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 isomorphic to the ring of integral piecewise Laurent polynomials on the associated fan. Analogues of this description hold for other complex-oriented equivariant cohomology theories, as we confirm in the case of homotopical complex cobordism, which is the universal example. We also prove that the Borel versions of the equivariant K-theory and complex cobordism rings of more general singular toric varieties, namely those whose integral cohomology is concentrated in even dimensions, are isomorphic to rings of appropriate piecewise formal power series. Finally, we confirm the corresponding descriptions for any smooth, compact, projective toric variety, and rewrite them in a face ring context. In many cases our results agree with those of Vezzosi and Vistoli for algebraic K-theory, Anderson and Payne for operational K-theory, Krishna and Uma for algebraic cobordism, and Gonzalez and Karu for operational cobordism; as we proceed, we summarize the details of these coincidences.

Abstract. We make explicit Poincaré duality for the equivariant K-theory of equivariant complex projective spaces. The case of the trivial group provides a new approach to the K-theory orientation [3].2000 Mathematics Subject Classification. 57R91, 55N15, 55N91. Introduction.In well behaved cases one expects the cohomology of a finite complex to be a contravariant functor of its homology. However, orientable manifolds have the special property that the cohomology is covariantly isomorphic to the homology, and hence in particular the cohomology ring is self-dual. More precisely, Poincaré duality states that taking the cap product with a fundamental class gives an isomorphism between homology and cohomology of a manifold.Classically, an n-manifold M is a topological space locally modelled on ‫ޒ‬ n , and the fundamental class of M is a homology class in H n (M). Equivariantly, it is much less clear how things should work. If we pick a point x of a smooth G-manifold, the tangent space V x is a representation of the isotropy group G x , and its G-orbit is locally modelled on G × G x V x ; both G x and V x depend on the point x. It may happen that we have a W -manifold, in the sense that there is a single representation W so that V x is the restriction of W to G x for all x, but this is very restrictive. Even if there are fixed points x, the representations V x at different points need not be equivalent. It is therefore not clear even in which dimension we should hope to find a fundamental class. In general one needs complicated apparatus to provide a suitable context [6], and ordinary cohomology is especially complicated. Fortunately, particular examples can be better behaved.The purpose of the present paper is to look at the very concrete example of linear complex projective spaces: these are not usually W -manifolds for any W , but we observe that in equivariant K-theory there is a natural choice of fundamental class, and we make the resulting Poincaré duality isomorphism explicit. In the non-equivariant case this gives an elementary approach to the classical K-theory fundamental class [3].

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 Hararda, 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.

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