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

Dave Anderson; Stephen Griffeth; Ezra Miller

Journal of the European Mathematical Society (2011)

- Volume: 013, Issue: 1, page 57-84
- ISSN: 1435-9855

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topAnderson, 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 -

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