Vanishing theorems in algebraic-arithmetic geometry

Classically they are of two sorts: Andreotti-Frankel for Stein manifolds, analogously Artin vanishing on affine varieties, and Artin-Schreier theory on proper varieties in positive characteristic. We ‘ll explain Scholze’s vanishing theorem which holds in characteristic zero and mixes the two theories, and mention our proof of it, which holds in positive characteristic (and thus answers Scholze’s conjecture).