It has been suggested that this article be merged into Arnold conjecture. (Discuss) Proposed since June 2024. |
The Arnold–Givental conjecture, named after Vladimir Arnold and Alexander Givental, is a statement on Lagrangian submanifolds. The conjecture gives a lower bound on the number of intersection points of two Lagrangian submanifolds L and in terms of the Betti numbers of , given that intersects L transversally and is Hamiltonian isotopic to L.
Statement
editLet be a compact -dimensional symplectic manifold, let be a compact Lagrangian submanifold of , and let be an anti-symplectic involution, that is, a diffeomorphism such that and , whose fixed point set is .
Let , be a smooth family of Hamiltonian functions on . This family generates a 1-parameter family of diffeomorphisms by flowing along the Hamiltonian vector field associated to . The Arnold–Givental conjecture states that if intersects transversely with , then
.[1]
One version of the Arnold conjecture can be obtained from the Arnold–Givental conjecture by considering the diagonal as a Lagrangian submanifold of for a compact symplectic manifold and as the anti-symplectic involution switching the two factors of .[1]
Status
editThe Arnold–Givental conjecture has been proved for certain special cases.
- Givental proved it for .[2]
- Yong-Geun Oh proved it for real forms of compact Hermitian spaces with suitable assumptions on the Maslov indices.[3]
- 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 is semi-positive.[4]
- Urs Frauenfelder proved it in the case when is a certain symplectic reduction, using gauged Floer theory.[1]
See also
editReferences
editCitations
edit- ^ a b c (Frauenfelder 2004)
- ^ (Givental 1989b)
- ^ (Oh 1995)
- ^ (Fukaya et al. 2009)
Bibliography
edit- 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.
- 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