# An update for Fanography.info: semisimplicity of quantum cohomology

Recall that Dubrovin's conjecture states the equivalence between

- the existence of a full exceptional collection in $\mathbf{D}^{\mathrm{b}}(X)$
- the generic semisimplicity of (big) quantum cohomology $\mathrm{BQH}(X)$

for a smooth projective variety $X$. One can study Dubrovin's conjecture for special classes of varieties, and I already added its status for $G/P$ to Grassmannian.info.

The conjecture is particularly interesting (and originally phrased for) Fano varieties, and for del Pezzo surfaces it is known to hold by Bayer's results. That brings us to Fano 3-folds, where I was wondering how much was known. Whilst figuring this out, I decided to add this information to Fanography.info, and here we are.

On Dubrovin's conjecture I made an overview of the Fano 3-folds for which

- expect (and in fact know) the existence of a full exceptional collection, for which a necessary condition (and in fact sufficient) condition is that the cohomology is of Hodge-Tate type, i.e. $\mathrm{h}^{1,2}=0$
- expect (and only
*partially*know) the semisimplicity

It turns out that in all these (known) cases, the *small* quantum cohomology is already generically semisimple. At the time of writing there seem to be 17 Fano 3-folds for which semisimplicity is not yet known, and I'll try to make some progress on this. Please let me know if I've missed some literature.

I should make things prettier at some point, and also add information on semiorthogonal decompositions. And figure out the original references for the quantum cohomology of $\rho=1$ Fano 3-folds. And add information on extremal contractions (a tiny piece of it is already available if you know where to look). Let me know if you have any other suggestions and requests!