April 18, Wednesday, 15:30, room 211

Grigory Solomadin (Moscow State University)

K-theory of toric varieties, convex chains and scissors-congruence

The complex K-theory ring $K(X)$ of a (nice) compact topological space $X$ is by definition the Grothendieck ring of the semigroup of vector bundles (up to isomorphism) over $X$ with respect to the Whitney sum and tensor product. Choosing a topological group $G$ and taking into account different good actions of $G$ on $G$-vector bundles over $X$ one obtains the ring $K_{G}(X)$ of equivariant K-theory. For a smooth projective toric variety $X^{2n}$ and complex torus $T^n$ the ring $K_{T^n}(X)$ has the embedding to the McMullen polytopal algebra $L(\Z^n)$. Forgetting the action structure on vector bundles leads to the embedding of the ring $K(X)$ to the coinvariants of the translation group acting on $L(\Z^n)$, i.e. ring generated by scissors-congruence classes of convex polytopes in $\R^n$. In the talk, we will discuss these embeddings.