Images of homogeneous vector bundles and varieties of complexes

The proof of this theorem uses the Borel-Weil-Bott theorem on the cohomology of homogeneous vector bundles [1] together with some facts surrounding the theory of rational resolutions [5]. The application that I have in mind for this theorem is the study of the singularities of the varieties of complexes introduced by Buchsbaum and Eisenbud PL I will first state what these varieties are. Let K°, . . . , K be a sequence of vector spaces. Let F be the direct sum of~Hom(A:, À: 1 ) , . . . , Yiom(K~, K). A point a in V is denoted (a1, . . . , an), where at G Hom^ 1 " 1 , K). A point a in V represents a complex if ai+ x ° at: = 0 for 0 < i < n. The rank of a is the sequence of integers, (rank ax, . . . , rank an), where rank b is the dimension of the image of the homomorphism b. If (m1, . . . , mn) is the rank of a complex, then mx < d imK° ,m n < dimK , and mi + mi+ x < dimK l for 0 < / < n. Conversely, any such sequence is the rank of a complex. Let M be the set of such sequences. If m G M, define the variety of Buchsbaum and Eisenbud, B-E(m), to be the variety of complexes a, such that rank at < mt for 1 < i < n.