Asymptotic behaviour and the moduli space of doubly-periodic instantons

We study doubly-periodic instantons, i.e. instantons on the product of a 1-dimensional complex torus T with a complex line ℂ, with quadratic curvature decay. We determine the asymptotic behaviour of these instantons, constructing new asymptotic invariants. We show that the underlying holomorphic bundle extends to T×ℙ¹. The converse statement is also true, namely a holomorphic bundle on T×ℙ¹ which is flat on the torus at infinity, and satisfies a stability condition, comes from a doubly-periodic instanton. Finally, we study the hyperk?hler geometry of the moduli space of doubly-periodic instantons, and prove that the Nahm transform previously defined by the second author is a hyperk?hler isometry with the moduli space of certain meromorphic Higgs bundles on the dual torus. Received June 8, 2000 / final version received February 1, 2001?Published online April 3, 2001