A proof of the Arnold-Givental conjecture

Abstract

We prove the Arnold-Givental conjecture in full generality: given a closed symplectic manifold $(X, ω)$, an anti-symplectic involution $τ_X: X \to X$ with fixed point set $L={\rm Fix}(τ_X)$, and a Hamiltonian diffeomorphism $ϕ: X \to X$ such that $ϕ(L)$ intersects transversely with $L$, the following inequality holds: \[ \# \big( ϕ(L) \cap L \big) \geq {\mathrm dim}_{{\mathbb F}_2} H_*(L; {\mathbb F}_2).\] The proof combines the methods of integral Floer theory of the first and fourth authors, a reduction to Hamiltonian Floer cohomology due to Lu, and a new idea related to localization in a $\mathbb Z/2$-equivariant Floer theory tailored to the problem.