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 conjectured that on a compact 4-manifold any hypersymplectic structure can be deformed through cohomologous hypersymplectic structures to a hyperk\"ahler 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 $\mathbb{T}^4$.