Let $X$ be a smooth algebraic variety over an algebraically closed field $k$ of characteristic $p>\dim X$. The motivation for my question comes from the following results.



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> **1.** If $X$ is *Frobenius split* (the $p$-th power map $\mathcal{O}\_X \to F\_\* \mathcal{O}\_X$ admits n $\mathcal{O}\_X$-linear splitting) then the Kodaira vanishing theorem holds for $X$.
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The proof uses nothing but Serre vanishing and the projection formula.



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> **2.** If the complex $F\_\* \Omega^\bullet\_X$ is quasi-isomorphic to a complex with zero differentials, then the Kodaira-Akizuki-Nakano vanishing theorem holds for $X$.
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The proof uses Cartier isomorphism, hypercohomology spectral sequences, Serre vanishing and the projection formula and is similar to that of **1.**



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> **3 (Deligne-Illusie 1987).** If $X$ lifts to $W\_2(k)$, then the complex $F\_\* \Omega^\bullet\_X$ is quasi-isomorphic to a complex with zero differentials.
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> **4 (Buch-Thomsen-Lauritzen-Mehta 1995).** If $X$ is strongly Frobenius split (that is, $0\to B\_1\to Z\_1\to \Omega^1\_X\to 0$ splits, where $Z\_i$ and $B\_i$ are cocycles/coboundaries in $F\_\* \Omega^\bullet\_X$), then $X$ and $F$ lift to $W\_2(k)$ and the Bott vanishing theorem holds for $X$.
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My (maybe incorrect) feeling is that strong Frobenius splitting and lifting of the Frobenius to $W\_2(k)$ are quite uncommon, Frobenius splitting is a common behavior "on the Fano side" and that lifting of $X$ to usually $W\_2(k)$ exists.



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> **Question.** Are there examples of Frobenius split varieties for which $F\_\* \Omega^\bullet\_X$ is not quasi-isomorphic to a complex with zero differentials (for example, because the Hodge spectral sequence does not degenerate, see also [this question on the Hodge spectral sequence](https://mathoverflow.net/questions/58834/degeneration-of-the-hodge-spectral-sequence))? If yes (that's my intuition here), does Frobenius splitting imply some weaker property of $F\_\* \Omega^\bullet\_X$ which implies Kodaira vanishing?
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**Edit.** Note that Frobenius splitting just states that the complex $F\_\* \Omega^\bullet\_X$ is quasi-isomorphic to a complex whose first differential $C^0 \to C^1$ is zero.



Frobenius splitting and derived Cartier isomorphism
Keeping the notation of the question $X$ is a smooth variety over an algebraically closed field $k$ of characteristic $p>\dim X$. Just following along from Decomposition of the de Rham Complex by V Srinivas, we see that the obstruction to lifting the pair $(X,F)$ to a pair $(X^{(2)}, F^{(2)})$ where $X^{(2)}$ is a lift of the variety to $W\_2(k)$ and $F^{(2)}$ is a lift of Frobenius consistent with all diagrams is exactly the class $\zeta\in \mathrm{Ext}^1(\Omega\_{X/k}^1, B\_X^1)$ that corresponds to the sequence $0\to B\_X^1\to Z\_X^1\to \Omega\_{X/k}^1\to 0$.


If we look at the sequence $0\to \mathcal{O}\_X\to F\_\* \mathcal{O}\_X\to B\_X^1\to 0$ and take $\mathrm{Hom}(\Omega^1, -)$ we get a connecting homomorphism in the long exact sequence $\mathrm{Ext}^1(\Omega^1, B^1)\stackrel{\delta}{\to} \mathrm{Ext}^2(\Omega^1, \mathcal{O}\_X)$. It is well known that the obstruction to lifting lies in $\mathrm{Ext}^2(\Omega^1, \mathcal{O}\_X)\simeq H^2(X, \mathcal{T}\_X)$, but what is not well-known is that the obstruction class in this case is exactly the image of $\zeta$ under $\delta$. So $\delta$ acts as sort of a forgetful map for obstruction to lifting the pair to obstruction for lifting the variety without lifting Frobenius.


While it is possible that a Frobenius split variety has non-zero obstruction class $\zeta$ (no example comes to mind right now) and hence the pair doesn't lift, this splitting assumption actually gives us lots of information when coupled with the above information.


We see that $0\to \mathcal{O}\_X\to F\_\* \mathcal{O}\_X\to B\_X^1\to 0$ splitting gives us $\mathrm{Ext}^1(\Omega^1, F\_\*\mathcal{O}\_X)\twoheadrightarrow \mathrm{Ext}^1(\Omega^1, B^1)\stackrel{\delta}{\to} \mathrm{Ext}^2(\Omega^1, \mathcal{O}\_X)$, so in fact $\delta=0$. This means that it doesn't matter whether or not we can lift the pair, all we had to know was that the obstruction to lifting $X$ was the image of $\zeta$ under $\delta$ which is $0$.


Thus any smooth Frobenius split variety lifts to $W\_2(k)$ and since we assumed $p>\dim X$ we also get that the Hodge-de Rham spectral sequence degenerates at $E\_1$ by work of Deligne and Illusie.


