A cubic fourfold is a hypersurface of complex dimension 4 defined by a homogeneous cubic polynomial in
. In the smooth case, rationality
means birational equivalence over 
with 
, or equivalently a purely transcendental function
field of transcendence degree 4.
Its bounded derived category of coherent sheaves has a Kuznetsov component complementary
to 
, 
, and 
. The rationality conjecture
relating this component to a K3 surface concerns an
exact equivalence with the ordinary derived category
of a smooth projective K3 surface. The claim is about
the cubic fourfold's rationality and cannot be inferred from the existence of a K3 surface category alone.
