Nonrationality of the generic cubic fourfold
I could wait for the fortnightly links, but the proof of the conjectured rationality of the generic cubic fourfold has seen lots of unsuccessful approaches over the past years and a proof certainly is big news. For an overview of the state-of-the-art (except for the claimed proof of course) there are recent lectures notes by Alexander Kuznetsov and Brendan Hassett.
Now there is a preprint by Mingmin Shen that proves the nonrationality of the generic cubic fourfold! The proof uses new techniques regarding stable rationality introduced by Claire Voison, about which I can't say too much at the moment. It'll be interesting to see what this approach to the proof implies for the other approaches (Hodge theory, and derived categories, especially the more precise conjecture regarding subcategories related to K3 surfaces).