Arnold conjecture

The Arnold conjecture, named after mathematician Vladimir Arnold, is a mathematical conjecture in the field of symplectic geometry, a branch of differential geometry.

Strong Arnold conjecture

Let $(M, ω)$ be a closed (compact without boundary) symplectic manifold. For any smooth function $H : M \to ℝ$ , the symplectic form $ω$ induces a Hamiltonian vector field $X_{H}$ on $M$ defined by the formula

ω (X_{H}, \cdot) = d H .

The function $H$ is called a Hamiltonian function.

Suppose there is a smooth 1-parameter family of Hamiltonian functions $H_{t} \in C^{\infty} (M)$ , $t \in [0, 1]$ . This family induces a 1-parameter family of Hamiltonian vector fields $X_{H_{t}}$ on $M$ . The family of vector fields integrates to a 1-parameter family of diffeomorphisms $φ_{t} : M \to M$ . Each individual $φ_{t}$ is a called a Hamiltonian diffeomorphism of $M$ .

The strong Arnold conjecture states that the number of fixed points of a Hamiltonian diffeomorphism of $M$ is greater than or equal to the number of critical points of a smooth function on $M$ .

Weak Arnold conjecture

Let $(M, ω)$ be a closed symplectic manifold. A Hamiltonian diffeomorphism $φ : M \to M$ is called nondegenerate if its graph intersects the diagonal of $M \times M$ transversely. For nondegenerate Hamiltonian diffeomorphisms, one variant of the Arnold conjecture says that the number of fixed points is at least equal to the minimal number of critical points of a Morse function on $M$ , called the Morse number of $M$ .

In view of the Morse inequality, the Morse number is greater than or equal to the sum of Betti numbers over a field $𝔽$ , namely $\sum_{i = 0}^{2 n} \dim H_{i} (M; 𝔽)$ . The weak Arnold conjecture says that

# {fixed points of φ} \geq \sum_{i = 0}^{2 n} \dim H_{i} (M; 𝔽)

for $φ : M \to M$ a nondegenerate Hamiltonian diffeomorphism.

Kenji Fukaya and Kaoru Ono proved the case $𝔽 = ℚ$ in 1999. Mohammed Abouzaid and Andrew J. Blumberg proved the case $𝔽 = 𝔽_{p}$ in 2021. Semon Rezchikov, Shaoyun Bai, and Guangbo Xu proved the case $𝔽 = ℤ$ in 2022.

Arnold–Givental conjecture

The Arnold–Givental conjecture, named after Vladimir Arnold and Alexander Givental, gives a lower bound on the number of intersection points of two Lagrangian submanifolds $L$ and $L^{'}$ in terms of the Betti numbers of $L$ , given that $L^{'}$ intersects $L$ transversally and $L^{'}$ is Hamiltonian isotopic to $L$ .

Let $(M, ω)$ be a compact $2 n$ -dimensional symplectic manifold, let $L \subset M$ be a compact Lagrangian submanifold of $M$ , and let $τ : M \to M$ be an anti-symplectic involution, that is, a diffeomorphism $τ : M \to M$ such that $τ^{*} ω = - ω$ and $τ^{2} = {id}_{M}$ , whose fixed point set is $L$ .

Let $H_{t} \in C^{\infty} (M)$ , $t \in [0, 1]$ be a smooth family of Hamiltonian functions on $M$ . This family generates a 1-parameter family of diffeomorphisms $φ_{t} : M \to M$ by flowing along the Hamiltonian vector field associated to $H_{t}$ . The Arnold–Givental conjecture states that if $φ_{1} (L)$ intersects transversely with $L$ , then

# (φ_{1} (L) \cap L) \geq \sum_{i = 0}^{n} \dim H_{i} (L; ℤ / 2 ℤ)

.

Status

Shaoyun Bai, Egor Shelukhin, Yi Wang, and Guangbo Xu announced a proof of the Arnold–Givental conjecture in August 2026. Before that, it had been proved for several special cases.

Alexander Givental proved it for $(M, L) = ({ℂ ℙ}^{n}, {ℝ ℙ}^{n})$ .
Yong-Geun Oh proved it for real forms of compact Hermitian spaces with suitable assumptions on the Maslov indices.
Lazzarini proved it for negative monotone case under suitable assumptions on the minimal Maslov number.
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono proved it for $(M, ω)$ semi-positive.
Urs Frauenfelder proved it in the case when $(M, ω)$ is a certain symplectic reduction, using gauged Floer theory.

