Hamiltonian circle actions on complete intersections

We study the problem of determining which diffeomorphism classes of Kahler manifolds admit a Hamiltonian circle action. Our main result is the following: Let M$M$ be a closed symplectic manifold, diffeomorphic to a complete intersection with complex dimension 4k$4k$, having a Hamiltonian circle action such that each component of the fixed-point set is an isolated fixed point or has dimension 2mod4$2 \mod {4}$. Then M$M$ is diffeomorphic to CP4k$\mathbb {CP}{4k}$, a quadric Q subset of CP4k+1$Q \subset \mathbb {CP}{4k+1}$ or an intersection of two quadrics Q1 boolean AND Q2 subset of CP4k+2$Q_1 \cap Q_2 \subset \mathbb {CP}{4k+2}$