Hypersymplectic Structures Invariant Under an Effective Circle Action

Abstract A hypersymplectic structure on a 4-manifold is a triple of symplectic forms for which any non-zero linear combination is again symplectic. In 2006, Donaldson [5] conjectured that on a compact 4-manifold any hypersymplectic structure can be deformed through cohomologous hypersymplectic structures to a hyperkähler triple. We prove this under the assumption that the initial structure is invariant under an effective $S^{1}$-action. In particular, we show that the underlying 4-manifold is diffeomorphic to $\mathbb{T}^{4}$.