Today's arXiv update includes Joe Waldron and Jakub Witaszek's paper, “An irregular smooth Fano fourfold in positive characteristic”. Here, irregular means that $\mathrm{H}^1(X,\mathcal{O}_X)\neq 0$. This is not possible for Fano varieties in characteristic zero, by Kodaira vanishing. But the authors construct a Fano fourfold over the algebraic closure of $\mathbb{F}_2$ with $\mathrm{H}^1(X,\mathcal{O}_X)\cong\overline{\mathbb{F}}_2$.

Whilst this is very interesting for questions about rationality, I am always thinking about derived categories. And, unlike for Fano varieties in characteristic zero, the isomorphism $\mathrm{H}^1(X,\mathcal{O}_X)\cong\overline{\mathbb{F}}_2$ implies that $\mathcal{O}_X$ (or for that matter, any line bundle!) is not an exceptional object. This is unfortunate because we like to start decomposing derived categories of Fano varieties by considering an exceptional sequence of line bundles and then continuing from there.

So what's up with the derived category of this Fano fourfold? It turns out there is an exceptional object, and the verification is almost done by the authors already! Line bundles are out, but there is a rank-2 vector bundle in their writeup that we can consider. Let's recall their setup a bit, although I will omit the precise construction, as I do not feel qualified to talk about regular foliations in characteristic two.

An exceptional rank-2 bundle

In Proposition 3.4 they consider a finite flat morphism $\pi\colon H\to X$ of degree 2, where $H$ is a complete intersection of a quadric and a cubic in $\mathbb{P}^6$ with 63 singular points. In Lemma 4.2 they construct a short exact sequence \[ 0\to\mathcal{O}_X\to\pi_*\mathcal{O}_H\to K\to 0 \] where $K$ is defined as $\ker(\pi_*\mathcal{O}_H(1)\to\pi_*\mathcal{O}_H(2))$, for a derivation that arises from their construction. The final ingredient for us is Proposition 4.5, which gives the isomorphism $\pi^*K\cong\mathcal{O}_H(1)$. We have now seen the main player for our observation.

Claim $\pi_*\mathcal{O}_H$ is exceptional.

Whilst munching on my lunch, I realized that there is a shorter proof.

New and shorter proof. Let us write $E=\pi_*\mathcal{O}_H$. We have, using that $\pi$ is both finite and flat, that \[ \mathrm{Ext}_X^\bullet(E,E) =\mathrm{Ext}_X^\bullet(E,\pi_*\mathcal{O}_H) \cong\mathrm{Ext}_H^\bullet(\pi^*E,\mathcal{O}_H) \cong\mathrm{Ext}_H^\bullet(\mathcal{O}_H,\pi^*E^\vee). \] The long exact sequence in cohomology attached to the dual of the sequence defining $E$ pulled back to $H$ (again using that $\pi$ is flat), together with $\mathrm{H}^\bullet(H,\mathcal{O}_H)\cong\overline{\mathbb{F}}_2[0]$ and $\mathrm{H}^\bullet(H,\mathcal{O}_H(-1))=0$ (using that $H$ is a complete intersection) gives us the sought-after exceptionality.$\square$

Original proof. Let us write $E=\pi_*\mathcal{O}_H$. Because $E$ has rank two, we have \[ E^\vee\cong E\otimes\det(E)^\vee\cong\pi_*\mathcal{O}_H\otimes K^\vee\cong\pi_*(\mathcal{O}_H\otimes\mathcal{O}_H(-1))\cong\pi_*\mathcal{O}_H(-1) \] by the usual pairing for vector bundles, the projection formula, and the properties recalled above. This gives us \[ \mathrm{H}^\bullet(X,E^\vee)\cong\mathrm{H}^\bullet(H,\mathcal{O}_H(-1))\cong 0 \] because $H$ is a complete intersection, so we can use the Koszul resolution. Note that, as in the proof of Lemma 4.1, this also gives us that \[ \mathrm{H}^\bullet(X,E)\cong\mathrm{H}^\bullet(H,\mathcal{O}_H)\cong\overline{\mathbb{F}}_2[0]. \]

Now, consider the defining short exact sequence for $E$ and tensor it with $E^\vee$ to get \[ 0\to E^\vee\to E^\vee\otimes E\to E^\vee\otimes K\to 0. \] Again, because $E$ has rank two, we have $E^\vee\otimes K\cong E$. We have computed the cohomology of the outer terms, and we thus get \[ \mathrm{H}^\bullet(X,E^\vee\otimes E)\cong\overline{\mathbb{F}}_2[0], \] which proves the claim. $\square$

Hence, we have \[ \mathbf{D}^{\mathrm{b}}(X) = \langle E^\perp,E\rangle \] where $E^\perp=\{F\in\mathbf{D}^{\mathrm{b}}(X)\mid\mathbf{R}\mathrm{Hom}(E,F)=0\}$. Note that, by the acyclicity used above, we have $\mathcal{O}_X\in E^\perp$.

Questions

I need to start preparing my lecture for next week, so I'm leaving you (and myself) with the following questions:

  • What are the Hodge numbers of this Fano fourfold? Note that using Proposition 5.1, we can actually compute that $\mathrm{h}^0(X,\Omega_X^1)\geq 6$, e.g., using my cohomology tables for complete intersections;
  • Does the Hochschild–Kostant–Rosenberg decomposition for Hochschild homology hold for this variety? It is not automatic for fourfolds in characteristic 2!
  • How can we decompose the derived category further? Note that $E^\vee$ is also exceptional, but we cannot fit it in a sequence together with $E$.