Bogomolov-Sommese type vanishing for globally \(F\)-regular threefolds (Q2664660)

The Bogomolov-Sommese (BS) vanishing theorem is a fundamental result about the positivity properties of sheaves of differential forms on smooth projective varieties in characteristic zero, closely related to the Kodaira vanishing theorem. It bounds the Kodaira dimension of an invertible subsheaf of the sheaf of \(p\)-forms by \(p\) (from above). The BS theorem has been generalised to the singular setup of the Minimal Model Program during the last decade. While it is known that Kodaira vanishing fails in general even for smooth varieties over fields of positive characteristic, it continues to hold for \(F\)-split varieties. In the paper under discussion, the author investigates the question whether the BS-theorem continues to hold for \(F\)-split varieties or varieties fulfilling related conditions. Indeed, a corresponding vanishing result is established for \(F\)-split surfaces and invertible subsheaves in the cotangent sheaves of globally \(F\)-regular threefolds (over perfect fields). In arbitrary dimension, under a ``high codimension'' condition on the singularities of a pair \((X, \Delta)\), a vanishing result is established for big and nef subsheaves of the logarithmic cotangent bundle \(\Omega^{[1]}_X (\log\Delta)\) of a globally \(F\)-regular \(n\)-fold pair \((X, \Delta)\).