The action on cohomology by compositions of rational maps

TL;DR

This paper investigates how compositions of rational maps affect cohomology, providing criteria for functoriality and revisiting established conditions with new proofs, supported by algebraic topology and intersection theory techniques.

Contribution

It introduces a simple criterion for the functoriality of composed rational maps on cohomology and offers new proofs of existing criteria using algebraic topology.

Findings

Established a sufficient criterion for functoriality of rational map compositions.

Reproved criteria of Diller-Favre, Bedford-Kim, and Dinh-Sibony.

Provided a cautionary example illustrating potential pitfalls.

Abstract

We use intuitive results from algebraic topology and intersection theory to clarify the pullback action on cohomology by compositions of rational maps. We use these techniques to prove a simple sufficient criterion for functoriality of a composition of two rational maps on all degrees of cohomology and we then reprove the criteria of Diller-Favre, Bedford-Kim, and Dinh-Sibony. We conclude with a cautionary example.