References

Citations

↑ Asselle, L.; Izydorek, M.; Starostka, M. (2022). "The Arnold conjecture in $ℂ ℙ^{n}$ and the Conley index". arXiv:2202.00422 [math.DS].
1 2 Rizell, Georgios Dimitroglou; Golovko, Roman (2017-01-05). "The number of Hamiltonian fixed points on symplectically aspherical manifolds". arXiv:1609.04776 [math.SG].
1 2 Arnold, Vladimir I. (2004). "1972-33". In Arnold, Vladimir I. (ed.). Arnold's Problems. Berlin: Springer-Verlag. p. 15. doi:10.1007/b138219. ISBN 3-540-20614-0. MR 2078115. See also comments, pp. 284–288.
↑ Fukaya, Kenji; Ono, Kaoru (1999-09-01). "Arnold conjecture and Gromov–Witten invariant". Topology. 38 (5): 933–1048. doi:10.1016/S0040-9383(98)00042-1. ISSN 0040-9383.
↑ Abouzaid, Mohammed; Blumberg, Andrew J. (2021-03-02). "Arnold Conjecture and Morava K-theory". arXiv:2103.01507v1 [math.SG].
↑ Rezchikov, Semon (2022-09-22). "Integral Arnol'd Conjecture". arXiv:2209.11165 [math.SG].
↑ Bai, Shaoyun; Xu, Guangbo (2022-09-18). "Arnold conjecture over integers". arXiv:2209.08599 [math.SG].
1 2 Frauenfelder 2004
↑ Bai, Shaoyun; Shelukhin, Egor; Wang, Yi; Xu, Guangbo (2026-08-27). "A proof of the Arnold-Givental conjecture". arXiv:2608.27242 [math.SG].
↑ Givental 1989b
↑ Oh 1995
↑ Fukaya et al. 2009

Bibliography

Frauenfelder, Urs (2004), "The Arnold–Givental conjecture and moment Floer homology", International Mathematics Research Notices, 2004 (42): 2179–2269, arXiv:math/0309373, doi:10.1155/S1073792804133941, MR 2076142{{citation}}: CS1 maint: unflagged free DOI (link).
Fukaya, Kenji; Oh, Yong-Geun; Ohta, Hiroshi; Ono, Kaoru (2009), Lagrangian intersection Floer theory - anomaly and obstruction, International Press, ISBN 978-0-8218-5253-8
Givental, A. B. (1989a), "Periodic maps in symplectic topology", Funktsional. Anal. I Prilozhen, 23 (4): 37–52
- Givental, A. B. (1989b), "Periodic maps in symplectic topology (translation from Funkts. Anal. Prilozh. 23, No. 4, 37-52 (1989))", Functional Analysis and Its Applications, 23 (4): 287–300, doi:10.1007/BF01078943, S2CID 123546007, Zbl 0724.58031
Oh, Yong-Geun (1992), "Floer cohomology and Arnol'd-Givental's conjecture of [on] Lagrangian intersections", Comptes Rendus de l'Académie des Sciences, 315 (3): 309–314, MR 1179726.
Oh, Yong-Geun (1995), "Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks, III: Arnold-Givental Conjecture", The Floer Memorial Volume, pp. 555–573, doi:10.1007/978-3-0348-9217-9_23, ISBN 978-3-0348-9948-2

[Asselle_Izydorek_Starostka_p.-1] Asselle, L.; Izydorek, M.; Starostka, M. (2022). "The Arnold conjecture in $ℂ ℙ^{n}$ and the Conley index". arXiv:2202.00422 [math.DS].

[:0-2] 1 2 Rizell, Georgios Dimitroglou; Golovko, Roman (2017-01-05). "The number of Hamiltonian fixed points on symplectically aspherical manifolds". arXiv:1609.04776 [math.SG].

[:1-3] 1 2 Arnold, Vladimir I. (2004). "1972-33". In Arnold, Vladimir I. (ed.). Arnold's Problems. Berlin: Springer-Verlag. p. 15. doi:10.1007/b138219. ISBN 3-540-20614-0. MR 2078115. See also comments, pp. 284–288.

[4] Fukaya, Kenji; Ono, Kaoru (1999-09-01). "Arnold conjecture and Gromov–Witten invariant". Topology. 38 (5): 933–1048. doi:10.1016/S0040-9383(98)00042-1. ISSN 0040-9383.

[5] Abouzaid, Mohammed; Blumberg, Andrew J. (2021-03-02). "Arnold Conjecture and Morava K-theory". arXiv:2103.01507v1 [math.SG].

[6] Rezchikov, Semon (2022-09-22). "Integral Arnol'd Conjecture". arXiv:2209.11165 [math.SG].

[7] Bai, Shaoyun; Xu, Guangbo (2022-09-18). "Arnold conjecture over integers". arXiv:2209.08599 [math.SG].

[Frauenfelder-8] 1 2 Frauenfelder 2004

[9] Bai, Shaoyun; Shelukhin, Egor; Wang, Yi; Xu, Guangbo (2026-08-27). "A proof of the Arnold-Givental conjecture". arXiv:2608.27242 [math.SG].

[10] Givental 1989b

[11] Oh 1995

[12] Fukaya et al. 2009