Unitary local systems, multiplier ideals, and polynomial periodicity of Hodge numbers (Q1016262)

Consider the situation of a smooth complex projective variety over the complex numbers. For any divisor \(D\) on \(X\), the author studies the space of unitary local systems on the complement of \(D\) (from here on denoted by \(U\)). He considers this space in terms of \textit{parabolic line bundles} (also called \textit{realizations of boundaries of \(X\) on \(D\)}), see Theorem 1.2 in the paper and the list of references where it was proved in various generalities. He then stratifies this space by using multiplier ideals (see the Motivation section of the introduction). For a precise statement see Theorem 1.3 in the paper. This generalizes the results of a number of people including \textit{M. Green} and \textit{R. Lazarsfeld} [Invent. Math. 90, 389--407 (1987; Zbl 0659.14007) and J. Am. Math. Soc. 4, No. 1, 87--103 (1991; Zbl 0735.14004)], \textit{D. Arapura} [Bull. Am. Math. Soc., New Ser. 26, No. 2, 310--314 (1992; Zbl 0759.14016)], \textit{R. Pink} and \textit{D. Roessler} [Math. Ann. 330, No. 2, 293--308 (2004; Zbl 1064.14048)], and \textit{C. Simpson} [Ann. Sci. Éc. Norm. Supér. (4) 26, No. 3, 361--401 (1993; Zbl 0798.14005)]. Although, as the author points out, he relies on Simpson's result in his proof (see section 6). As an application, the author uses these ideas to show the polynomial periodicity of Hodge numbers \(h^{q,0}\) of congruence covers. Further generalizations in this direction (for the Hodge rank function) are also obtained. See in particular Theorems 1.8, 1.9 and the surrounding discussion for a history of this problem. Finally, the author uses some of these ideas to prove characterizations of abelian covers (and explains how some these results even follow immediately from the more classical results). See Corollaries 1.10 through 1.13.