Conic decomposition of a toric variety and its application to cohomology

TL;DR

This paper introduces conic sequences for convex polytopes and applies them to toric varieties, enabling new cohomology vanishing results and Poincaré polynomial calculations for singular cases.

Contribution

It presents a novel conic decomposition method for polytopes and applies it to derive cohomological properties of toric varieties.

Findings

Established vanishing results in rational cohomology

Calculated Poincaré polynomials for many singular toric varieties

Developed an iterated cofibration structure on toric varieties

Abstract

We introduce the notion of a \emph{conic sequence} of a convex polytope. It is a way of building up a polytope starting from a vertex and attaching faces one by one with certain regulations. We apply this to a toric variety to obtain an iterated cofibration structure on it. This allows us to prove several vanishing results in the rational cohomology of a toric variety and to calculate Poincar\'e polynomials for a large class of singular toric varieties.