  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 $S^1$-action, and discuss
the relation to symplectic rationality problems.
