Monotone Symplectic Six-Manifolds That Admit a Hamiltonian GKM Action are Diffeomorphic to Smooth Fano Threefolds

Let (M,ω) be a compact symplectic manifold with a Hamiltonian GKM torus action. We formulate a positive condition on the space; this condition is satisfied if the underlying symplectic manifold is monotone. The main result is that if a Hamiltonian GKM space is positive and six-dimensional, then it is diffeomorphic to a smooth Fano threefold. We first deduce from results of Goertsches, Konstantis, and Zoller that if the complexity of the action is zero or one, then the equivariant and ordinary cohomologies with integer coefficients are determined by the GKM graph. If the dimension of the manifold is six, then the diffeotype is also determined by the graph. Then, specializing results of Godinho and Sabatini on linear relations on the weights and extending results of Goldin and Tolman on special Kirwan classes, we compute the list of GKM graphs of positive Hamiltonian GKM spaces of dimension six. We deduce that each is isomorphic to the GKM graph of a holomorphic T-action on a smooth Fano threefold.