This research proves that hypersymplectic structures on 4-manifolds can deform to a hyperkähler triple, showing a connection to effective circle actions.
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 S¹-action. In particular, we show that the underlying 4-manifold is diffeomorphic to T⁴.
No takes yet. Share an insight, caveat, or question.
Fine et al. (2025) studied this question.
Synapse has enriched 4 closely related papers on similar clinical questions. Consider them for comparative context: