Last week Burt Totaro uploaded the preprint Bott vanishing for Fano 3-folds to the arXiv, after earlier having studied Bott vanishing for del Pezzo surfaces. The results are now More below the fold.

We say that Bott vanishing holds for a smooth projective variety $X$ if $\mathrm{H}^j(X,\Omega_X^i\otimes\mathcal{L})=0$ for all $j\geq 1$, $i\geq 0$ and $\mathcal{L}\in\operatorname{Pic}(X)$ ample. This is a curious property, that we don't understand very well yet:

  • it holds for toric Fano varieties
  • for a Fano variety it implies $X$ is rigid, thus there are only finitely many Fano varieties in each dimension satisfying Bott vanishing

These two properties suggest there might be some combinatorial classification, as alluded to by Burt, but I don't think anyone has a good idea here? More on Bott vanishing:

  • it is expected to fail for every partial flag variety $G/P$ which is not $\mathbb{P}^n$ (see more in Hochschild cohomology of generalised Grassmannians)
  • the most curious thing from Totaro's paper is that it is possible that $X$ does not satisfy Bott vanishing, but the blowup of $X$ in some subvariety $Y$ does!

Now that your interest is piqued, go take a look at the table of all Fano 3-folds satisfying Bott vanishing!