A hyperbolicity conjecture for adjoint bundles

Let $X$ be a $n$-dimensional smooth projective variety and $L$ be an ample Cartier divisor on $X$. We conjecture that a very general element of the linear system $\mathrm{\mid }{K}_X+(3n+1)L\mathrm{\mid }$ is a hyperbolic algebraic variety. This conjecture holds for some classical varieties: surfaces, products of projective spaces, and Grassmannians. In this article, we investigate the conjecture for $X$ a toric variety. We confirm the conjecture in the case of smooth projective toric varieties. When $X$ is a Gorenstein toric variety, we show that $\mathrm{\mid }{K}_X+(3n+1)L\mathrm{\mid }$ is pseudo hyperbolic. For a Gorenstein toric threefold $X$, we show that $\mathrm{\mid }{K}_X+9L\mathrm{\mid }$ is hyperbolic.