Arnold–Givental conjecture

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 $L'$ in terms of the Betti numbers of $L$, given that $L'$ intersects L transversally and $L'$ is Hamiltonian isotopic to L.

Statement

Let $(M,\omega )$ be a compact $2n$-dimensional symplectic manifold, let $L\subset M$ be a compact Lagrangian submanifold of $M$, and let $\tau :M\to M$ be an anti-symplectic involution, that is, a diffeomorphism $\tau :M\to M$ such that $\tau ^{*}\omega =-\omega$ and $\tau ^{2}={\text{id}}_{M}$, 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 $\varphi _{t}:M\to M$ by flowing along the Hamiltonian vector field associated to $H_{t}$. The Arnold–Givental conjecture states that if $\varphi _{1}(L)$ intersects transversely with $L$, then

$\#(\varphi _{1}(L)\cap L)\geq \sum _{i=0}^{n}\dim H_{i}(L;{\mathbb {Z} }_{2})$.

One version of the Arnold conjecture can be obtained from the Arnold–Givental conjecture by considering the diagonal $\Delta$ as a Lagrangian submanifold of $(M\times M,\omega \oplus -\omega )$ for $M$ a compact symplectic manifold and $\tau :M\to M$ as the anti-symplectic involution switching the two factors of $M\times M$.

Status

The Arnold–Givental conjecture has been proved for certain special cases.

• Givental proved it for $(M,L)=(\mathbb {CP} ^{n},\mathbb {RP} ^{n})$.
• 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 $(M,\omega )$ is semi-positive.
• Urs Frauenfelder proved it in the case when $(M,\omega )$ is a certain symplectic reduction, using gauged Floer theory.

See also

• Arnold conjecture

References

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.
• 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