Cohomology bases of toric surfaces

TL;DR

This paper compares two cohomology bases for compact toric surfaces, showing that the matrices representing their intersection and cup products are inverses, thus linking geometric and algebraic descriptions.

Contribution

It introduces and compares Poincaré dual and cellular bases for cohomology, proving their intersection and cup product matrices are inverses.

Findings

Matrices for intersection and cup products are inverses

Establishes a correspondence between geometric and algebraic cohomology descriptions

Provides explicit bases for cohomology of toric surfaces

Abstract

Given a compact toric surface, the multiplication of its rational cohomology can be described in terms of the intersection products of Weil divisors, or in terms of the cup products of cohomology classes representing specific cells. In this paper, we aim to compare these two descriptions. More precisely, we define two different cohomology bases, the \emph{Poincar\'{e} dual basis} and the \emph{cellular basis}, which give rise to matrices representing the intersection product and the cup product. We prove that these representing matrices are inverse of each other.