Geodesic
Positivity and Kleiman transversality in equivariant $K$-theory of homogeneous spaces
Journal of the European Mathematical Society, Tome 13 (2011) no. 1, pp. 57-84 Cet article a éte moissonné depuis la source EMS Press

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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.
DOI : 10.4171/jems/244
Classification : 19-XX, 00-XX
Keywords: Flag variety, equivariant K-theory, Kleiman transversality, homological transversality, Schubert variety, Borel mixing space, rational singularities, Bott–Samelson resolution 
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     author = {Dave Anderson and Stephen Griffeth and Ezra Miller},
     title = {Positivity and {Kleiman} transversality in equivariant $K$-theory of homogeneous spaces},
     journal = {Journal of the European Mathematical Society},
     pages = {57--84},
     year = {2011},
     volume = {13},
     number = {1},
     doi = {10.4171/jems/244},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/244/}
}
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Dave Anderson; Stephen Griffeth; Ezra Miller. Positivity and Kleiman transversality in equivariant $K$-theory of homogeneous spaces. Journal of the European Mathematical Society, Tome 13 (2011) no. 1, pp. 57-84. doi: 10.4171/jems/244

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