Frobenius splitting and derived Cartier isomorphism - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T11:00:25Z http://mathoverflow.net/feeds/question/59067 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/59067/frobenius-splitting-and-derived-cartier-isomorphism Frobenius splitting and derived Cartier isomorphism Piotr Achinger 2011-03-21T15:05:49Z 2011-09-29T01:13:13Z <p>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.</p> <blockquote> <p><strong>1.</strong> If $X$ is <em>Frobenius split</em> (the $p$-th power map <code>$\mathcal{O}_X \to F_* \mathcal{O}_X$</code> admits n $\mathcal{O}_X$-linear splitting) then the Kodaira vanishing theorem holds for $X$. </p> </blockquote> <p>The proof uses nothing but Serre vanishing and the projection formula.</p> <blockquote> <p><strong>2.</strong> If the complex <code>$F_* \Omega^\bullet_X$</code> is quasi-isomorphic to a complex with zero differentials, then the Kodaira-Akizuki-Nakano vanishing theorem holds for $X$.</p> </blockquote> <p>The proof uses Cartier isomorphism, hypercohomology spectral sequences, Serre vanishing and the projection formula and is similar to that of <strong>1.</strong></p> <blockquote> <p><strong>3 (Deligne-Illusie 1987).</strong> If $X$ lifts to $W_2(k)$, then the complex <code>$F_* \Omega^\bullet_X$</code> is quasi-isomorphic to a complex with zero differentials.</p> <p><strong>4 (Buch-Thomsen-Lauritzen-Mehta 1995).</strong> If $X$ is strongly Frobenius split (that is, <code>$0\to B_1\to Z_1\to \Omega^1_X\to 0$</code> splits, where $Z_i$ and $B_i$ are cocycles/coboundaries in <code>$F_* \Omega^\bullet_X$</code>), then $X$ and $F$ lift to $W_2(k)$ and the Bott vanishing theorem holds for $X$.</p> </blockquote> <p>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.</p> <blockquote> <p><strong>Question.</strong> Are there examples of Frobenius split varieties for which <code>$F_* \Omega^\bullet_X$</code> is not quasi-isomorphic to a complex with zero differentials (for example, because the Hodge spectral sequence does not degenerate, see also <a href="http://mathoverflow.net/questions/58834/degeneration-of-the-hodge-spectral-sequence" rel="nofollow">this question on the Hodge spectral sequence</a>)? If yes (that's my intuition here), does Frobenius splitting imply some weaker property of $F_* \Omega^\bullet_X$ which implies Kodaira vanishing? </p> </blockquote> <p><strong>Edit.</strong> Note that Frobenius splitting just states that the complex <code>$F_* \Omega^\bullet_X$</code> is quasi-isomorphic to a complex whose first differential $C^0 \to C^1$ is zero.</p> http://mathoverflow.net/questions/59067/frobenius-splitting-and-derived-cartier-isomorphism/66514#66514 Answer by Matt for Frobenius splitting and derived Cartier isomorphism Matt 2011-05-31T02:51:23Z 2011-05-31T02:51:23Z <p>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 $0\to B_X^1\to Z_X^1\to \Omega_{X/k}^1\to 0$.</p> <p>If we look at the sequence <code>$0\to \mathcal{O}_X\to F_* \mathcal{O}_X\to B_X^1\to 0$</code> 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.</p> <p>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.</p> <p>We see that <code>$0\to \mathcal{O}_X\to F_* \mathcal{O}_X\to B_X^1\to 0$</code> 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 $0$.</p> <p>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.</p> http://mathoverflow.net/questions/59067/frobenius-splitting-and-derived-cartier-isomorphism/76714#76714 Answer by Piotr Achinger for Frobenius splitting and derived Cartier isomorphism Piotr Achinger 2011-09-29T01:13:13Z 2011-09-29T01:13:13Z <p>I just found a nice reference for this question: K. Joshi "Exotic Torsion, Frobenius Splitting and the Slope Spectral Sequence" (Canad. Math. Bull. Vol. 50 (4), 2007), section 9.</p> <p><strong>Theorem 9.1</strong> (unpublished work of V. B. Mehta). Let X be a smooth, projective, F-split variety over an algebraically closed field $k$ of characteristic $p>0$. Then for all $i+j &lt; p$, the Hodge to de Rham spectral sequence degenerates at $E^{i,j}_1$. In particular, for $i+j = 1$ we have the following exact sequence $$0\to H^0(X, \Omega^1_X)\to H^1_{DR}(X/k)\to H^1(X, \mathcal{O}_X)\to 0.$$ Moreover, any F-split variety with $dim(X) <p><strong>Corollary 9.2.</strong> Let$X/k$be a smooth, projective, Frobenius split variety. Then$X$admits a flat lifting to$W_2\$. </p>