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 divisorial fans.

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Last updated: 22 Sep 2023