Hodge Theory and Moduli of Finite Group Schemes in Positive Characteristic

This thesis studies finite locally free commutative group schemes and their applications to algebraic geometry over fields of positive characteristic. First, we explore the relationship between group schemes and smooth projective varieties with certain pathological Hodge-theoretic properties. By constructing a degeneration of the constant group scheme $\mathbb{Z}/p^2\mathbb{Z}$ to $\alpha_p \oplus \alpha_p$, we give examples of smooth projective families of algebraic varieties in characteristic $p$ such that the dimensions of the de Rham and Hodge cohomology groups of the fibers can be made to jump by an arbitrarily large amount. Along the way, we give a self-contained exposition of the construction of Godeaux--Serre varieties. In joint work with Joshua Mundinger, we then study deperfections of Dieudonné theory in positive characteristic. For all $n \geq 1$, there is a notion of an $n$-smooth group scheme over any $\mathbb{F}_p$-algebra $R$, which may be thought of as a "Frobenius analogue" of an $n$-truncated Barsotti--Tate group over $R$. We prove that the category of $n$-smooth commutative group schemes over $R$ is equivalent to an explicit full subcategory of Dieudonné modules over $R$. As a consequence, we show that the moduli stack $\text{Sm}_n$ of $n$-smooth commutative group schemes is smooth over $\mathbb{F}_p$ and that the natural truncation morphism $\text{Sm}_{n+1} \to \text{Sm}_n$ is smooth and surjective. These results affirmatively answer conjectures of Drinfeld. We conclude by using Dieudonné theory to concretely study moduli stacks of finite locally free group schemes; in particular we give an explicit presentation for the moduli stack of $n$-smooth commutative group schemes of order $p^n$ for any $n \geq 1$.