Positivity and Kleiman transversality in equivariant $K$-theory of homogeneous spaces

Anderson, Dave, Griffeth, Stephen, and Miller, Ezra. "Positivity and Kleiman transversality in equivariant $K$-theory of homogeneous spaces." Journal of the European Mathematical Society 013.1 (2011): 57-84. <http://eudml.org/doc/277804>.

@article{Anderson2011,
abstract = {We prove the conjectures of Graham–Kumar [GrKu08] and Griffeth–Ram [GrRa04] concerning the alternation of signs in the structure constants for torus-equivariant $K$-theory of generalized flag varieties $G/P$. These results are immediate consequences of an equivariant homological Kleiman transversality principle for the Borel mixing spaces of homogeneous spaces, and their subvarieties, under a natural group action with finitely many orbits. The computation of the coefficients in the expansion of the equivariant $K$-class of a subvariety in terms of Schubert classes is reduced to an Euler characteristic using the homological transversality theorem for non-transitive group actions due to S. Sierra. A vanishing theorem, when the subvariety has rational singularities, shows that the Euler characteristic is a sum of at most one term-the top one-with a well-defined sign. The vanishing is proved by suitably modifying a geometric argument due to M. Brion in ordinary $K$-theory that brings Kawamata–Viehweg vanishing to bear.},
author = {Anderson, Dave, Griffeth, Stephen, Miller, Ezra},
journal = {Journal of the European Mathematical Society},
keywords = {flag variety; equivariant $K$-theory; Kleiman transversality; homological transversality; Schubert variety; Borel mixing space; rational singularities; Bott–Samelson resolution; flag variety; equivariant -theory; Borel mixing space},
language = {eng},
number = {1},
pages = {57-84},
publisher = {European Mathematical Society Publishing House},
title = {Positivity and Kleiman transversality in equivariant $K$-theory of homogeneous spaces},
url = {http://eudml.org/doc/277804},
volume = {013},
year = {2011},
}

TY - JOUR
AU - Anderson, Dave
AU - Griffeth, Stephen
AU - Miller, Ezra
TI - Positivity and Kleiman transversality in equivariant $K$-theory of homogeneous spaces
JO - Journal of the European Mathematical Society
PY - 2011
PB - European Mathematical Society Publishing House
VL - 013
IS - 1
SP - 57
EP - 84
AB - We prove the conjectures of Graham–Kumar [GrKu08] and Griffeth–Ram [GrRa04] concerning the alternation of signs in the structure constants for torus-equivariant $K$-theory of generalized flag varieties $G/P$. These results are immediate consequences of an equivariant homological Kleiman transversality principle for the Borel mixing spaces of homogeneous spaces, and their subvarieties, under a natural group action with finitely many orbits. The computation of the coefficients in the expansion of the equivariant $K$-class of a subvariety in terms of Schubert classes is reduced to an Euler characteristic using the homological transversality theorem for non-transitive group actions due to S. Sierra. A vanishing theorem, when the subvariety has rational singularities, shows that the Euler characteristic is a sum of at most one term-the top one-with a well-defined sign. The vanishing is proved by suitably modifying a geometric argument due to M. Brion in ordinary $K$-theory that brings Kawamata–Viehweg vanishing to bear.
LA - eng
KW - flag variety; equivariant $K$-theory; Kleiman transversality; homological transversality; Schubert variety; Borel mixing space; rational singularities; Bott–Samelson resolution; flag variety; equivariant -theory; Borel mixing space
UR - http://eudml.org/doc/277804
ER -