Summary: PERVERSE COHERENT SHEAVES AND THE GEOMETRY OF
SPECIAL PIECES IN THE UNIPOTENT VARIETY
PRAMOD N. ACHAR AND DANIEL S. SAGE
Abstract. Let X be a scheme of finite type over a Noetherian base scheme S
admitting a dualizing complex, and let U X be an open set whose comple-
ment has codimension at least 2. We extend the theory of perverse coherent
sheaves, due to Deligne and disseminated by Bezrukavnikov, by showing that a
coherent middle extension (or intersection cohomology) functor from perverse
sheaves on U to perverse sheaves on X may be defined for a much broader class
of perversities than has previously been known. We also introduce a derived
category version of the coherent middle extension functor.
Under suitable hypotheses, we introduce a construction (called "S2-exten-
sion") in terms of perverse coherent sheaves of algebras on X that takes a finite
morphism to U and extends it in a canonical way to a finite morphism to X.
In particular, this construction gives a canonical "S2-ification" of appropriate
X. The construction also has applications to the "Macaulayfication" problem,
and it is particularly well-behaved when X is Gorenstein.
Our main goal, however, is to address a conjecture of Lusztig on the ge-
ometry of special pieces (certain subvarieties of the unipotent variety of a
reductive algebraic group). The conjecture asserts in part that each special