Higher direct images of dualizing sheaves III

TL;DR

This paper proves that for flat morphisms between varieties with rational singularities, the higher direct images of the structure sheaf are locally free, leading to smoothness of the relative Picard scheme's identity component.

Contribution

It establishes the local freeness of higher direct images of the structure sheaf under specific morphisms, correcting a previous lemma and confirming the main theorem.

Findings

Higher direct images are locally free for certain morphisms.

The identity component of the relative Picard scheme is smooth.

Correction of a previous lemma does not affect the main results.

Abstract

We show that for flat morphisms between varieties with rational singularities, the higher direct images of the structure sheaf are locally free. As a consequence, the identity component of the relative Picard scheme is a smooth algebraic group scheme. v.2. Bhatt pointed out that Lemma 9 was incorrect; it is now replaced. The main theorem is unchanged.