This blogpost is about Failure of Bott vanishing for (co)adjoint partial flag varieties. If you just want mathematics and not some of the backstory, you can click the link to immediately get to the preprint.

Background

Bott vanishing is a very strong cohomological vanishing property for a smooth projective variety $X$, which says that \[ \mathrm{H}^p(X,\Omega_X^q\otimes\mathcal{L})=0 \] for $p\geq 1$, $q\geq 0$ and $\mathcal{L}$ an ample line bundle on $X$. It is satisfied only for very special varieties. Buch–Thomsen–Lauritzen–Mehta showed that over fields of positive characteristic it is implied by liftability of $X$ modulo $p^2$ including their Frobenius.

Without asking the Frobenius lifts along, it is precisely the setting of the famous Deligne–Illusie paper, and the lifting modulo $p^2$ gives the (weaker) Kodaira–Akizuki–Nakano vanishing \[ \mathrm{H}^p(X,\Omega_X^q\otimes\mathcal{L})=0 \] for $p+q\geq\dim X+1$ and an ample line bundle $\mathcal{L}$ on $X$. Every variety arising from characteristic zero admits a lifting modulo $p^2$. This is the key insight used by Deligne–Illusie to give an algebraic proof of Kodaira-type vanishing.

Conjecture

Varieties which lift together with their Frobenius are much rarer, because they must satisfy Bott vanishing. In the case of partial flag varieties, Buch–Thomsen–Lauritzen–Mehta conjectured the following.

Conjecture (Buch–Thomsen–Lauritzen–Mehta) Let $X=\mathrm{G}/\mathrm{P}$ be a partial flag variety which is not isomorphic to a product of projective spaces Then $X$ does not satisfy Bott vanishing.

This shows that such partial flag varieties cannot be lifted together with their Frobenius morphism.

Results

In the short paper that this blogpost is concerned with, I prove this conjecture for (co)adjoint partial flag varieties (see Grassmannian.info for an overview of these varieties). The (co)minuscule case was already established by Buch–Thomsen–Lauritzen–Mehta. Note that this gives a new proof of the non-liftability of (co)adjoint partial flag varieties: recently, and using a very different obstruction, this was established (for all partial flag varieties) by Achinger–Witaszek–Zdanowicz.

The failure of Bott vanishing outside the (co)minuscule and (co)adjoint case can also be related to an observation Maxim Smirnov and I made about the Hochschild–Kostant–Rosenberg decomposition of Hochschild cohomology of partial flag varieties: we conjectured it to always have some non-zero higher cohomology, and we computationally verified it in many cases.

The cool thing is that Anton Fonarev proved exactly that! The preprint also appeared today. So taken all together, the Buch–Thomsen–Lauritzen–Mehta conjecture is almost settled for generalised Grassmannians.

Methods

The method of proof is easy to explain. Because of how I think of this problem, I prefer to think about $\bigwedge^q\mathrm{T}_X\cong\Omega_X^{\dim X-q}\otimes\omega_X^\vee$, where $\omega_X^\vee$ is ample. It turns out that to prove Bott non-vanishing, it suffices to twist everything by $\mathcal{O}_X(-1)$, so that the line bundle remains ample.

The recipe thus becomes:

  1. write down the associated graded of the Jordan–Hölder filtration of the tangent bundle: this is annoying in general (although in the classical types B, C and D there is a nice description and this is what Anton uses too), and can be done on a computer on a case-by-case basis amenable to these types of calculations, but the (co)adjoint case allows for a really good description;
  2. search for the smallest exterior power (which will depend on the Dynkin type), such that, when twisting by $\mathcal{O}_X(-1)$ you get a non-zero higher sheaf cohomology
  3. prove that it is indeed non-vanishing.

For parts (2) and (3), one can use a computer calculation, taking care that the method indeed works for all Dynkin types, and not just the ones that you happened to have checked computationally.

Recently, Bott (non-)vanishing was studied by several other authors. This prompted me to make this observation on the non-vanishing for (co)adjoint varieties (which in some premature form dates back to discussions with Maxim Smirnov when we were working on Hochschild cohomology of generalised Grassmannians) available as a short article.

The preprint contains full details, including an explicit construction for each type. Comments and questions welcome!