This fortnight's links are all about fun things you can do with derived categories of smooth projective vareties.

  • Mattia Ornaghi, Laura Pertusi: Voevodsky's conjecture for cubic fourfolds and Gushel-Mukai fourfolds via noncommutative K3 surfaces discusses Voevodsky's conjecture relating smash-nilpotence to numerical equivalence for Chow groups to semiorthogonal decompositions of derived categories. By the work of Bernardara–Marcolli–Tabuada this geometric conjecture is equivalent to a noncommutative version (i.e. for the derived category) relating two notions of equivalence relation on the Grothendieck group.

    They show Voevodsky's conjecture for cubic fourfolds and Gushel–Mukai fourfolds (who have seen considerable interest regarding rationality questions, as discussed in earlier fortnightly links) by showing that a particular blowup of these fourfolds can be described as a quadric fibration over a well-understood base. But this means that the derived category of the blowup has both a description as a blowup (containing the derived category of the original fourfold as a component) and as a quadric fibration. And for all the terms in the semiorthogonal decomposition as a quadric fibration the conjecture is known, which implies the conjecture for the blowup, hence also for the original category. Fun!

  • Jeff Achter, Sebastian Casalaina-Martin, Charles Vial: Derived equivalent threefolds, algebraic representatives, and the coniveau filtration gives further evidence for Orlov's conjecture, which says that derived equivalent varieties have isomorphic Chow motives (when taken with $\mathbb{Q}$-coefficients), by showing that an invariant (the intermediate Jacobian) which can be deduced from the Chow motive is actually a derived invariant, for threefolds.