Cohomological localization for Hamiltonian $S^1$-actions and symmetries of complete intersections

To begin the paper we revisit a cohomological localization result of Jones-Rawnsley which was subsequently improved by Farber, further generalizing the result. We then proceed to improve a previous result of the author on complete intersections of dimension $8k$ with a Hamiltonian $S^1$-action in two directions. Firstly, in dimension $8$ we remove the assumption on the fixed point set. Secondly, in any dimension we prove the result under an analogous assumption on the fixed point set. We also give some applications towards the unimodality of Betti numbers of symplectic manifolds having a Hamiltonian $S^1$-action, and discuss the relation to symplectic rationality problems.