Purity, formality, and arrangement complements

TL;DR

This paper proves that the rational homotopy type of complements of certain algebraic varieties, like toric arrangements, is determined by their cohomology, extending classical results for hyperplane arrangements.

Contribution

It establishes a 'purity implies formality' result in rational homotopy theory and applies it to show that complements of toric arrangements are formal.

Findings

Complement of a toric arrangement is formal

Generalizes classical hyperplane arrangement results

Connects purity with formality in algebraic topology

Abstract

We prove a "purity implies formality" statement in the context of the rational homotopy theory of smooth complex algebraic varieties, and apply it to complements of hypersurface arrangements. In particular, we prove that the complement of a toric arrangement is formal. This is analogous to the classical formality theorem for complements of hyperplane arrangements, due to Brieskorn, and generalizes a theorem of De Concini and Procesi.