Mayer–Vietoris sequences and equivariant $K$-theory rings of toric varieties

We apply a Mayer–Vietoris sequence argument to identify the Atiyah–Segal equivariant complex $K$-theory rings of certain toric varieties with rings of integral piecewise Laurent polynomials on the associated fans. We provide necessary and sufficient conditions for this identification to hold for toric varieties of complex dimension $2$, including smooth and singular cases. We prove that it always holds for smooth toric varieties, regardless of whether or not the fan is polytopal or complete. Finally, we introduce the notion of fans with “distant singular cones” and prove that the identification holds for them. The identification has already been made by Harada, Holm, Ray and Williams in the case of divisive weighted projective spaces; in addition to enlarging the class of toric varieties for which the identification holds, this work provides an example in which the identification fails. We make every effort to ensure that our work is rich in examples.

Keywords

toric variety, fan, equivariant $K$-theory, piecewise Laurent polynomial

2010 Mathematics Subject Classification

Primary 19L47. Secondary 14M25, 55N15, 55N91, 57R18.

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TH was partially supported by the Simons Foundation through Grants #266377 and #79064 and by the National Science Foundation through Grant #DMS-1711317. GW was partially supported by a Research in Pairs grant from the London Mathematical Society.

Received 9 April 2018

Received revised 31 July 2018

Accepted 12 July 2018

Published 28 November 2018