On the length of perverse sheaves on hyperplane arrangements

Abstract

In this article we address the length of perverse sheaves arising as direct images of rank one local systems on complements of hyperplane arrangements. In the case of a cone over an essential line arrangement with at most triple points, we provide combinatorial formulas for these lengths. As by-products, we also obtain in this case combinatorial formulas for the intersection cohomology Betti numbers of rank one local systems on the complement with same monodromy around the planes.

Length and decomposition of the cohomology of the complement to a hyperplane arrangement

“…Let A be an arrangement in C n , and j :Ũ → C n the inclusion of the open complement of the hyperplanes. We will now see that (5) means that the number of decomposition factors behaves as the Poincaré polynomial of the set of flats. We follow [13].…”

“…The case of central line arrangements and a rank 1 locally constant sheaf is treated in [1,3]. See also [5] for more general results.…”

“…We first note that all composition factors are of the form [N F ], where F is a flat. This follows by induction on the number of hyperplanes of the arrangement from the sequence(5), as in the proof of Theorem 4.3. The base cases are C X [n] = N X and N F…”