Fermat varieties of Hodge-Witt type (Q5917776)

Let \(X\) be a smooth projective variety over a perfect field \(k\) of characteristic \(p>0\) and \(W\) the ring of Witt vectors on \(k\). Suppose that \(X\) is of Hodge Witt type, i.e. that \(H^j(X,W\Omega^i)\) is of finite type over \(W\) for all \((i,j)\). The author considers the Fermat variety \[ F_{n,m,p}: X_0^m+X_1^m+ \cdots+ X_{n+1}^m=0 \] of dimension \(n>1\), degree \(m>2\) where \(p\nmid m\), and \(p\not\equiv 1\pmod m\). By using a results of \textit{N. Suwa} [J. Math. Soc. Japan 45, No. 2, 295-300 (1993; Zbl 0815.14013)] he proves that \(F_{n,m,p}\) is of Hodge-Witt type if and only if \[ \begin{aligned} \text{either }&(n,m,p)=\begin{cases} (2,3,p) &\text{with }p\equiv 2\pmod 3\\ (3,3,p) &\text{with }p\equiv 2\pmod 3\\ (3,4,p)&\text{with }p\equiv 3\pmod 4\\ (5,3,p) &\text{with }p \equiv 2\pmod 3 \end{cases}\\ \text{or } &(n,m,p)=(2,7,p) \quad\text{with }p\equiv 2,4 \pmod 7.\end{aligned} \] {}.