Local cohomology of Du Bois singularities and applications to families

In this paper we study the local cohomology modules of Du Bois singularities. Let (R, m) be a local ring; we prove that if R-red is Du Bois, then H-m(i) (R) -> H-m(i) (R-red) is surjective for every i. We find many applications of this result. For example, we answer a question of Kovacs and Schwede [Inversion of adjunction for rational and Du Bois pairs, Algebra Number Theory 10 (2016), 969-1000; MR 3531359] on the Cohen-Macaulay property of Du Bois singularities. We obtain results on the injectivity of Ext that provide substantial partial answers to questions in Eisenbud et al. [Cohomology on toric varieties and local cohomology with monomial supports, J. Symbolic Comput. 29 (2000), 583-600] in characteristic 0. These results can also be viewed as generalizations of the Kodaira vanishing theorem for Cohen-Macaulay Du Bois varieties. We prove results on the set-theoretic Cohen-Macaulayness of the defining ideal of Du Bois singularities, which are characteristic-0 analogs and generalizations of results of Singh-Walther and answer some of their questions in Singh and Walther [On the arithmetic rank of certain Segre products, in Commutative algebra and algebraic geometry, Contemporary Mathematics, vol. 390 (American Mathematical Society, Providence, RI, 2005), 147-155]. We extend results on the relation between Koszul cohomology and local cohomology for F-injective and Du Bois singularities first shown in Hochster and Roberts [The purity of the Frobenius and local cohomology, Adv. Math. 21 (1976), 117-172; MR 0417172 (54 #5230)]. We also prove that singularities of dense F-injective type deform.