Trihyperkahler reduction and instanton bundles on CP^3

Abstract

A trisymplectic structure on a complex 2n-manifold is a triple of holomorphic symplectic forms such that any linear combination of these forms has constant rank 2n, n or 0, 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 hyperkaehler manifold, and define a trisymplectic reduction of a trisymplectic manifold, which is a complexified form of a hyperkaehler reduction. We prove that the trisymplectic reduction in the space of regular rational curves on the twistor space of a hyperkaehler manifold M is compatible with the hyperkaehler 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 CP^3 is a smooth, connected, trisymplectic manifold of complex dimension 4rc. In particular, it follows that the moduli space of rank 2, charge c instanton bundles on CP^3 is a smooth complex manifold dimension 8c-3, thus settling a 30-year old conjecture.

42 pages, v. 3.2, changes in section 3.1: the notion of trisymplectic structure stated differently, Clifford algebra action introduced

References in corpus (2)

• Moduli of mathematical instanton vector bundles with odd c_2 on projective space
• Moduli spaces of framed perverse instantons on P^3

Cited by in corpus (12)

• Two infinite series of moduli spaces of rank 2 sheaves on ${\mathbb{P}}^3$
• Moduli spaces of rank 2 instanton sheaves on the projective space
• Kuga-Satake construction and cohomology of hyperkahler manifolds
• Instanton bundles on the blow up of the projective $3$-space at a point
• Singular locus of instanton sheaves on ${\mathbb{P}}^3$
• New moduli components of rank 2 bundles on projective space
• Compactification of the moduli space of instanton sheaves on the Fano threefold $V_5$
• k-symplectic structures and absolutely trianalytic subvarieties in hyperkahler manifolds
• Compactification of the moduli space of minimal instantons on the Fano threefold $V_4$
• Yang-Mills Instanton Sheaves
• On the fixed locus of framed instanton sheaves on ${\mathbb{P}}^3$
• Moduli of autodual instanton bundles