Local terms for transversal intersections

Abstract

The goal of this note is to show that in the case of ‘transversal intersections’ the ‘true local terms’ appearing in the Lefschetz trace formula are equal to the ‘naive local terms’. To prove the result, we extend the strategy used in our previous work, where the case of contracting correspondences is treated. Our new ingredients are the observation of Verdier that specialization of an étale sheaf to the normal cone is monodromic and the assertion that local terms are ‘constant in families’. As an application, we get a generalization of the Deligne–Lusztig trace formula.