Here's a question I have had for a while, ever since reading Noncommutative rigidity of the moduli stack of stable pointed curves by Shinnosuke Okawa and Taro Sano.

They prove the cool result that $\mathrm{HH}^2(\overline{\mathcal{M}}_{g,n})=0$ (except when $(g,n)=(0,5)$, and possibly when $(g,n)=(4, 0), (3, 1), (3, 0), (2, 2), (2, 1), (2, 0), (1, 3), (1, 2)$). Here $\mathrm{HH}^2$ is the second Hochschild cohomology, and it measures the deformations of the category of coherent sheaves. This is the noncommutative rigidity in the title of their paper. Note that they are considering a smooth projective Deligne–Mumford stack whenever $g\geq 1$.

Finding noncommutatively rigid varieties

Because I like Fano varieties so much, I was wondering whether we can find a Fano variety $X$ for which the second Hochschild cohomology vanishes, i.e., does there exist a noncommutatively rigid Fano variety?

Note that it is in fact easy to find a non-Fano example of a noncommutatively rigid smooth projective surface: a fake projective plane is of this shape. But I wanted a Fano variety!

Finding noncommutatively rigid Fano varieties

In dimension 2, we have that $\overline{\mathcal{M}}_{0,5}$ is the del Pezzo surface of degree 5, and whilst it is rigid as a variety, it has a 6-dimensional space of Poisson structures, and thus lots of interesting noncommutative deformations. The same holds for the other rigid del Pezzo surfaces: lots of interesting noncommutative deformations, which are truly at the heart of noncommutative algebraic geometry.

In dimension 3, we can use Fanography, which contains the data from the paper Polyvector fields for Fano 3-folds that I wrote a few years ago with Enrico Fatighenti and Fabio Tanturri, to conclude that there are also no noncommutatively rigid Fano 3-folds.

In an ongoing project with many people on classifying Fano 4-folds we are also computing polyvector cohomology, but none of the cases studied so far seem to satisfy the required vanishings.

The example

Now, to dimension 5! PartialFlagVarieties.jl is a useful first selection mechanism, as it makes it possible to consider many zero loci in partial flag varieties at once when looking for candidates. This is how I found a complete intersection in $\mathbb{P}^2\times\mathbb{P}^2\times\mathbb{P}^2\times\mathbb{P}^3$, of degrees $(1,1,0,0)$, $(0,1,0,1)$, $(0,0,1,1)$, $(0,0,1,1)$, which had the potential of being an example. With a little help from an LLM, I could translate the remainder of the problem into a Macaulay2 computation, which confirmed that it is indeed an example.

I was originally writing this up as a blogpost, but it got longer and longer, so I felt like a pdf was more appropriate: A noncommutatively rigid Fano fivefold. It contains all the Julia and Macaulay2 code needed to verify the claims, so that you can check them yourself, and the code is also available in the repository noncommutatively-rigid-fano-fivefold. For now I have no plans to submit it for publication anywhere, but feel free to convince me otherwise.