Hyperbolicity of adjoint linear series on varieties with positive tangent bundle

Let $X$ be a smooth projective variety of dimension $n\geq 3$, and let $L$ be an ample line bundle on $X$. In this article, we study the algebraic hyperbolicity of a very general section of the adjoint linear series $|K_X+mL|$ when the tangent bundle $T_X$ of $X$ has suitable positivity properties. As a consequence, we show that the linear system $|K_X+mL|$ is hyperbolic (or pseudo-hyperbolic) for $m\geq 3n+1$, for various classes of polarized pairs $(X,L)$, thus providing new evidence of a conjecture that was proposed by the second and fourth authors. Moreover, when $X$ is abelian, we show that the linear system $|mL|$ is hyperbolic for $m\geq n$, and the same holds when $m\geq n-1$, if $|L|$ has no base divisors. It turns out that these bounds for abelian varieties are sharp. We also prove analogous statements for Kummer varieties and certain classes of hyperelliptic varieties.