Weak Akizuki-Nakano vanishing theorem for singular globally \(F\)-split 3-folds (Q6564483)

It is well-known that the (Kodaira-)Akizuki-Nakano vanishing theorem fails in positive characteristic, but under additional hypotheses, such as being smooth and globally \(F\)-split, it may hold.\N\NThe present paper proves the following version of KAN vanishing: Let \(X\) be a projective globally \(F\)-regular variety over a perfect field of characteristic \(p > 0\) and \(L\) be a globally generated ample line bundle on \(X\) such that the map defined by \(L\) is generically étale onto its image. If \(\dim X \le 3\) and \(X\) has only isolated singularities, then \(X\) satisfies KAN vanishing (for the sheaves of reflexive differentials \(\Omega_X^{[p]}\)).\N\NThere is also a version for the sheaves of Kähler differentials \(\Omega_X^{p}\), under stronger hypotheses. Two applications are given: one to the deformation theory of globally \(F\)-split Fano threefolds and the other to Kodaira vanishing on infinitesimal thickenings of lci subvarieties of projective space. The latter recovers a result by \textit{B. Bhatt} et al. [Am. J. Math. 141, No. 2, 531--561 (2019; Zbl 1425.13008)].