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

TL;DR

This paper proves that six-dimensional monotone symplectic manifolds with a Hamiltonian GKM action are diffeomorphic to smooth Fano threefolds, using GKM graph classification and cohomology analysis.

Contribution

It establishes a classification linking positive Hamiltonian GKM spaces in six dimensions to smooth Fano threefolds via GKM graph analysis.

Findings

Six-dimensional positive Hamiltonian GKM spaces are diffeomorphic to smooth Fano threefolds.

GKM graphs of these spaces correspond to those of smooth Fano threefolds.

The diffeomorphism type is determined by the GKM graph.

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.