On vanishing of higher direct images of the structure sheaf

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.

We strengthened the main theorem and the corollary. We added some references. We corrected some typos. to appear in Michigan Mathematical Journal