Higher direct images of dualizing sheaves III

In a series of extremely influential papers that appeared 40 years ago, Kollár showed several surprising properties of higher direct images of dualizing sheaves via projective morphisms. In this talk I will report on recent joint work with Kollár that should be considered a sequel to his original papers (hence the counter in the title). We show that for a flat morphism between varieties with rational singularities even stronger statements are true. In particular, Kollár’s splitting theorem from 40 years ago holds for the higher direct images of the relative dualizing sheaf and some other results. For instance we obtain that the higher direct images of the structure sheaf are locally free. As an application in another direction, using an argument of Y. Kim, this leads to the statement that the identity (or neutral) component of the relative Picard scheme of a flat morphism between varieties with rational singularities is a smooth algebraic group scheme.