On intersection cohomology with torus actions of complexity one

Abstract

The purpose of this article is to investigate the intersection cohomology for algebraic varieties with torus action. Given an algebraic torus \({{\mathbb {T}}}\), one of our result determines the intersection cohomology Betti numbers of any normal projective \(\mathbb {T}\)-variety admitting an algebraic curve as global quotient. The calculation is expressed in terms of a combinatorial description involving a divisorial fan which is the analogous of the defining fan of a toric variety. Our main tool to obtain this computation is a description of the decomposition theorem in this context.