Intersection cohomology is a cohomology theory for describing the topology of
singular algebraic varieties. We are interested in studying intersection cohomology
of complete complex algebraic varieties endowed with an action of an algebraic
torus. An important invariant in the classification of torus actions is the complexity.
It is defined as the codimension of a general torus orbit. Classification of torus
actions is intimately related to questions of convex geometry.
In this talk, we focus on the calculation of the (rational) intersection cohomology
Betti numbers of complex complete normal algebraic varieties with a torus action of
complexity one. Intersection cohomology for the surface and toric cases was studied
by Stanley, Fieseler–Kaup, Braden–MacPherson and many others. We suggest a natural
generalisation using the geometric and combinatorial approach of Altmann, Hausen,
and Süß for normal varieties with a torus action in terms of the language of