Vanishing theorems for constructible sheaves. II (Q1282086)

In part I of this paper [J. Reine Angew. Math. 471, 115-138 (1996; Zbl 0876.14012)] the authors proved theorems on the vanishing of higher cohomology of direct images of complexes of constructible sheaves. In this paper they take the ``dual'' point of view and derive results concerning the vanishing of lower cohomology of direct images of such complexes. The main result is a statement which yields lower bounds for the vanishing and applies to \(q\)-complete maps in the sense of Andreotti-Grauert. As an application several very general Lefschetz type theorems and Zariski-Lefschetz type theorems (for open varieties) for the cohomoloyg of constructible sheaves are given. In an appendix a base change theorem for weakly constructible complexes is derived.