Let be a smooth algebraic group acting on a variety . Let and be coherent sheaves on . We show that if all the higher sheaves of against -orbits vanish, then for generic , the sheaf vanishes for all . 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.
"A general homological Kleiman–Bertini theorem." Algebra Number Theory 3 (5) 597 - 609, 2009. https://doi.org/10.2140/ant.2009.3.597