On the topology of rational $\mathbb{T}$-varieties of complexity one

We generalize classical results about the topology of toric varieties to the case of projective $\mathbb{Q}$-factorial $\mathbb{T}$-varieties of complexity one using the language of divisorial fans. We describe the Hodge–Deligne polynomial in the smooth case, the cohomology ring and the Chow ring in the contraction-free case.