Hamiltonian $S^1$-actions on complete intersections

Abstract

We study the problem of determining which diffeomorphism classes of Kähler 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 $2\mathrm{mod}4$. Then $M$ is diffeomorphic to ${\mathbb{CP}}^{4k}$, a quadric $Q\subset {\mathbb{CP}}^{4k+1}$ or an intersection of two quadrics $Q_1\cap Q_2\subset {\mathbb{CP}}^{4k+2}$.

Accepted version, to appear in Bulletin of the London Mathematical Society. 7 Pages

References in corpus (2)

• Realization of GKM fibrations and new examples of Hamiltonian non-Kähler actions
• Symplectic and Kähler structures on $\mathbb{C}{P}^1$-bundles over $\mathbb{C}{P}^2$