Arnold–Givental conjecture

Let ${\displaystyle (M,\omega )}$ be a compact ${\displaystyle 2n}$-dimensional symplectic manifold. An anti-symplectic involution is a diffeomorphism ${\displaystyle \tau :M\to M}$ such that ${\displaystyle \tau ^{*}\omega =-\omega }$. The fixed point set ${\displaystyle L\subset M}$ of ${\displaystyle \tau }$ is necessarily a Lagrangian submanifold.

Let ${\displaystyle H_{t}\in C^{\infty }(M),0\leq t\leq 1}$ be a smooth family of Hamiltonian functions on ${\displaystyle M}$ which generates a 1-parameter family of Hamiltonian diffeomorphisms ${\displaystyle \varphi _{t}:M\to M}$. The Arnold–Givental conjecture says, suppose ${\displaystyle \varphi _{1}(L)}$ intersects transversely with ${\displaystyle L}$, then

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

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

Givental proved it for the case when ${\displaystyle (M,L)=(\mathbb {CP} ^{n},\mathbb {RP} ^{n})}$ (see [1]).

Yong-Geun Oh proved it for real forms of compact Hermitian spaces with suitable assumptions on the Maslov indices (see [2]).

Lazzarini proved it for negative monotone case under suitable assumptions on the minimal Maslov number.

Kenji Fukaya, Yong-Geun Oh, Ohta, and Ono proved for the case when ${\displaystyle (M,\omega )}$ is semi-positive (see [3]).

Frauenfelder proved it for the situation when ${\displaystyle (M,\omega )}$ is a certain symplectic reduction, using gauged Floer theory (see [4]).

• Arnold conjecture

• 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.
• 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.