A general homological Kleiman–Bertini theorem

Abstract

Let [math] be a smooth algebraic group acting on a variety [math] . Let [math] and [math] be coherent sheaves on [math] . We show that if all the higher [math] sheaves of [math] against [math] -orbits vanish, then for generic [math] , the sheaf [math] vanishes for all [math] . This generalizes a result of Miller and Speyer for transitive group actions and a result of Speiser, itself generalizing the classical Kleiman–Bertini theorem, on generic transversality, under a general group action, of smooth subvarieties over an algebraically closed field of characteristic 0.