Revista Matemática Iberoamericana

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Volume 30, Issue 2, 2014, pp. 373–404
DOI: 10.4171/RMI/785

Published online: 2014-07-08

Depth of cohomology support loci for quasi-projective varieties via orbifold pencils

Enrique Artal Bartolo[1], José Ignacio Cogolludo-Agustín[2] and Anatoly Libgober[3]

(1) Universidad de Zaragoza, Spain
(2) Universidad de Zaragoza, Spain
(3) University of Illinois at Chicago, USA

We describe several relations between a homological invariant of characters of fundamental groups of projective manifolds called depth and maps onto orbicurves. This extends previously studied relations between families of local systems and holomorphic maps onto hyperbolic curves. First, we derive the existence of characters whose depth is bounded below by the number of independent orbifold pencils. Conversely, for some class of characters, we deduce the existence of as many independent pencils as the depth of the character. Second, we show a new relation between depth of characters of the fundamental group and solutions of a certain Diophantine equation (related to the Pell equation) over the field of rational functions. Finally we give a Hodge theoretical characterization of essential coordinate characters of the fundamental groups of the complements to plane curves, i.e., characters whose existence cannot be detected by considering the homology of branched abelian covers.

Keywords: Algebraic curves, characteristic varieties, Albanese maps, orbifold pencils, Pell’s equation on function fields

Artal Bartolo Enrique, Cogolludo-Agustín José Ignacio, Libgober Anatoly: Depth of cohomology support loci for quasi-projective varieties via orbifold pencils. Rev. Mat. Iberoam. 30 (2014), 373-404. doi: 10.4171/RMI/785