We present the Nahm transform of the doubly-periodic instantons previously introduced by the author, converting them into certain meromorphic solutions of Hitchin's equations over an elliptic curve. doubly-periodic instantons 7 7 singular Higgs pairs o o / / spectral curves Acknowledgements. This work is part of my Ph.D. project [11], which was funded by CNPq, Brazil. I am grateful to my supervisors, Simon Donaldson and Nigel Hitchin, for their constant support and guidance. I also thank Brian Steer and Olivier Biquard for valuable suggestions in the later stages of this project. Extensibility and asymptotic behaviourWe now use the extensibility hypothesis to study the compatibility between the instanton connection A and the extended bundle E → T × P 1 . More precisely, we first want to show that the holomorphic type of the restriction of the extended bundle to the added divisor T ∞ = T × {∞} is indeed directly determined by the asymptotic behaviour of the instanton connection A. Then we argue that the topology of E is fixed by the energy (L 2 -norm) of A. Before that, we must fix an appropriate trivialisation at infinity.Gauge fixing at infinity. Let B R denote a closed ball in C of radius R, and let V R be its complement. Also, consider the obvious projection p : T × V R → T . We shall need the following technical proposition, which follows from the gauge-fixing result established in [2] (see also the appendix in [11]).Proposition 2 If |F A | ∼ O(r −2 ), then, for R sufficiently large, there is a gauge over T × V R and a constant flat connection Γ on a topologically trivial rank two bundle over the elliptic curve such that:A − p * Γ = α ∼ O(r −1 · log r)

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 CP 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 CP 3 is a smooth complex manifold dimension 8c − 3, thus settling part of a 30-year-old conjecture.

We construct finite-energy instanton connections over R 4 which are periodic in two directions via an analogue of the Nahm transform for certain singular solutions of Hitchin's equations defined over a 2-torus.

Abstract. We study the irreducible components of the moduli space of instanton sheaves on P 3 , that is rank 2 torsion free sheaves E with c 1 (E) = c 3 (E) = 0 satisfying h 1 (E(−2)) = h 2 (E(−2)) = 0. In particular, we classify all instanton sheaves with c 2 (E) ≤ 4, describing all the irreducible components of their moduli space. A key ingredient for our argument is the study of the moduli space T (d) of stable sheaves on P 3 with Hilbert polynomial P (t) = dt, which contains, as an open subset, the moduli space of rank 0 instanton sheaves of multiplicity d; we describe all the irreducible components of T (d) for d ≤ 4.

We investigate Yang-Mills instanton theory over four dimensional asymptotically locally flat (ALF) geometries, including gravitational instantons of this type, by exploiting the existence of a natural smooth compactification of these spaces introduced by Hausel-Hunsicker-Mazzeo.First referring to the codimension 2 singularity removal theorem of Sibner-Sibner and Råde we prove that given a smooth, finite energy, self-dual SU(2) connection over a complete ALF space, its energy is congruent to a Chern-Simons invariant of the boundary threemanifold if the connection satisfies a certain holonomy condition at infinity and its curvature decays rapidly.Then we introduce framed moduli spaces of self-dual connections over Ricci flat ALF spaces. We prove that the moduli space of smooth, irreducible, rapidly decaying selfdual connections obeying the holonomy condition with fixed finite energy and prescribed asymptotic behaviour on a fixed bundle is a finite dimensional manifold. We calculate its dimension by a variant of the Gromov-Lawson relative index theorem.As an application, we study Yang-Mills instantons over the flat R 3 × S 1 , the multi-Taub-NUT family, and the Riemannian Schwarzschild space.Proof of Theorem 3.1. All we have to do is to show that the right hand sides of (24) and (25) are equal.