A rational Hodge structure of integer weight 
is a finite-dimensional
rational vector space 
together with a decomposition
 |
where 
and complex conjugation interchanges 
and 
. A morphism
of Hodge structures is a rational linear map
whose complexification preserves these types. The cohomology 
of a smooth
variety that is a projective variety over
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 
. A weight-2 structure
with 
unless 
is 
, 
, or 
and 
is of K3 type (Huybrechts 2016).
