Cohomology of Local systems

This survey is intended to provide a background for the authors paper [23]. The latter was the subject of the talk given by the second author at the Arrangement Workshop. The central theme of this survey is the cohomology of local systems on quasi-projective varieties, especially on the complements to algebraic curves and arrangements of lines in P 2 . A few of the results of [23] are discussed in section 4 while the first part of this paper contains some of highlights of Deligne's theory [7] and several examples from the theory of Alexander invariants developed mostly by the first author in the series of papers [17] [22]. We also included several problems indicating possible further development. The second author uses the opportunity to thank M. Oka and H. Terao for the hard labor of organizing the Arrangement Workshop.