  We study Brou\'e's abelian defect group conjecture for groups of Lie type
using the recent theory of perverse equivalences and Deligne--Lusztig
varieties. Our approach is to analyze the perverse equivalence induced by
certain Deligne--Lusztig varieties (the geometric form of Brou\'e's conjecture)
directly; this uses the cohomology of these varieties, together with
information from the cyclotomic Hecke algebra. We start with a conjecture on
the cohomology of these Deligne--Lusztig varieties, prove various desirable
properties about it, and then use this to prove the existence of the perverse
equivalences predicted by the geometric form of Brou\'e's conjecture whenever
the defect group is cyclic (except possibly for two blocks whose Brauer tree is
unknown). This is a necessary first step to proving Brou\'e's conjecture in
general, as perverse equivalences are built up inductively from various Levi
subgroups.
  This article is the latest in a series by Raphael Rouquier and the author
with the eventual aim of proving Brou\'e's conjecture for unipotent blocks of
groups of Lie type.
