Hodge theory of cyclic covers branched over a union of hyperplanes

TL;DR

This paper investigates the Hodge theory of cyclic covers of projective space branched over hyperplane arrangements, proving cases of the Beilinson-Hodge and generalized Hodge conjectures, especially for covers with normal crossings and prime degrees.

Contribution

It establishes new cases where the Beilinson-Hodge and generalized Hodge conjectures hold for cyclic covers, including when the branch divisor has normal crossings and the degree is prime.

Findings

Beilinson-Hodge conjecture holds under certain coprimality conditions.

Generalized Hodge conjecture holds for toroidal resolutions when degree is prime.

Partial extensions to nonabelian covers are discussed.

Abstract

Suppose that Y is a cyclic cover of projective space branched over a hyperplane arrangement D, and that U is the complement of the ramification locus in Y. The first theorem implies that the Beilinson-Hodge conjecture holds for U if certain multiplicities of D are coprime to the degree of the cover. For instance this applies when D is reduced with normal crossings. The second theorem shows that when D has normal crossings and the degree of the cover is a prime number, the generalized Hodge conjecture holds for any toroidal resolution of Y. The last section contains some partial extensions to more general nonabelian covers.