Steenbrink-type vanishing for surfaces in positive characteristic (Q6647818)

Let \((X, B)\) be a pair (in the sense of birational geometry) over a field \(k\), \(f \colon Y \to X\) a log resolution, \(E \subset Y\) the reduced exceptional divisor and \(B_Y := f^{-1}_* \lfloor B \rfloor + E\).\N\NSteenbrink vanishing [\textit{J. H. M. Steenbrink}, Astérisque 130, 330--341 (1985; Zbl 0582.32039)] is the following statement: if \(k = \mathbb C\), then\N\[\NR^q f_* \big( \Omega_Y^p (\log B_Y) (-B_Y) \big) = 0\N\]\Nfor \(p + q > n := \dim X\).\N\NA variant of this due to \textit{D. Greb} et al. [Publ. Math., Inst. Hautes Étud. Sci. 114, 87--169 (2011; Zbl 1258.14021)] says: if \(k = \mathbb C\) and \((X, B)\) is log canonical, then\N\[\NR^{n-1} f_* \big( \Omega_Y^p (\log B_Y) (-B_Y) \big) = 0\N\]\Nfor \textit{all} \(p \ge 0\).\N\NThe author deals with the case where \(k\) is a perfect field of characteristic \(p > 0\) and \(n = 2\). He proves that if \((X, B)\) is log canonical and \(p \ge 7\) or if \((X, B)\) is \(F\)-pure, then\N\[\NR^1 f_* \big( \Omega_Y^p (\log B_Y) (-B_Y) \big) = 0\N\]\Nfor all \(p \ge 0\).\N\NThis gives a classification-free proof of the main result of [\textit{P. Graf}, J. Lond. Math. Soc., II. Ser. 104, No. 5, 2208--2239 (2021; Zbl 1507.14007)] in even greater generality (said result did not cover the \(F\)-pure case).