# Fortnightly links (116)

Mykola Matviichuk, Brent Pym, Travis Schedler: A local Torelli theorem for log symplectic manifolds gives, aside from many important theoretical results, a classification of the deformation theory of $\mathbb{P}^4$. In particular, there are 40 different components (for $\mathbb{P}^2$ there is a single one, for $\mathbb{P}^3$ there are 6), and I'm looking forward to their more detailed analysis they have announced!

Daniel Halpern-Leistner: Derived $\Theta$-stratifications and the D-equivalence conjecture is a long-awaited preprint of Daniel discussing an instance of the "birational Calabi-Yau implies derived equivalence" conjecture in

*arbitrary*dimension (it is known in dimension 3 by Bridgeland), for varieties which are birational to moduli spaces of sheaves on K3 surfaces. Bridgeland comes into this picture again, now by virtue of Bridgeland stability conditions, as the proof goes via wall-crossing. In Bonn we've run a seminar on this topic 2 years ago, and it is a really interesting machinery behind the proof.Tanya Kaushal Srivastava: Pathologies of Hilbert scheme of points of supersingular Enriques surface shows that Hilbert schemes of points on supersingular Enriques surfaces (in characteristic 2 the $\mathbb{Z}/2\mathbb{Z}$ one has as torsion in the Picard group can be come either $\mu_2$, the

*singular*case or $\alpha_2$, the*supersingular*case) are examples of varieties showing that the pretty description of the Hodge numbers fails, and more importantly that the Beauville–Bogomolov classification of varieties with trivial canonical bundle and no étale covers is richer than in characteristic 0. Cool!