Higher-rank instantons sheaves on Fano threefolds

We define higher-rank instanton sheaves on smooth Fano threefolds X of Picard rank one and show that their topological type depends on two integers, namely the rank n (or half of it if the Fano index of X is odd) and the charge k.With the possible exception of Fano threefolds of index one and genus at most 3, or genus 4 contained in a singular quadric, we determine the minimal charge k 0 of slope-stable n-instanton bundles as an integer depending only on H 3 X and n, and we prove the existence of slope-stable n-instanton bundles of charge k ≥ k 0 .Next, we study several properties of a general element of an irreducible component, which we call the main component, of the moduli space of instanton sheaves.This component is defined inductively on the rank and the charge, a natural procedure based on elementary transformations along rational curves, starting with minimal instantons.The questions we address include the generic splitting over certain rational curves contained in X, as well as stable restriction to a K3 section S of X.We obtain applications to Lagrangian subvarieties of moduli spaces of sheaves on S.Finally, we study the acyclic extension associated with an instanton sheaf on Fano threefolds with curvilinear Kuznetsov component and give a monadic description in the case H 3 (X, ‫)ޚ‬ = 0.