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

@article{Holm2019MayerVietorisSA, title={Mayer–Vietoris sequences and equivariant \$K\$-theory rings of toric varieties}, author={Tara S. Holm and Gareth Williams}, journal={Homology, Homotopy and Applications}, year={2019} }

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…

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## References

SHOWING 1-10 OF 33 REFERENCES

The equivariant $K$-theory and cobordism rings of divisive weighted projective spaces

- Mathematics
- 2013

We apply results of Harada, Holm and Henriques to prove that the Atiyah-Segal equivariant complex $K$-theory ring of a divisive weighted projective space (which is singular for nontrivial weights) is…

Describing toric varieties and their equivariant cohomology

- Mathematics
- 2010

Topologically, compact toric varieties can be constructed as identification spaces: they are quotients of the product of a compact torus and the order complex of the fan. We give a detailed proof of…

Toric vector bundles, branched covers of fans, and the resolution property

- Mathematics
- 2006

We associate to each toric vector bundle on a toric variety X(∆) a “branched cover” of the fan ∆ together with a piecewise-linear function on the branched cover. This construction generalizes the…

Operational K-Theory

- Mathematics
- 2015

We study the operational bivariant theory associated to the covariant theory of Grothendieck groups of coherent sheaves, and prove that it has many geometric properties analogous to those of…

On the splitting type of an equivariant vector bundle over a toric manifold

- Mathematics, Physics
- 2000

From the work of Lian, Liu, and Yau on ”Mirror Principle”, in the explicit computation of the Euler data Q = { Q0, Q1, � � � } for an equivariant concavex bundle E over a toric manifold, there are…

Twisted K-theory

- Mathematics
- 2004

Twisted complex K-theory can be defined for a space X equipped with a bundle of complex projective spaces, or, equivalently, with a bundle of C ∗ -algebras. Up to equivalence, the twisting…

On the Integral Cohomology Ring of Toric Orbifolds and Singular Toric Varieties

- Mathematics
- 2015

We examine the integral cohomology rings of certain families of $2n$-dimensional orbifolds $X$ that are equipped with a well-behaved action of the $n$-dimensional real torus. These orbifolds arise…

Introduction to toric varieties

- Mathematics
- 2004

The course given during the School and Workshop “The Geometry and Topology of Singularities”, 8-26 January 2007, Cuernavaca, Mexico is based on a previous course given during the 23o Coloquio…

Higher algebraic K-theory for actions of diagonalizable groups

- Mathematics
- 2001

We study the K-theory of actions of diagonalizable group schemes on noetherian regular separated algebraic spaces: our main result shows how to reconstruct the K-theory ring of such an action from…

Torus-equivariant vector bundles on projective spaces

- Mathematics
- 1988

Introduction For a free Z-module N of rank n, let T = TN be an ^-dimensional algebraic torus over an algebraically closed field k defined by N. Let X = TN emb (A) be a smooth complete toric variety…