Commuting Varieties and Cohomological Complexity

TL;DR

This paper investigates the structure and dimension of nilpotent commuting varieties in general linear Lie algebras, linking these geometric properties to the cohomological complexity of modules over Frobenius kernels of algebraic groups.

Contribution

It determines the maximal dimension components of nilpotent commuting varieties for large r, and connects these to module complexities, extending previous inequalities to all simple algebraic groups.

Findings

Dimension of nilpotent commuting varieties is (r+1) * floor(n^2/4) for large r and n≥4.

Explicit complexity values for modules over Frobenius kernels of GL_n.

Established inequalities between complexities of modules restricted to different subgroup schemes.

Abstract

In this paper we determine, for all $r$ sufficiently large, the irreducible component(s) of maximal dimension of the variety of commuting $r$-tuples of nilpotent elements of $\mathfrak{gl}_n$. Our main result is that in characteristic $\neq 2,3$, this nilpotent commuting variety has dimension $(r+1)\lfloor \frac{n^2}{4}\rfloor$ for $n\geq 4$, $r\geq 7$. We use this to find the dimension of the (ordinary) $r$-th commuting varieties of $\mathfrak{gl}_n$ and $\mathfrak{sl}_n$ for the same range of values of $r$ and $n$. Our principal motivation is the connection between nilpotent commuting varieties and cohomological complexity of finite group schemes, which we exploit in the last section of the paper to obtain explicit values for complexities of a large family of modules over the $r$-th Frobenius kernel $({\rm GL}_n)_{(r)}$. These results indicate an inequality between the complexities of a rational $G$-module $M$ when restricted to $G_{(r)}$ or to $G(\mathbb F_{p^r})$; we subsequently establish this inequality for every simple algebraic group $G$ defined over an algebraically closed field of good characteristic, significantly extending a result of Lin and Nakano.