The integer cohomology of toric Weyl arrangements

TL;DR

This paper investigates the integer cohomology of the complement of toric Weyl arrangements, establishing torsion freeness under certain conditions, though a key proof contains an error leaving some conjectures unresolved.

Contribution

We prove torsion freeness of the integer cohomology for complements of toric Weyl arrangements defined by cocharacters lattices, advancing understanding in this area.

Findings

Integer cohomology of the complement is torsion free under specific conditions.

The proof of torsion freeness relies on properties of the cocharacters lattice.

A key lemma's proof contains an error, leaving some conjectures unproven.

Abstract

A referee found an error in the proof of the Theorem 2 that we could not fix. More precisely, the proof of Lemma 2.1 is incorrect. Hence the fact that integer cohomology of complement of toric Weyl arrangements is torsion free is still a conjecture. ----- A toric arrangement is a finite set of hypersurfaces in a complex torus, every hypersurface being the kernel of a character. In the present paper we prove that if $\Cal T_{\wdt W}$ is the toric arrangement defined by the \textit{cocharacters} lattice of a Weyl group $\wdt W$, then the integer cohomology of its complement is torsion free.