Let $f\colon X \to S$ be a smooth proper morphism of schemes. If $S$ is of characteristic zero (i.e., $S$ is a $\mathbb Q$-scheme), then Deligne has shown:

$R^af_*\Omega^b_{X/S}$ is locally free for all $a,b \geq 0$.

The Hodge-De Rham spectral sequence $E^{ab}_1 = R^af_*\Omega^b(X/S) \Rightarrow H_{\rm DR}^{a+b}(X/S)$ degenerates in $E_1$.

This is known to fail in positive characteristic. Mumford gave examples of smooth projective surfaces over algebraically closed fields. Nevertheless there are several interesting cases of schemes $X \to S$ in characteristic $p > 0$ where I know this to be true:

a. $X$ is an abelian scheme, a relative curve, a global complete intersection in projective space, or a K3-surface over $S$.

b. $X$ is a smooth projective toric variety over a field.

c. There is also a criterion of Deligne and Illusie which in particular shows 1. und 2. to hold if $\dim(X/S) < p$ and $X$ can be lifted to $W_2(S)$.

**Question**: What are other examples in positive characteristic, where 1. and 2. hold?

There is also a variant of the result of Deligne for logarithmic schemes. In particular I would be also interested for examples where the logarithmic analogue of 1. and 2. hold.

ADDITION: I am taking the risk to name two examples of smooth projective schemes over a field, where I would not be too surprised if (1. and) 2. hold, but where I know of no results:

d. $X$ is Calabi-Yau (i.e., its canonical bundle is trivial).

Edit: As Torsten Ekedahl pointed out below this definition of "Calabi-Yau" is not the "right" one (not even in char. $> 2$ as I wrote in an earlier edit) and does not imply in general that the Hodge-De Rham spectral sequence degenerates.

e. $X$ is $G$-spherical for a reductive group $G$ (i.e., $X$ carries a $G$-action such that there exists a dense $B$-orbit, where $B$ is a Borel subgroup of $G$).

Edit: Again one might to have exclude some small primes depending on the Dynkin type of $G$.

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