Homogeneous spaces over a field of positive characteristic

Across all areas of mathematics, we try to classify objects up to isomorphism. For an algebraic geometer, such objects are algebraic varieties, namely zero sets of polynomial equations, in this case with coefficients in some algebraically closed field $k$. Such a goal is far too difficult: a natural strategy is then to focus on special classes of objects that have a lot of symmetries. In this talk, we concentrate on projective (in a sense, compact) varieties that are homogeneous: a linear algebraic group (a subgroup of some $\mathrm{GL}_N$) acts transitively on them. Classically, they are called “flag varieties” and are described as quotients $G/P$, where $G$ is a product of simple factors, and $P$ contains a maximal solvable subgroup. Over the complex numbers, these objects and their classification are well understood. However, when the characteristic of $k$ is positive, the situation becomes more complicated, due to the presence of the Frobenius morphism. Through basic examples in $\mathrm{SL}_2$ and $\mathrm{SL}_3$, we will illustrate the classification of all projective homogeneous varieties for $p \geq 5$. Then we will discuss a first geometric application, describing the automorphism group of $G/P$, which generalizes a classical result of Demazure.