A Symplectic Fixed Point Theorem for Complex Projective Spaces

1. Arnold's conjecture. An automorphism ^ of a symplectic manifold (P,u;) is homologous to the identity if there is a smooth family tyt (* € [0,1]) of automorphisms such that the time-dependent vector field £t defined by di^t/dt = & ° ^t is globally hamiltonian; i.e. if there is a smooth family Ht of real-valued functions on P such that £*JCÜ = dHt. It was conjectured by Arnold [1], as an extension of the Poincaré-Birkhoff annulus theorem [3, 7], that every automorphism of a compact symplectic manifold P, homologous to the identity, has at least as many fixed points as a function on P has critical points. Arnold's conjecture was proven by Conley and Zehnder [4] for the torus T « R 2 n / Z 2 n with its usual symplectic structure. They show that every symplectic automorphism of T 2 n , homologous to the identity, has at least n + 1 fixed points, and at least 2 n if all are nondegenerate. Their method was extended in [8] to prove a version of Arnold's conjecture for arbitrary P under the additional assumption that the hamiltonian vector field £t is sufficiently C° small. In this note we announce a proof of Arnold's conjecture for the complex projective space C P n with its standard symplectic structure. We prove that a symplectic diffeomorphism of C P n , homologous to the identity, has at least n+\ distinct fixed points. (By the Lefschetz fixed point theorem, any continuous map from C P n to itself, homotopic to the identity, has at least n + 1 fixed points counted with multiplicities.) For n = 1 (CP 1 « S) the result was already known [1], but with a proof which worked only in this two-dimensional case. The proof for T 2 n in [4] made use of a variational principle in which the fixed points of the map were identified with periodic solutions of a timedependent hamiltonian system and then identified with critical points of a functional on the space of contractible loops on T n . The corresponding functional in the case of C P n is multiple valued, and there are other difficulties connected with the curved geometry of C P n , so we need a new approach. Our trick is to consider the hamiltonian system on C P n as the reduction, in the sense of [6], of a hamiltonian system on C n + 1 and then adapt recently developed methods [2] for finding periodic orbits in C n + 1 . This method is similar to that of Conley and Zehnder in that a problem on a compact manifold is lifted to a problem on euclidean space invariant under a group of transformations.