A Realization Theorem for Modules of Constant Jordan Type and Vector Bundles

Let E be an elementary abelian p-group of rank r and let k be a field of characteristic p. We introduce functors Fi from finitely generated kE-modules of constant Jordan type to vector bundles over projective space Pr−1. The fibers of the functors Fi encode complete information about the Jordan type of the module. We prove that given any vector bundle F of rank s on Pr−1, there is a kE-module M of stable constant Jordan type [1] such that F1(M) ∼= F if p = 2, and such that F1(M) ∼= F ∗(F) if p is odd. Here, F : Pr−1 → Pr−1 is the Frobenius map. We prove that the theorem cannot be improved if p is odd, because if M is any module of stable constant Jordan type [1] then the Chern numbers c1, . . . , cp−2 of F1(M) are divisible by p.