The KH-Theory of Complete Simplicial Toric Varieties and the Algebraic K-Theory of Weighted Projective Spaces

TL;DR

This paper develops a method to compute the homotopy $ ext{KH}$-theory of complete simplicial toric varieties using open covers, and applies it to weighted projective spaces, revealing their $ ext{K}$-theoretic properties.

Contribution

It introduces a new approach to determine the $ ext{KH}$-theory of toric varieties and establishes conditions for $ ext{KH}$-theory equivalences, also analyzing $ ext{K}$-regularity of toric surfaces and spaces.

Findings

Homotopy $ ext{KH}$-theory of complete simplicial toric varieties can be computed via open covers.

Rational $ ext{KH}$-theory of weighted projective spaces is determined.

Weighted projective spaces are not $ ext{K}_1$-regular and, in higher dimensions, not $ ext{K}_0$-regular.

Abstract

We show that, for a complete simplicial toric variety $X$, we can determine its homotopy $\KH$-theory entirely in terms of the torus pieces of open sets forming an open cover of $X$. We then construct conditions under which, given two complete simplicial toric varieties, the two spectra $\KH(X) \otimes \Q$ and $\KH(Y) \otimes \Q$ are weakly equivalent. We apply this result to determine the rational $\KH$-theory of weighted projective spaces. We next examine $\K$-regularity for complete toric surfaces; in particular, we show that complete toric surfaces are $\K_{0}$-regular. We then determine conditions under which our approach for dimension 2 works in arbitrary dimensions, before demonstrating that weighted projective spaces are not $\K_{1}$-regular, and for dimensions bigger than 2 are also not in general $\K_{0}$-regular.