Subsheaves in bundles on ℙ_n and the penrose transform

Abstract

Let E be a holomorphic vector bundle on the n-dimensional complex projective space ℙ_n, n ≥ 2, of generic splitting type (a₁,…, a_r), a₁ ≥ … ≥a_r. Let, for some 1≤i≤r−1, a_i − a_i+1 = 1. Then the obstruction for the existence of a coherent subsheaf of type (a₁,…, a_i) can be interpreted as a "part" P_i (E) of the Penrose transform. We obtain that, for n≥3, the property "E does not contain a coherent subsheaf of generic splitting type (a₁,…,a_i)" is preserved under restriction to general hyperplanes. If a_j − a_j+1 = 1 for all j = 1,…, r − 1, then this property (fulfilled for j = 1,…, r−1) is equivalent to stability in the sense of Mumford and Takemoto.