Monotone Symplectic Six-Manifolds that admit a Hamiltonian GKM Action are diffeomorphic to Smooth Fano Threefolds

Abstract

Let $(M,\omega)$ be a compact symplectic manifold with a Hamiltonian GKM action of a compact torus. We formulate a positive condition on the space; this condition is satisfied if the underlying symplectic manifold is monotone. The main result of this article is that the underlying manifold of a positive Hamiltonian GKM space of dimension six is diffeomorphic to a smooth Fano threefold. We prove the main result in two steps. In the first step, we deduce from results of Goertsches, Konstantis, and Zoller that if the complexity of the action is zero or one then the equivariant and the ordinary cohomology with integer coefficients are determined by the GKM graph. This result, in combination with a classification result by Jupp, Wall and Zubr for certain six-manifolds, implies that the diffeomorphism type of a compact symplectic six-manifold with a Hamiltonian GKM action is determined by the associated GKM graph. In the second step, based on results by Godinho and Sabatini, we compute the complete list of the GKM graphs of positive Hamiltonian GKM spaces of dimension six. We deduce that any such GKM graph is isomorphic to a GKM graph of a smooth Fano threefold.