A generalized Kummer variety 
, for an integer 
, is the fiber over the origin
of the summation morphism ![A^([n+1])->A](/images/equations/GeneralizedKummerVariety/Inline3.svg)
associated with a two-dimensional complex abelian variety 
. Here ![A^([n+1])](/images/equations/GeneralizedKummerVariety/Inline5.svg)
is the Hilbert scheme
of length-
subschemes of 
, and the summation morphism adds
their support points with multiplicities. The resulting smooth complex
manifold has complex dimension 
(Beauville 1983).
A complex manifold is of generalized Kummer type if it is deformation equivalent to such a fiber. This larger
class is not restricted to the fibers themselves. For projective
members of this class with 
and 
prime, the Hodge
conjecture holds for every self-power (Floccari and Fu 2026).
