Degenerate symplectic fixed points and Gromov-Witten invariants

Abstract

We establish a connection between Gromov-Witten invariants and the number of fixed points of Hamiltonian diffeomorphisms on a closed rational symplectic manifold via deformed Hamiltonian spectral invariants. We generalize Givental's symplectic fixed point theorem for Fano toric manifolds to closed rational symplectic manifolds which admit nonzero Gromov-Witten invariants with fixed marked points and one point insertion. We prove a new cuplength estimate of symplectic fixed points involved in deformed spectral invariants. We extend Schwarz's quantum cuplength to the notion of deformed quantum cuplength for symplectic periods and employ it to estimate the number of fixed points of Hamiltonian diffeomorphisms on monotone symplectic manifolds with nonzero mixed Gromov-Witten invariants.