On a cohomological property of the center of a resolution

Abstract

In this note, we explore the cohomological property of the codimension of the center of a resolution. In particular, we define a resolution $f:X^{\prime }\to X$ to be $q$-birational if the center of $f$ satisfies $\mathrm{codim}\mathrm{Cent}(f)\ge q+1$, and we prove that $R^i{f}_{\ast }{\mathcal{O}}_X(E)=0$ for every $1\le i\le q-1$ and every $f$-anti-nef effective $f$-exceptional divisor $E$ on $X^{\prime }$ if $X$ is $(R_q)$ and $(S_{q+1})$. We also discuss a partial converse of the theorem.

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