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

By David Anderson, Stephen Griffeth, Ezra Miller. 2008

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We prove the conjectures of Graham-Kumar and Griffeth-Ram 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bb9281915abcef77beab474fba18eee58ac92e5a

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

A Chevalley formula for the equivariant quantum $K$-theory of cominuscule varieties (2016, background): “This has been proved for the structure constants Nw,0u,v of KT (X) by Anderson, Griffeth, and Miller [3].”