The nonrationality conjecture for cubic, $X_{10}$, and Kuechle fourfolds

Consider a four-dimensional cubic, a four-dimensional $X_{10}$ variety, or a Kuechle manifold. A commutative K3 component means the derived category of a commutative K3 surface occurring in a semiorthogonal decomposition of the variety's derived category.

Nonrationality conjecture. These fourfolds are not rational if their semiorthogonal decompositions do not contain the derived category of a commutative K3 surface.

The conjecture is motivated by the Hasset–Kuznetsov–Tschinkel program and by gaps in the Orlov spectra; the source notes that the case with a commutative K3 component is more delicate.