Title: On the Humphreys conjecture on support varieties of tilting modules,
Abstract: Let $G$ be a simply-connected semisimple algebraic group over an
algebraically closed field of characteristic $p$, assumed to be larger than the
Coxeter number. The "support variety" of a $G$-module $M$ is a certain closed
subvariety of the nilpotent cone of $G$, defined in terms of cohomology for the
first Frobenius kernel $G_1$. In the 1990s, Humphreys proposed a conjectural
description of the support varieties of tilting modules; this conjecture has
been proved for $G = \mathrm{SL}_n$ in earlier work of the second author.
In this paper, we show that for any $G$, the support variety of a tilting
module always contains the variety predicted by Humphreys, and that they
coincide (i.e., the Humphreys conjecture is true) when $p$ is sufficiently
large. We also prove variants of these statements involving "relative support
varieties."