On subadditivity of Kodaira dimension in positive characteristic

We show that for a surjective, separable morphism f of smooth projective varieties over an algebraically closed field of positive characteristic such that $f_* \mathcal{O}_X = \mathcal{O}_Y$ subadditivity of Kodaira dimension holds, provided the base is of general type and the Hasse-Witt matrix of the geometric general fiber is not nilpotent.Comment: 24 pages, comments are welcome

Frobenius techniques in birational geometry

This is a survey for the 2015 AMS Summer Institute on Algebraic Geometry about the Frobenius type techniques recently used extensively in positive characteristic algebraic geometry. We first explain the basic ideas through simple versions of the fundamental definitions and statements, and then we survey most of the recent algebraic geometry results obtained using these techniques

Ampleness of the CM line bundle on the moduli space of canonically polarized varieties

We prove that the CM line bundle is ample on the proper moduli space which parametrizes KSBA stable varieties