Essential p-dimension and Chern numbers

I will explore how the geometry of a smooth projective variety X constrains the actions of a finite p-group G on it. The main result is that when X has a Chern number prime to p, any G-action on X factors through a subgroup of GL_n(k), where n = dim X and k is the base field. Many properties of G-representations thus become properties of G-actions on X. One consequence is a fixed-point theorem which, unlike previous results of this kind, is sensitive to the arithmetic of the base field. The proof combines the theory of essential p-dimension with Karpenko and Merkurjev's computation of the essential p-dimension of p-groups.