The Kuznetsov component of a smooth cubic fourfold 
is the full triangulated subcategory
of 
defined by
![Ku(X)={F:Hom(O_X(i),F[k])=0 for i=0,1,2 and every k in Z}.](/images/equations/KuznetsovComponent/NumberedEquation1.svg) |
It is the right orthogonal to the three indicated line bundles and appears in the associated semiorthogonal decomposition.
An equivalence of this component with the ordinary derived category of a projective K3 surface is called a
geometric K3 category in this setting. This is an exact 
-linear categorical equivalence, not a birational map of the
cubic to the surface. The Kuznetsov
rationality conjecture predicts a connection between this equivalence and rationality
of the cubic.
