A realization theorem for modules of constant Jordan type and vector bundles. (Q2844846)

Let \(k\) be a field of prime characteristic \(p\). Consider an operator \(x\) on a finite dimensional vector space \(M\) with \(x^p=0\). The Jordan canonical form of \(x\) consists of blocks of size at most \(p\) and is determined by the number of blocks of each size. This information is encoded by saying that the Jordan type of the action is \([p]^{a_p}[p-1]^{a_{p-1}}\cdots[2]^{a_2}[1]^{a_1}\) where there are \(a_i\) blocks of size \(i\). Let \(E\) be an elementary Abelian group. Given a \(kE\)-module \(M\), \(M\) is said to be of \textit{constant Jordan type} if the Jordan types of certain nilpotent operators \(x\in KE\) over all field extensions \(K/k\) are in fact the same. This concept was introduced (in the more general setting of finite group schemes) by \textit{J. F. Carlson, E. M. Friedlander}, and the second author [J. Reine. Angew. Math. 614, 191-234 (2008; Zbl 1144.20025)]. These modules have seen extensive study and application in recent years. One of these applications has been to the construction of algebraic vector bundles on projective varieties, which is the goal of the paper under review.NEWLINENEWLINE For each \(1\leq i\leq p\), the authors construct a functor \(\mathcal F_i\) from the category of finitely generated \(kE\)-modules to the category of coherent sheaves on the projective space \(\mathbb P^{r-1}\), where \(r\) is the rank of \(E\). For a module \(M\) of constant Jordan type \([p]^{a_p}[p-1]^{a_{p-1}}\cdots[2]^{a_2}[1]^{a_1}\), \(\mathcal F_i(M)\) is a locally free sheaf (equivalently, algebraic vector bundle) of rank \(a_i\). The authors consider how the functors behave with respect to syzygies of modules and Serre shifts of sheaves. In particular, it is shown, for \(1\leq i\leq p-1\), that \(\mathcal F_i(M)(-p+i)\cong\mathcal F_{p-i}(\Omega M)\) where the sheaf on the left denotes the \((-p+i)\)-Serre shift of \(\mathcal F_i(M)\). It is also shown that \(\mathcal F_i\) takes the dual of a module to a shift of the dual sheaf.NEWLINENEWLINE The main result of the paper is that the functor \(\mathcal F_1\) can be used to almost realize vector bundles on \(\mathbb P^{r-1}\). In particular, for \(p=2\) and any vector bundle of \(\mathcal F\) of rank \(s\) on \(\mathbb P^{r-1}\), there exists a finitely generated \(kE\)-module \(M\) of Jordan type \([p]^t[1]^s\) for some \(t\) (or what is known as \textit{stable} Jordan type \([1]^s\)) such that \(\mathcal F_1(M)\cong\mathcal F\). For odd primes, it is shown that the pullback of \(\mathcal F\) along the Frobenius morphism \(F\colon\mathbb P^{r-1}\to\mathbb P^{r-1}\) can similarly be realized.NEWLINENEWLINE In the odd prime case, the authors explain in a sense why \(\mathcal F\) itself cannot be realized. If a module \(M\) has stable Jordan type \([1]^s\), it is shown that the Chern numbers \(c_m(\mathcal F_1(M))\) are divisible by \(p\) for \(1\leq m\leq p-2\). The authors note that the divisibility criterion does not necessarily hold for modules that have other constant Jordan types. Indeed, the authors provide examples throughout the paper to illustrate the ideas and results.