Cohomology of abelian arrangements

Abstract

An abelian arrangement is a finite set of codimension one abelian subvarieties (possibly translated) in a complex abelian variety. In this paper, we study the cohomology of the complement of an abelian arrangement. For unimodular abelian arrangements, we provide a combinatorial presentation for a differential graded algebra whose cohomology is isomorphic to the rational cohomology of the complement. Moreover, this DGA has a bi-grading that allows us to compute the mixed Hodge numbers.

The integer cohomology algebra of toric arrangements

“…It has already been noticed in [2] that the analogous spectral sequence over the rationals collapses at the third page in the more general case of smooth connected divisors intersecting like hyperplanes in a smooth complex projective variety.…”

“…This type of operation has been investigated by Bibby in [2] and by Deshpande and Sutar in [15]. Here we discuss how some of their results generalize to cohomology with integer coefficients, and start with a remark on degeneration of spectral sequences.…”

“…The spectral sequence above is equivalent to the one used by Bibby in [2], induced by the inclusion M pAq ãÑ T . In fact the inclusions T c ãÑ T and SalpAq ãÑ M pAq are homotopy equivalences and the following square is homotopy commutative.…”

“…Among these let us mention the work of Dupont [16] developing algebraic models for complements of divisors with hyperplane-like crossings and of Bibby [2] studying the rational cohomology of complements of arrangements in abelian varieties. Both apply indeed to the case of interest to us, that of toric arrangements.…”

“…This raises the question of whether, as is the case with the OS-Algebra of hyperplane arrangements, the integer cohomology ring is combinatorially determined. The work of Dupont [16] and Bibby [2] mentioned earlier, although more general in scope, does include the case of toric arrangements but falls slightly short of our aim in that on the one hand it uses field coefficients 1 and on the other hand computes only the bigraded module associated to a filtration of the cohomology algebra obtained as the abutment of a spectral sequence.…”