Perverse Sheaves and Hyperplane Arrangements

The category of Perverse Sheaves is known to be an Abelian and Artinian category. As a result, we can talk about the length and decomposition factors of a perverse sheaf. In this thesis we look at a class of perverse sheaves arising from local systems on a complement of a hyperplane arrangement and give a concrete (combinatorial) formula to identify the simple ones; that is, those of length one. By the Riemann-Hilbert correspondence, there is an equivalence of categories between perverse sheaves and regular holonomic D-modules. Therefore, we also obtain the analogous criterion for a class of regular holonomic D-modules on the complement of a hyperplane arrangement.

The above result is obtained by studying the cohomology jump loci on the complement of a hyperplane arrangement and relating it with the support of Sabbah's specialization complex, which generalizes the nearby cycles complex of Deligne.