Here we study extensions 0-•£->F->Q->0 of vector bundles over a projective variety X with trivial middle V (hence V -> Q induces a map h from X to a Grassmannian G). For fixed X and Q and moving V -• Q we study the induced local deformations of S. This gives morphisms h with suitable h*(TG). Let X be a complete variety and £2 a vector bundle on X set r := rank(β). Assume that Q is spanned (i.e. generated by its global sections). Hence there is a trivial vector bundle V on X and a surjection q:

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... -* Q set m := rank(F) and S := Ker(#). Thus we have the following exact sequence on X. (1) 0->S^V-+Q^0. The map q induces a morphism h from X to the Grassmannian G := G(r, m). By the description of the tangent bundle TG on G we have /z*(Γ(?) = S* β. Since the bundle A*(ΓG) reflects very much the geometry of h and X, it was an intensive object of study, in particular in the case r = 1, i.e. G = P" 2 " 1 . Several questions are natural and their answer is known for certain X, m, r (e.g. X = P 1 , r = 1, see [GIS] and [R]). Fix X, m, and r what are the possible h*(TG)Ί Fix also h and h*(TG); what is the relation between the deformations of h*(TG) as abstract bundle on X and its deformations coming from deformations of h ? For instance, if X is a curve, the set of nearby bundles can be stratified according to the numerical invariants of an Harder-Narasimhan filtration of the bundles ("Shatz stratification" in the sense of [B2]; see also [He]) (if X = P 1 this is exactly the stratification according to isomorphism classes). Here we study a refined problem: fix X,m 9 r 9 and Q, and study the possible S obtained from different surjections q: V -> Q. The method is very simple: study the differential of the corresponding map of functors; under very strong cohomological conditions we will get a surjectivity 201 202 EDOARDO BALLICO PUBLISHED BY PACIFIC JOURNAL OF MATHEMATICS, A NON-PROFIT CORPORATION