{
  "id": "0808.2785",
  "version": "v4",
  "published": "2008-08-20T18:50:24.000Z",
  "updated": "2017-03-13T16:58:07.000Z",
  "title": "Positivity and Kleiman transversality in equivariant K-theory of homogeneous spaces",
  "authors": [
    "Dave Anderson",
    "Stephen Griffeth",
    "Ezra Miller"
  ],
  "comment": "28 pages; v2 has slightly expanded exposition and fixes an error in v1 that treated dualizing sheaves of Schubert varieties as if they were line bundles; v3 is the published version, but includes corrections of the signs of weights in Section 2.3 and the definition of a torus action in Section 6; v4 corrects and simplifies the proofs of Proposition 8.1 and Lemma 10.2",
  "journal": "J. Eur. Math. Soc. (JEMS) 13 (2011), no. 1, 57-84",
  "categories": [
    "math.AG"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2012-02-17T23:57:15.000Z",
      "comment": "29 pages; second version has slightly expanded exposition and fixes an error in the first version that treated dualizing sheaves of Schubert varieties as if they were line bundles; v3 is the published version, but includes corrections of the signs of weights in Section 2.3 and the definition of a torus action in Section 6",
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2017-03-13T16:58:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14M17",
      "14N15",
      "19E08",
      "14F43",
      "32M10",
      "14F17",
      "14F05",
      "14M15",
      "14L30",
      "14L35",
      "57T15",
      "51N30",
      "51N35",
      "14C40",
      "14J17",
      "14C35"
    ],
    "keywords": [
      "homogeneous spaces",
      "equivariant k-theory",
      "euler characteristic",
      "equivariant homological kleiman transversality principle",
      "positivity"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0808.2785A"
    }
  }
}