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In the paper "Cohomology of the complement of an elliptic arrangement", the authors (Levin and Varchenko) consider the complement to an arrangement of (elliptic) hyperplanes in a cartesian power of an elliptic curve and describe its cohomology with coefficients in a non-trivial rank one local system (Actually, their result concerns generic rank one local systems).

What about the case of the trivial rank one local system $\mathbb C$?

I'm mainly interested in the case of the square of an elliptic curve. Maybe this is already known. In this case, a good reference would be welcome

Thanks for your answers.

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The following preprint of Christin Bibby seems to solve your problem completely: http://arxiv.org/abs/1310.4866

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  • $\begingroup$ Well, it gives an explicit cdga model for the complement of an elliptic arrangement, under a certain unimodularity assumption. But this does not automatically yield a closed formula (say, generators and relators) for the cohomology ring of the complement... $\endgroup$ – Alex Suciu Jul 18 '16 at 1:46

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