B. Rudyak and John Oprea, On the Lusternik–Schnirelmann category of symplectic manifolds and the Arnold conjecture, Mathematische Zeitschrift 230(4), 673–678 (1999), DOI 10.1007/PL00004709, https://doi.org/10.1007/PL00004709
8. Maciej Starostka and Nils Waterstraat, The $E$-Cohomological Conley Index, Cup-Lengths and the Arnold Conjecture on $\mathbb T^{2n}$, Advanced Nonlinear Studies 19(3), 519–528 (2019), DOI 10.1515/ans-2019-2044, https://doi.org/10.1515/ans-2019-2044
9. Thomas Rot, Maciej Starostka, and Nils Waterstraat, The relative cup-length in local Morse cohomology, Topological Methods in Nonlinear Analysis 64(1), 1–15 (2024), DOI 10.12775/tmna.2024.002, https://doi.org/10.12775/tmna.2024.002
10. Jianxun Hu, A symplectic fixed-point theorem for $T^{2k}\times\mathbb{CP}^n\times\mathbb{CP}^m$, Proceedings of the Royal Society of Edinburgh: Section A Mathematics 125(3), 465–492 (1995), DOI 10.1017/S0308210500032625, https://doi.org/10.1017/S0308210500032625
11. Hao Jiao, Estimate of Periodic Orbits of Degenerate Hamiltonians, arXiv:2606.15103 (2026), DOI 10.48550/arXiv.2606.15103, https://arxiv.org/abs/2606.15103