# Fortnightly links (67)

Summer heat has caught up with me, and this one is half a week late. If you want a (not very enlightening) sneak peek at something I've been up to lately, check out the Twitter page for this cute rodent. Also, the Fields (and other) medals have been awarded yesterday, congratulations on all!

Emanuele Macrì, Paolo Stellari: Lectures on non-commutative K3 surfaces, Bridgeland stability, and moduli spaces are lecture notes for the Birational geometry of hypersurfaces school, which has featured before in fortnightly links for Colliot-Thélène's notes. These set of notes gives a wonderful overview of how the geometry of cubic fourfolds interacts with their derived categories, and how the Kuznetsov component controls so much of the geometry. They are an awesome read.

Federico Caucci, Giuseppe Pareschi: Derived invariants arising from the Albanese map is a preprint close to what I like to think about: how much of the geometry is controlled by the derived category? In particular, how much numerical information can we extract? The most famous open question in this direction is the invariance of Hodge numbers. In this preprint it is shown that if the Albanese varieties of derived equivalent $X$ and $Y$ are of maximal dimension, then the Hodge numbers $\mathrm{h}^{0,i}$ are equal. This is closely related to the conjectured derived invariance of the homological units, from Abuaf.

Because I keep forgetting the location of these resources, let me record them here. Hodge numbers of Calabi–Yau threefolds can be found on Rhys Davies' (outdated) website. There is also Benjamin Jurke's interactive version which I recall using several years ago, but it stopped working a while back, and now the whole website is (temporarily?) down unfortunately.