A Semicontinuity Result for Monodromy under Degeneration

We x a a prime number l. We denote by E a nite extension of Ql inside a chosen algebraic closure Ql of Ql, by O the ring of integers in E , by F its residue eld, and by F an algebraic closure of F . We take as coeÆcient eld A one of the elds on the following list: F , F , E , or Ql. We work over a eld k in which l is invertible. We are given a smooth connected k-scheme S=k, separated and of nite type, of dimension r 1. In S, we are given a reduced and irreducible closed subscheme Z, of some dimension d 0. We assume that an open dense set V1 Z is smooth over k (a condition which is automatic if the ground eld k is perfect). On S, we are given a constructible A-sheaf F . Because F is constructible, its restriction to S Z is constructible, so there exists a dense open set U in S Z on which F is lisse. Similarly, the restriction of F to V1 is lisse, so there exists a dense open set V in V1 on which F is lisse. Let us denote by j the inclusion of U into S, and by i the inclusion of V into S. Thus we have a lisse A-sheaf jF on U , and a lisse A-sheaf iF on V . In this generality, there is absolutely nothing one can say relating the monodromy of the lisseA-sheaf iF on V to the monodomy of the lisseA-sheaf jF on U:However, there is a class of constructible A-sheaves F on S for which these monodromies are related, namely those \of perverse origin". We say that a constructible A-sheaf F on S is of perverse origin if there exists a perverse A-sheaf M on S such that