A rational Hodge structure of weight 
is a finite-dimensional rational
vector space 
together with a decomposition
 |
where complex conjugation interchanges 
and 
. A morphism of Hodge structures is a rational linear
map whose complexification preserves these types. The cohomology 
of a smooth projective complex
variety provides an example through its Hodge
decomposition. A sub-Hodge structure is a rational subspace on which that decomposition
restricts. The Hodge group is the smallest rational
algebraic group containing the Hodge circle, which
acts on 
by 
.
For weight 2 with 
,
the structure is of K3 type.
