Let $X$ be a smooth projective variety defined over an algebraically closed field of positive characteristic $p$ whose tangent bundle is nef. We prove that $X$ admits a smooth morphism $X \to M$ such that the fibers are Fano varieties with nef tangent bundle and $T_M$ is numerically flat. We also prove that extremal contractions exist as smooth morphisms. As an application, we prove that, if the Frobenius morphism can be lifted modulo $p^2$, then $X$ admits, up to a finite étale Galois cover, a smooth morphism onto an ordinary abelian variety whose fibers are products of projective spaces.

Let $P$ be a maximal proper parabolic subgroup of a connected simple linear algebraic group $G$, defined over $\mathbb{C}$, such that $n\,:=\,{{\dim}_{\mathbb{C}}}\,G/P\,\ge \,4$. Let $\iota :\,Z\,\to \,G/P$ be a reduced smooth hypersurface of degree at least $\left( n\,-\,1 \right)\,.\,\deg \text{ree}\left( T\left( G/P \right) \right)/n$. We prove that the restriction of the tangent bundle ${{\iota }^{*}}\,TG/P$ is semistable.

A line bundle over a complex projective variety is called big and 1-ample if a large multiple of it is generated by global sections and a morphism induced by the evaluation of the spanning sections is generically finite and has at most 1-dimensional fibers. A vector bundle is called big and 1-ample if the relative hyperplane line bundle over its projectivisation is big and 1-ample.

The main theorem of the present paper asserts that any complex projective manifold of dimension 4 or more, whose tangent bundle is big and 1-ample, is equal either to a projective space or to a smooth quadric. Since big and 1-ample bundles are ‘almost’ ample, the present result is yet another extension of the celebrated Mori paper ‘Projective manifolds with ample tangent bundles’ (Ann. of Math. 110 (1979) 593–606).

The proof of the theorem applies results about contractions of complex symplectic manifolds and of manifolds whose tangent bundles are numerically effective. In the appendix we re-prove these results.

For a smooth projective variety $X$ of dimension $n$ in a projective space ${\bb P}^N$ defined over an algebraically closed field $k$, the Gauss map is a morphism from $X$ to the Grassmannian of $n$-plans in ${\bb P}^N$sending $x \in X$ to the embedded tangent space $T_xX \subset {\bb P}^N$. The purpose of this paper is to prove the generic injectivity of Gauss maps in positive characteristic for two cases; (1) weighted complete intersections of dimension $n \geqslant 3$ of general type; (2) surfaces or 3-folds with $\mu$-semistable tangent bundles; based on a criterion of Kaji by looking at the stability of Frobenius pull-backs of their tangent bundles. The first result implies that a conjecture of Kleiman-Piene is true in case $X$ is of general type of dimension $n \geqslant 3$. The second result is a generalization of the injectivity for curves.