# Fortnightly links (85)

Mark Green, Yoon-Joo Kim, Radu Laza, Colleen Robles: The LLV decomposition of hyper-Kaehler cohomology gives the weight decomposition of the cohomology of a hyperkähler variety under the action of the Looijenga–Lunts–Verbitsky Lie algebra, and uses this to determine the Betti numbers of O'Grady 10 (previously featured on this blog) in a what feels to me more conceptual way, starting from the Euler number and the vanishing of odd cohomology. This method works for O'Grady 10 because it has a relatively large second Betti number (so a relatively large Lie algebra).

Lenny Taelman: Derived equivalences of hyperkähler varieties shows how the Lie algebra which featured in the previous fortnightly link is actually a derived invariant (at least for a hyperkähler variety). Having this new derived invariant allows one to study the action of auto-equivalences on it, and this is what is used to describe the auto-equivalence group of hyperkähler varieties (a complete answer is only known for K3 surfaces).

Arnaud Beauville, Vector bundles on Fano threefolds and K3 surfaces shows how moduli of vector bundles on K3 surfaces are related to moduli of vector bundles on Fano 3-folds for which they are anticanonical sections.

Maxim Kontsevich, Birational invariants from quantum cohomology is the video of a talk of Kontsevich at the recent homological mirror symmetry conference in Moscow. The other videos are also available, but let me point out that he suggests their new results are strong enough to prove non-rationality of the very general cubic fourfold (this is mentioned around 2:20). For the flavor of their invariants on the level of the Grothendieck group, one can go to 56:00, to see the definition of a new invariant $\mathrm{K}_0(\mathrm{Var}/k)\to\mathbb{Z}$.