Intersection cohomology on nonrational polytopes

TL;DR

This paper extends intersection cohomology theory to nonrational polytopes by developing sheaf categories on fans, proving a decomposition theorem, and establishing duality principles, thus broadening the understanding of toric varieties beyond rational cases.

Contribution

It introduces a sheaf-theoretic framework for intersection cohomology on nonrational polytopes, including a decomposition theorem and duality theory, generalizing existing concepts for rational fans.

Findings

Decomposition theorem for fan subdivisions

Development of Borel-Moore Verdier duality

Framework for intersection cohomology on nonrational polytopes

Abstract

Viewing a fan as a partially ordered set (of cones) we consider a category of sheaves on the fan which corresponds to a category of equivariant sheaves on the corresponding toric variety if the fan is rational. In this category we define an object which corresponds to the equivariant intersection cohomology complex. Our first main result is the ``elementary'' decomposition theorem for the direct image under subdivision of fans We also develop the Borel-Moore- Verdier duality in the derived category of sheaves on the fan.