The Frobenius trace kernel is ample on projective spaces and, for curves, surfaces and threefolds, only on Fano varieties of Picard rank 1.
Kawakami: On Kawamata–Viehweg type vanishing for three dimensional Mori fiber spaces in positive characteristic, Trans
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Varieties with ample Frobenius-trace kernel
The Frobenius trace kernel is ample on projective spaces and, for curves, surfaces and threefolds, only on Fano varieties of Picard rank 1.