On intersection cohomology with torus action of complexity one, II

TL;DR

This paper investigates the intersection cohomology of varieties with complexity one torus actions, revealing structural properties and explicit calculations for Betti numbers, especially for affine trinomial hypersurfaces.

Contribution

It establishes that components in the decomposition theorem are intersection cohomology complexes of even codimensional subvarieties and provides methods to compute intersection cohomology from weight matrices.

Findings

Odd-dimensional intersection cohomology vanishes for rational complete varieties with complexity one torus action.

Betti numbers of affine trinomial hypersurfaces are explicitly determined.

Structural results on linear torus actions facilitate intersection cohomology computations.

Abstract

We show that the components, appearing in the decomposition theorem for contraction maps of torus actions of complexity one, are intersection cohomology complexes of even codimensional subvarieties. As a consequence, we obtain the vanishing of the odd dimensional intersection cohomology for rational complete varieties with torus action of complexity one. The article also presents structural results on linear torus action in order to compute the intersection cohomology from the weight matrix. In particular, we determine the intersection cohomology Betti numbers of affine trinomial hypersurfaces in terms of their defining equation.