Perverse coherent sheaves and the geometry of special pieces in the unipotent variety

TL;DR

This paper extends the theory of perverse coherent sheaves to broader contexts, introduces a canonical S2-ification process, and applies these methods to prove a conjecture about the structure of special pieces in unipotent varieties.

Contribution

It generalizes the perverse coherent sheaves framework, introduces a derived category version of the middle extension, and constructs a canonical S2-ification to address Lusztig's conjecture.

Findings

Extended perverse coherent sheaves to broader perversities.

Introduced a derived category version of the middle extension.

Provided a uniform construction of varieties for special pieces.

Abstract

Let X be a scheme of finite type over a Noetherian base scheme S admitting a dualizing complex, and let U be an open subset whose complement has codimension at least 2. We extend the Deligne-Bezrukavnikov theory of perverse coherent sheaves 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-extension") 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 geometry of special pieces (certain subvarieties of the unipotent variety of a reductive algebraic group). The conjecture asserts in part that each special piece is the quotient of some variety (previously unknown in the exceptional groups and in positive characteristic) by the action of a certain finite group. We use S2-extension to give a uniform construction of the desired variety.