Let X be a proper smooth variety over an algebraically closed field k of characteristic p > 0. This thesis studies the p-primary component of the Brauer group of X, denoted by Br(X)[p^∞]. In the first part, we start by recalling that Br(X)[p^∞]≅(Q_p/Z_p)^{r−ρ}⊕H^3(X,Z_p(1))[p^∞], where r denotes the multiplicity of slope 1 in H^2_{cris}(X/W ), and ρ is the Picard number of X. The term H^3(X, Z_p(1))[p∞] denotes the torsion part of the projective limit lim H^3_{fppf}(X, μ_{p^n}). By applying Illusie’s theory of logarithmic de Rham-Witt sheaves, we show that H^3(X,Z_p(1))[p^∞] fits into a short exact sequence 0→U(k)→H^3(X,Z_p(1))[p^∞]→J→0, where U is a connected commutative unipotent group over k and J is a finite group. We then review Crew's formula and Ekedahl's inequality, which provide information about the dimension of U.
In the second part, we employ a formula introduced by Skorobogatov to explicitly compute Br(A)[p^n] for certain abelian varieties. Using this approach, we determine the dimension of U [p], the p-torsion subgroup of U, for an arbitrary principally polarized abelian variety in terms of the Ekedahl-Oort type of A, when p ≠ 2. Combined with a calculation for non-polarized abelian varieties using Kraft’s cycles, this allows us to identify the isogeny class of U for all abelian threefolds. Finally, we propose a conjectural description of the isogeny class of U for an arbitrary supergeneral abelian variety.
The third part contains further applications, some conjectures and partial results in the direction of these conjectures. We relate the fppf cohomology of μpn to the crystalline cohomology at level n. In particular, we obtain an injectivity criterion that generalizes a result of Ogus. We establish smoothness of flat Artin–Mazur formal groups for all abelian threefolds and for a large class of abelian fourfolds, without relying on Ekedahl’s diagonal t-structure.