A proof of the Arnold-Givental conjecture

Abstract

We prove the Arnold-Givental conjecture in full generality: given a closed symplectic manifold $(X,\omega )$, an anti-symplectic involution ${\tau }_X:X\to X$ with fixed point set $L=\mathrm{Fix}({\tau }_X)$, and a Hamiltonian diffeomorphism $\varphi :X\to X$ such that $\varphi (L)$ 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.

Version 1: 69 pages. Comments welcome!