# Fortnightly links (71)

Apologies to all of you who were eagerly waiting for their fortnightly links, I was feeling sick and didn't feel like writing this one. On the other hand, on March 18 this year I announced that I wanted to do something with Fano 3-folds, à la superficie.info, and in a few days I will release what I came up with.

Ivan Cheltsov, Victor Przyjalkowski, Constantin Shramov: Fano threefolds with infinite automorphism groups is a paper I didn't know was in progress, but that I've been waiting for. I had computed some odd cases myself, but it's great to see

*all of them*done carefully.And regarding the announcement from the introduction: I have incorporated all data on automorphism groups, so the information from this preprint will be included.

Ivan Cheltsov, Victor Przyjalkowski: Katzarkov-Kontsevich-Pantev conjecture for Fano threefolds is a monumental paper, checking a conjecture regarding the behavior of Hodge numbers in mirror symmetry.

In homological mirror symmetry for Fano varieties, there is a correspondence between the Fano variety and a Landau–Ginzburg mirror, a morphism $w\colon Y\to\mathbb{A}^1$ (or compactified to $Y\to\mathbb{P}^1$), such that

the derived category of $X$ corresponds to the Fukaya–Seidel category of $w$

the Fukaya category of $X$ corresponds to the category of matrix factorisations of $w$

As the derived category of $X$ in the Fano case completely determines $X$ (actually, it is conjectured that Hodge numbers are derived invariants) it is therefore interesting to see whether one can recover these numbers from the symplectic geometry of $w$. The conjecture suggests a method to compute Hodge numbers on the mirror side, and this is what is checked

**case-by-case**for all Fano 3-folds, by computing $\mathrm{h}^{1,2}$ and $\mathrm{h}^{1,1}=\operatorname{rk}\operatorname{Pic}(X)$ in this way.Joseph Karmazyn, Alexander Kuznetsov, Evgeny Shinder: Derived categories of singular surfaces is a paper I did know was in progress, and that I've been waiting for. This is as far as I know (one of the) first papers in which a systematic study of semiorthogonal decompositions in a singular setting is done, where one has to be much more careful about functors as not everything automatically has all adjoints. The smooth and proper setting really is a wonderland compared to the real world. This is a very interesting read!