In Birkenhake and Lange's book, they prove a version of the Nakai-Moishezon theorem for complex abelian varieties that says that if $L_0$ is an ample line bundle on a complex abelian variety $X$ of dimension $g$, then a line bundle $L$ is ample if and only if $(L^\nu\cdot L_0^{g-\nu})>0$ for $\nu=1,\ldots,g$ (Corollary 4.3.3 on page 77 of the second edition).

I would love for this to be true in arbitrary characteristic. The problem is that the proof over $\mathbb{C}$ explicitly uses the Hermitian forms associated to the line bundles, and I don't see how this could be adapted to arbitrary characteristic.

Does anyone know if this is true in arbitrary characteristic? Are there any references or at least some idea for a proof?