It's interesting how things can get postponed, it's like a psychological experiment I perform on myself seeing when I actually put up fortnightly links.
Michael Brown and Mark Walker: Standard conjecture D for matrix factorizations shows that the noncommutative analogue for Grothendieck's conjecture D (which says that numerical equivalence agrees with homological equivalence) holds for categories of matrix factorisations. The proof goes via lots of cool Hodge theory.
Ciro Ciliberto: The classification of complex algebraic surfaces is what the title says, a really nice set of lecture notes on the classification of algebraic surfaces.
Christian Schnell: The Fourier–Mukai transform made easy revisits the first result on Fourier–Mukai transforms. It shows that an alternative point-of-view on the equivalence of categories provided by the Poincaré line bundle, where sprinkling a bit of Grothendieck duality on the definition makes the statements cleaner and more conceptual.
Nicolas Addington, Benjamin Antieau, Sarah Frei, Katrina Honigs: Rational points and derived equivalence is a really cool paper, exploring how derived categories behave over non-algebraically closed fields. What makes it extra interesting for me is to see how an essential part is contained in a computer algebra computation, whose subtleties are discussed at the end of the paper.