Hamiltonian $S^1$-actions on complete intersections

We study the problem of determining which diffeomorphism classes of K"{a}hler manifolds admit a Hamiltonian circle action. Our main result is the following: Let $M$ be a closed symplectic manifold, diffeomorphic to a complete intersection with complex dimension $4k$, having a Hamiltonian circle action such that each component of the fixed point set is an isolated fixed point or has dimension $2\mathrm{m}\mathrm{o}\mathrm{d}4$. Then $M$ is diffeomorphic to ${\mathbb{C}\mathbb{P}}^{4k}$, a quadric $Q\subset {\mathbb{C}\mathbb{P}}^{4k+1}$ or an intersection of two quadrics $Q_1 \cap Q_2 \subset $\mathbb{C}\mathbb{P}$^{4k+2}$.