On intersection cohomology with torus actions of complexity one

TL;DR

This paper computes the intersection cohomology Betti numbers for certain algebraic varieties with torus actions of complexity one, using combinatorial methods related to divisorial fans and the decomposition theorem.

Contribution

It provides a combinatorial formula for intersection cohomology Betti numbers of normal projective torus-varieties with a curve as a global quotient, extending toric variety techniques.

Findings

Explicit Betti number formulas for these varieties.

A new application of the decomposition theorem in this context.

Bridging combinatorial data with intersection cohomology calculations.

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.