We study a class of torsion-free sheaves on complex projective spaces which generalize the much studied mathematical instanton bundles. Instanton sheaves can be obtained as cohomologies of linear monads and are shown to be semistable if its rank is not too large, while semistable torsion-free sheaves satisfying certain cohomological conditions are instanton. We also study a few examples of moduli spaces of instanton sheaves.

We establish a decomposability criterion for linear sheaves on ℙn. Applying it to instanton bundles, we show, in particular, that every rank 2n instanton bundle of charge 1 on ℙn is decomposable. Moreover, we provide an example of an indecomposable instanton bundle of rank 2n − 1 and charge 1, thus showing that our criterion is sharp.

We study doubly-periodic instantons, i.e. instantons on the product of a 1-dimensional complex torus $T$ with a complex line $\u2102$, 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 $T\times {\mathbb{P}}^{1}$. The converse statement is also true, namely a holomorphic bundle on $T\times {\mathbb{P}}^{1}$ which is flat on the torus at infinity, and satisfies a stability condition, comes from a doubly-periodic instanton....

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