Pure and Applied Mathematics Quarterly

Volume 11 (2015)

Number 2

$K$-theoretic and categorical properties of toric Deligne–Mumford stacks

Pages: 239 – 266

DOI: http://dx.doi.org/10.4310/PAMQ.2015.v11.n2.a3

Authors

Tom Coates (Department of Mathematics, Imperial College London, United Kingdom)

Hiroshi Iritani (Department of Mathematics, Graduate School of Science, Kyoto University, Sakyo-ku, Kyoto, Japan)

Yunfeng Jiang (Department of Mathematics, University of Kansas, Lawrence, Ks., U.S.A.)

Ed Segal (Department of Mathematics, Imperial College London, United Kingdom)

Abstract

We prove the following results for toric Deligne–Mumford stacks, under minimal compactness hypotheses: the Localization Theorem in equivariant $K$-theory; the equivariant Hirzebruch–Riemann–Roch theorem; the Fourier–Mukai transformation associated to a crepant toric wall-crossing gives an equivariant derived equivalence.

Keywords

toric Deligne–Mumford stacks, orbifolds, $K$-theory, localization, derived category of coherent sheaves, Fourier–Mukai transformation, flop, $K$-equivalence, equivariant, variation of GIT quotient

2010 Mathematics Subject Classification

Primary 14A20. Secondary 14F05, 19L47.

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