Trihyperkähler reduction and instanton bundles on

A trisymplectic structure on a complex $2n$ -manifold is a three-dimensional space Ω of closed holomorphic forms such that any element of Ω has constant rank $2n$ , n or zero, and degenerate forms in Ω belong to a non-degenerate quadric hypersurface. We show that a trisymplectic manifold is equipped with a holomorphic 3-web and the Chern connection of this 3-web is holomorphic, torsion-free, and preserves the three symplectic forms. We construct a trisymplectic structure on the moduli of regular rational curves in the twistor space of a hyperkähler manifold, and define a trisymplectic reduction of a trisymplectic manifold, which is a complexified form of a hyperkähler reduction. We prove that the trisymplectic reduction in the space of regular rational curves on the twistor space of a hyperkähler manifold M is compatible with the hyperkähler reduction on M . As an application of these geometric ideas, we consider the ADHM construction of instantons and show that the moduli space of rank r , charge c framed instanton bundles on CP³ is a smooth trisymplectic manifold of complex dimension $4rc$ . In particular, it follows that the moduli space of rank two, charge c instanton bundles on CP³ is a smooth complex manifold dimension $8c-3$ , thus settling part of a 30-year-old conjecture.