Liftability and vanishing theorems for Fano threefolds in positive characteristic I

In our series of papers, we prove that smooth Fano threefolds in positive characteristic lift to the ring of Witt vectors. Moreover, we show that they satisfy Akizuki-Nakano vanishing, $E_1$-degeneration of the Hodge to de Rham spectral sequence, and torsion-freeness of Crystalline cohomologies. In this paper, we establish these results for the case when $\mathrm{\mid }-K_X\mathrm{\mid }$ is very ample and the Picard group is generated by ${\omega }_X$.