Dear MO,

**Question 1.** Do you know of an example of a Fano variety which is **not** Frobenius split?

**Background**

(1) A variety $X$ in characteristic $p$ is called *Frobenius split* if there is a "$p$-th root" map $\sigma: \mathcal{O}_X \to \mathcal{O}_X$, that is, an additive map satisfying $\sigma(f^p g) = f\sigma(g)$ and $\sigma(1) = 1$ (in particular, $\sigma(f^p) = f$, so that $\sigma$ is an $\mathcal{O}_X$-linear splitting of the Frobenius map $F: \mathcal{O}_X \to F_* \mathcal{O}_X$). Such varieties enjoy very nice properties, for example, $H^i(X, L)=0$ for $i>0$ for every ample line bundle $L$ on $X$.

In case $X$ is smooth and projective, $X$ is Frobenius split if and only if the map $F: H^{\dim X}(X, \omega_X) \to H^{\dim X}(X, F^* \omega_X)$ is nonzero. Note that $F^* \omega_X = \omega_X^p$. In particular, by Serre duality, $H^{\dim X}(X, \omega_X^p)^\vee = H^{0}(X, \omega_X^{1-p})$ is nonzero, that is, $(1-p)K_X$ is effective - so Frobenius split varieties are ,,on the Fano side''.

(2) A smooth projective variety $X$ is called *Fano* if $\omega_X^{-1}$ is ample. One can prove (Brion, Kumar *Frobenius splitting methods in geometry and representation theory*, Exercise 1.6.E5) that if $X$ is a Fano variety in characteristic $0$, then for $p\gg 0$ the reduction $X_p$ of $X$ mod $p$ is Fano and Frobenius split. This means that counterexamples to Question 1 might be difficult to find.

**Further questions**

Therefore, I am almost sure that, if counterexamples appear in the answers, they will have $\dim X$ (or other invariants of $X$, for example the degree of $K_X$ or its index) big compared to $p$. So I would like to ask:

**Question 2.** Can you find an effective bound $M = M(X) = M(\dim X, \ldots)$, depending on the dimension of $X$ and maybe other relatively simple invariants, such as the degree or index, such that whenever $X$ is a Fano variety in characteristic $p>M(X)$ then $X$ is Frobenius split. For example, does $M = 0$ (this is Question 1) or $M = n$ work?

*Note.* The $M = n$ case reminds me of the requirement in the theorem of Deligne-Illusie about decompositions of the de Rham complex that $p$ has to be $>n$.

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