1,809 research outputs found
Perverse coherent sheaves and the geometry of special pieces in the unipotent variety
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.Comment: 30 pages; minor corrections and addition
Around -independence
In this article we study various forms of -independence (including the
case ) for the cohomology and fundamental groups of varieties over
finite fields and equicharacteristic local fields. Our first result is a strong
form of -independence for the unipotent fundamental group of smooth and
projective varieties over finite fields, by then proving a certain `spreading
out' result we are able to deduce a much weaker form of -independence for
unipotent fundamental groups over equicharacteristic local fields, at least in
the semistable case. In a similar vein, we can also use this to deduce
-independence results for the cohomology of semistable varieties from the
well-known results on -independence for smooth and proper varieties over
finite fields. As another consequence of this `spreading out' result we are
able to deduce the existence of a Clemens--Schmid exact sequence for formal
semistable families. Finally, by deforming to characteristic we show a
similar weak version of -independence for the unipotent fundamental group
of a semistable curve in mixed characteristic.Comment: 23 pages, comments welcom
On some partitions of a flag manifold
In an earlier paper by Kazhdan and the author, a map from the set of
unipotent classes in a reductive connected group over C to the conjugacy
classes in the Weyl group was defined. Here we present some experimental
evidence for a possibly alternative definition of that map in terms of the
pieces in certain partitions of the flag manifold.Comment: 9 page
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