### ?

## Trihyperkähler reduction and instanton bundles on CP^3

A trisymplectic structure on a complex 2n-manifold is a

three-dimensional space ${\rm\Omega}$ of closed holomorphic forms such

that any element of \Omega has constant rank 2n, n or zero, and

degenerate forms in \Omega 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 hyperkhler reduction. We

prove that the trisymplectic reduction in the space of regular

rational curves on the twistor space of a hyperkhler 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 \mathbb{C}\mathbb{P}^{3} 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 \mathbb{C}\mathbb{P}^{3} is a smooth complex manifold

dimension 8c-3, thus settling part of a 30-year-old conjecture.