Hamiltonian $S^1$-actions on complete intersections

TL;DR

This paper classifies certain symplectic manifolds with Hamiltonian circle actions, showing they are diffeomorphic to well-known complete intersections like projective spaces or quadrics.

Contribution

It provides a classification of complete intersection symplectic manifolds with specific Hamiltonian circle actions, identifying them as standard complex geometric objects.

Findings

Manifolds are diffeomorphic to projective space, quadrics, or intersections of quadrics.

The fixed point set components are either isolated points or have dimension 2 mod 4.

The classification applies to manifolds of complex dimension divisible by 4.

Abstract

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 $2 \mod 4$. Then $M$ is diffeomorphic to $\mathbb{CP}^{4k}$, a quadric $Q \subset \mathbb{CP}^{4k+1}$ or an intersection of two quadrics $Q_1 \cap Q_2 \subset \mathbb{CP}^{4k+2}$.