Generalized Minkowski weights and Chow rings of $T$-varieties

Abstract

We give a combinatorial characterization of Fulton’s operational Chow cohomology groups of a complete, $\mathbb{Q}$-factorial, rational $T$-variety of complexity one in terms of so called generalized Minkowski weights in the contraction-free case. We also describe the intersection product with Cartier invariant divisors in terms of the combinatorial data. In particular this provides a new way of computing top intersection numbers of invariant Cartier divisors combinatorially.