Trisymplectic manifolds and the moduli of instantons on $C P^3$

A trisymplectic structure on a complex n-manifold is a triple of holomorphic symplectic forms such that any linear combination of these forms has rank 2n, n or 0. 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. I will construct a trisymplectic structure on the moduli of regular rational curves in the twistor space of a hyperkaehler manifold. I will show that the moduli space of holomorphic vector bundles on $C P^3$ that are trivial along a line admits a trisymplectic structure. Using the ADHM construction of instantons, it is shown that the moduli space of rank 2, charge c instanton bundles on $CP^3$ is a smooth complex manifold of dimension 8c-3, thus settling a 30-year old conjecture.