We give applications of integral canonical models of orthogonal Shimura varieties and the Kuga-Satake morphism to the arithmetic of $K3$ surfaces over finite fields. We prove that every $K3$ surface of finite height over a finite field admits a characteristic $0$ lifting whose generic fibre is a $K3$ surface with complex multiplication. Combined with the results of Mukai and Buskin, we prove the Tate conjecture for the square of a $K3$ surface over a finite field. To obtain these results, we construct an analogue of Kisin’s algebraic group for a $K3$ surface of finite height and construct characteristic $0$ liftings of the $K3$ surface preserving the action of tori in the algebraic group. We obtain these results for $K3$ surfaces over finite fields of any characteristics, including those of characteristic $2$ or $3$ .

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We study deformations of rational curves and their singularities in positive characteristic. We use this to prove that if a smooth and proper surface in positive characteristic p is dominated by a family of rational curves such that one member has all δ-invariants (resp. Jacobian numbers) strictly less than {\frac{1}{2}(p-1)} (resp. p), then the surface has negative Kodaira dimension. We also prove similar, but weaker results hold for higher-dimensional varieties. Moreover, we show by example that our result is in some sense optimal. On our way, we obtain a sufficient criterion in terms of Jacobian numbers for the normalization of a curve over an imperfect field to be smooth.