On vanishing of higher direct images of the structure sheaf
Shihoko Ishii, Ken-ichi Yoshida
TL;DR
The paper studies vanishing phenomena for higher direct images $R^i \varphi_* \mathcal{O}_X$ of a projective birational map $\varphi:X\to Y$ under various singularity and depth hypotheses. It proves a vanishing theorem for $R^1$ when $Y$ is locally quasi-unmixed, has pseudo-rational singularities in codimension two, and satisfies $S_3$, with $X$ normal, using a reduction to the projective blow-up via Chow’s lemma and depth control of the Rees algebra; this yields $R^1 \varphi_* \mathcal{O}_X=0$ in broad cases, including regular $Y$. Conversely, the authors construct explicit higher-dimensional examples showing $R^2 \varphi_* \mathcal{O}_X\neq 0$, even with $Y$ regular and $X$ normal CM, by extending Cutkosky’s 3-fold construction through iterative Rees-algebra-based enlargements and Macaulayfication. These results delineate the limits of extending rational-singularity-type vanishing in positive characteristic and highlight the role of isolated CM behavior in higher direct images. Overall, the work clarifies when first-order vanishing holds and demonstrates the inevitability of higher-order obstructions in general, offering concrete constructions and a framework via Rees algebras and depth methods.
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.