De Rham cohomology of varieties in positive characteristic

Hodge theory provides an additional structure of Hodge decomposition on the cohomology of a smooth proper variety over complex numbers, and implies cohomology vanishing results such as Kodaira vanishing. For varieties in positive characteristic Hodge decomposition in general fails to exist, but Deligne and Illusie found a very satisfactory substitute for Hodge theory that applies to smooth proper varieties over $\mathbf{F}_p$ that lift to $\mathbf{Z}/p^2$ and have dimension $\leq p$. They proved that in this case the algebraic de Rham complex is quasi-isomorphic to the direct sum of its cohomology sheaves, which induces a decomposition of de Rham cohomology into the direct sum of Hodge cohomology groups, and implies Kodaira vanishing.For liftable varieties of larger dimension Hodge decomposition might still fail to exist, but there are more narrow classes of varieties of arbitrary dimension, such as Frobenius-split and quasi-Frobenius-split ones, for which the de Rham complex decomposes. I will discuss the proof of these decomposition results which relies on interpreting de Rham cohomology via the de Rham stack, introduced in positive characteristic by Drinfeld and Bhatt-Lurie.