Sections of fibrations onto curves in characteristic p>0.

This talk is based on joint work in progress with Ben Church. Using symplectic geometry, Pieloch showed that every smooth fibration $f\colon X\to \mathbb{P}^1$ of complex projective varieties always admits a section. I will explain how this theorem can be recovered using techniques from Hodge theory and the Minimal Model Program. An advantage of this approach is that it allows for a positive characteristic generalization, by replacing the Hodge theoretic input by a crystalline one. I will also give an example showing that Pieloch’s result can fail in characteristic p>0.