- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:06
Journal of the European Mathematical Society
J. Eur. Math. Soc.
JEMS
1435-9855
1435-9863
General
10.4171/JEMS
http://www.ems-ph.org/doi/10.4171/JEMS
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
13
2011
1
Positivity and Kleiman transversality in equivariant K-theory of homogeneous spaces
Dave
Anderson
University of Michigan, ANN ARBOR, UNITED STATES
Stephen
Griffeth
University of Minnesota, MINNEAPOLIS, UNITED STATES
Ezra
Miller
Duke University, DURHAM, UNITED STATES
Flag variety, equivariant K-theory, Kleiman transversality, homological transversality, Schubert variety, Borel mixing space, rational singularities, Bott–Samelson resolution
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.
$K$-theory
General
57
84
10.4171/JEMS/244
http://www.ems-ph.org/doi/10.4171/JEMS/244