Arnold–Givental conjecture
The Arnold–Givental conjecture, named after Vladimir Arnold and Alexander Givental, is a statement on Lagrangian submanifolds. It gives a lower bound in terms of the Betti numbers of a Lagrangian submanifold on the number of intersection points of with another Lagrangian submanifold which is obtained from by Hamiltonian isotopy, and which intersects transversally. Statement Let be a compact -dimensional symplectic manifold. An anti-symplectic involution is a diffeomorphism such that . The fixed point set of is necessarily a Lagrangian submanifold. Let be a smooth family of Hamiltonian functions on which generates a 1-parameter family of Hamiltonian diffeomorphisms . The Arnold–Givental conjecture says, suppose intersects transversely with , then Status The Arnold–Givental conjecture has been proved for certain special cases. Givental proved it for the case when (see ). Yong-Geun Oh proved it for real forms of compact Hermitian spaces with suitable assumptions on the Maslov indices (see ). 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 is semi-positive (see ). Frauenfelder proved it for the situation when is a certain symplectic reduction, using gauged Floer theory (see ). See also Arnold