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.