# Nakano vanishing in positive characteristic

Let $$X$$ be a smooth projective variety defined over a field $$k$$. In characteristic zero, the following is a special case of the (Kodaira-Akizuki-)Nakano vanishing theorem:

$$(\ast) \quad$$ $$\mathrm H^0 \big( X, \Omega_X^p \otimes \mathscr L^{-1} \big) = 0$$ for all $$p < \dim X$$ and ample line bundles $$\mathscr L$$.

I am interested in what happens in positive characteristic.

Question 1: Is there a counterexample to $$(\ast)$$ when $$\mathrm{char}(k) > 0$$?

Since the statement is obviously related to Kodaira vanishing, I would guess that a counterexample is already known. The following question, however, might be harder:

Question 2: Is there a counterexample as above, but additionally with $$K_X \sim_{\mathbb Q} 0$$ or $$X$$ Fano?

Actually, a log version would also be sufficient, with $$\Omega_X^p$$ replaced by $$\Omega_X^p(\log D)$$, where $$(X, D)$$ is an snc pair (and in the second question, $$K_X + D \sim_{\mathbb Q} 0$$ or $$(X, D)$$ log Fano).

• By Koll\'ar - Miyaoka - Mori, every Fano manifold is rationally chain connected. By the Bloch - Srinivas argument, there exists a "decomposition of the diagonal" after inverting an integer $N$ that Chatzistamatiou and Levine call the "torsion order". If $N$ is prime to $\text{char}(k)$, then regardless of liftings mod $p^2$, the cohomology groups above vanish. This is proved by Totaro in his article about stable irrationality of hypersurfaces. The torsion order divides all "enumerative 2-point, genus-0 Gromov-Witten invariants" and so can be bounded, cf. my work with Z. Tian and R. Zong. – Jason Starr Mar 6 '19 at 11:40
• Probably the best place to look for a counterexample is in the Lauritzen-Rao example (as appearing for example in Sandor's paper arxiv.org/abs/1703.02080) - it's a 6fold Fano in characteristic 2 whose construction is rather explicit, so it should be possible to check if the above vanishing holds. There's also the Totaro examples arxiv.org/abs/1710.04364 but checking here might involve more work – Frank Mar 6 '19 at 13:00
• You may already be aware of this, but I believe that Shepherd-Barron in the proof of Theorem 7 in "Unstable vector bundles and linear systems on surfaces in characteristic $p$" shows that on surfaces, the vanishing $(\ast)$ holds for $p = 1$ (with the weaker assumption that the divisor class $L$ of $\mathscr{L}$ is in "positive cone" $C_{++}(X)$) as long as the surface is neither quasi-elliptic nor of general type. In the general type case, if $(\ast)$ does not hold, the proof seems to say that the surface is purely inseparably uniruled. – Takumi Murayama Mar 6 '19 at 18:30

There exists singular Fano varieties of dimension $$n\ge 3$$ in characteristic $$p$$ ($$p$$ small when compared to $$n$$) with a non-zero section of $$\Omega^{n-1}_X\otimes \mathcal L^*$$ for an ample $$\mathcal L$$. The existence of these examples is established in Kollár’s paper Nonrational hypersurfaces. They are constructed as degree $$p$$ coverings of Fano manifolds ramified over smooth hypersurfaces. It is unclear to me if in any of his examples $$X$$ is actually smooth.

A general (logarithmic) Kodaira-Akizuki-Nakano vanishing theorem in characteristic $$p$$ is proven in $$\S$$ 11 of

H. Esnault, E. Viehweg: Lectures on vanishing theorems. Notes, grew out of the DMV-seminar on algebraic geometry, held at Reisensburg, October 13-19, 1991, DMV Seminar. 20. Basel: Birkhäuser Verlag. 164 p. (1992). ZBL0779.14003,

see in particular Corollary 11.3. The precise statement is the following

Theorem. Let $$k$$ be a perfect field, let $$X$$ be a proper smooth $$k$$-scheme and $$D \subset X$$ a normal crossing divisor, both admitting a lifting $$\tilde{D} \subset \tilde{X}$$ to $$W_2(k)$$.

For $$p+q < \mathrm{min}\{\mathrm{char}(k), \, \dim X\}$$ and $$\mathscr{L}$$ ample and invertible, we have $$H^q(X, \, \Omega^p(\log D) \otimes \mathscr{L}^{-1})=0.$$

So you cannot have a counterexample to $$(*)$$, unless $$k$$ is not perfect or the pair $$(X, \, D)$$ does not lift to the ring of Witt vectors. I guess that counterexamples in the last case could be obtained by using the same strategy of Reynaud's famous counterexample for Kodaira vanishing in positive characteristic (see the reference below), but unfortunately I do not know any of them.

M. Raynaud: Contre-exemple au ''vanishing theorem” en caracteristique $$p>0$$, C.P. Ramanujam. - A tribute. Collect. Publ. of C.P. Ramanujam and Pap. in his Mem., Tata Inst. fundam. Res., Stud. Math. 8, 273-278 (1978). ZBL0441.14006.

• Francesco, just to clarify, Raynaud gives an example where $\mathcal{L}$ is ample but $H^1(X, \mathcal{L}^{-1})\not=0$. It is possible that the actual question, where $q=0$, might be OK. – Donu Arapura Mar 6 '19 at 13:01
• @DonuArapura: you are right, thanks for the observation. I was actually talking about a counterexample to the theorem in my answer. – Francesco Polizzi Mar 6 '19 at 14:38