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## Trihyperkähler reduction and instanton bundles on CP^3

Compositio Mathematica. 2014. Vol. 150. No. 11. P. 1836-1868.
Jardim M., Verbitsky M.

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.