A realization theorem for modules of constant Jordan type and vector bundles

Abstract:

Let $E$ be an elementary abelian $p$-group of rank $r$ and let $k$ be a field of characteristic $p$. We introduce functors $\mathcal {F}_i$ from finitely generated $kE$-modules of constant Jordan type to vector bundles over projective space $\mathbb {P}^{r-1}$. The fibers of the functors $\mathcal {F}_i$ encode complete information about the Jordan type of the module.

We prove that given any vector bundle $\mathcal {F}$ of rank $s$ on $\mathbb {P}^{r-1}$, there is a $kE$-module $M$ of stable constant Jordan type $[1]^s$ such that $\mathcal {F}_1(M)\cong \mathcal {F}$ if $p=2$, and such that $\mathcal {F}_1(M) \cong F^*(\mathcal {F})$ if $p$ is odd. Here, $F\colon \mathbb {P}^{r-1}\to \mathbb {P}^{r-1}$ 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 $[1]^s$, then the Chern numbers $c_1,\dots ,c_{p-2}$ of $\mathcal {F}_1(M)$ are divisible by $p$.