6,474 research outputs found
Characteristic classes of singular toric varieties
In this paper we compute the motivic Chern classes and homology Hirzebruch
characteristic classes of (possibly singular) toric varieties, which in the
complete case fit nicely with a generalized Hirzebruch-Riemann-Roch theorem. As
special cases, we obtain new (or recover well-known) formulae for the
Baum-Fulton-MacPherson Todd (or MacPherson-Chern) classes of toric varieties,
as well as for the Thom-Milnor L-classes of simplicial projective toric
varieties. We present two different perspectives for the computation of these
characteristic classes of toric varieties. First, we take advantage of the
torus-orbit decomposition and the motivic properties of the motivic Chern and
resp. homology Hirzebruch classes to express the latter in terms of dualizing
sheaves and resp. the (dual) Todd classes of closures of orbits. This method
even applies to torus-invariant subspaces of a given toric variety. The
obtained formula is then applied to weighted lattice point counting in lattice
polytopes and their subcomplexes, yielding generalized Pick-type formulae.
Secondly, in the case of simplicial toric varieties, we compute our
characteristic classes by using the Lefschetz-Riemann-Roch theorem of
Edidin-Graham in the context of the geometric quotient description of such
varieties. In this setting, we define mock Hirzebruch classes of simplicial
toric varieties and investigate the difference between the (actual) homology
Hirzebruch class and the mock Hirzebruch class. We show that this difference is
localized on the singular locus, and we obtain a formula for it in which the
contribution of each singular cone is identified explicitly. Finally, the two
methods of computing characteristic classes are combined for proving several
characteristic class formulae originally obtained by Cappell and Shaneson in
the early 1990s.Comment: v2: new references added; many results hold now in greater
generality, e.g. for closed algebraic toric invariant subspaces of toric
varieties which have only DuBois singularities by work of Ishida; motivic
Chern classes are also computed; new examples are worked out in detai
Computing torus-equivariant K-theory of singular varieties
This expository note surveys some results on equivariant K-theory of
varieties with a torus action, focusing on recent work with Sam Payne and
Richard Gonzales. It is based on my contribution to the Clifford Lectures at
Tulane University in March 2015.Comment: 16 page
Computing toric degenerations of flag varieties
We compute toric degenerations arising from the tropicalization of the full
flag varieties and embedded in a
product of Grassmannians. For and we
compare toric degenerations arising from string polytopes and the FFLV polytope
with those obtained from the tropicalization of the flag varieties. We also
present a general procedure to find toric degenerations in the cases where the
initial ideal arising from a cone of the tropicalization of a variety is not
prime.Comment: 35 pages, 6 figure
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