Symplectic torus actions on complete intersections

In [20] the author classified complete intersections of dimension $8k$ admitting a Hamiltonian circle action, under a certain assumption on the fixed point set. In this paper, we improve this result 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 of our methods to the unimodality of Betti numbers of symplectic manifolds having a Hamiltonian S¹-action, and discuss the relation to symplectic rationality problems.