Vanishing of higher cohomology and cohomological rationality

Abstract: Our work begins by studying the vanishing of higher cohomology for the derived pushforward of a torsion free sheaf along a modification of integral Noetherian schemes. Leveraging this, we establish a relatively simple recipe for constructing varieties with (mild) rational or Du Bois singularities, some of which is applicable in both zero and nonzero characteristic. Furthermore, we explore notions of singularities that lie in the same cohomological veins of rational singularities for varieties in characteristic zero, including Bhatt's derived splinters and Kov\'{a}cs' birational derived splinters.