Geodesic
Mayer–Vietoris sequences and equivariant $K$-theory rings of toric varieties
Homology, homotopy, and applications, Tome 21 (2019) no. 1, pp. 375-401.

Voir la notice de l'article provenant de la source International Press of Boston

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.
DOI : 10.4310/HHA.2019.v21.n1.a18
Classification : 19L47, 14M25, 55N15, 55N91, 57R18
Keywords: toric variety, fan, equivariant $K$-theory, piecewise Laurent polynomial 
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     author = {Tara S. Holm and Gareth Williams},
     title = {Mayer{\textendash}Vietoris sequences and equivariant $K$-theory rings of toric varieties},
     journal = {Homology, homotopy, and applications},
     pages = {375--401},
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     url = {http://geodesic.mathdoc.fr/articles/10.4310/HHA.2019.v21.n1.a18/}
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Tara S. Holm; Gareth Williams. Mayer–Vietoris sequences and equivariant $K$-theory rings of toric varieties. Homology, homotopy, and applications, Tome 21 (2019) no. 1, pp. 375-401. doi : 10.4310/HHA.2019.v21.n1.a18. http://geodesic.mathdoc.fr/articles/10.4310/HHA.2019.v21.n1.a18/

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