An algebraic approach to the hyperbolicity of moduli stacks of Calabi-Yau varieties

The moduli stacks of Calabi-Yau varieties are known to enjoy several hyperbolicity properties. The best results have so far been proven using sophisticated analytic tools such as complex Hodge theory. Although the situation is very different in positive characteristic (e.g. the moduli stack of principally polarized abelian varieties of dimension at least 2 contains rational curves), we explain in this note how one can prove many hyperbolicity results by reduction to positive characteristic, relying ultimately on the nonabelian Hodge theory in positive characteristic developed by Ogus and Vologodsky.