On vanishing of higher direct images of the structure sheaf

TL;DR

This paper proves the vanishing of the first higher direct image of the structure sheaf for certain proper birational morphisms over regular schemes, and provides counterexamples for the second direct image in higher dimensions.

Contribution

It establishes new vanishing results for the first direct image and presents counterexamples for the second in dimensions greater than two.

Findings

First direct image vanishes for proper birational maps over regular schemes.

Counterexamples show second direct image does not vanish in dimensions > 2.

Second direct image can have isolated support in higher dimensions.

Abstract

We show the vanishing of the first direct image of the structure sheaf of a normal scheme $X$ which is mapped properly and birationally over a regular scheme of any dimension. On the other hand, for any dimension greater than two, we show examples of a proper birational morphism from a normal and Cohen-Macaulay scheme to a regular scheme such that the second direct image does not vanish and has an isolated support.