A Kuga-Satake correspondence is the standard Hodge-theoretic map from a polarized Hodge structure of K3 type into a tensor construction
on the first cohomology of an associated abelian
variety. Its algebraicity asks whether this specified map is induced by an algebraic correspondence. For a projective
K3 surface 
, put 
with its cup-product form 
.
In the convention 
, let 
be the full even Clifford
algebra, realized as 
. For a rational nonisotropic 
, the standard map is
 |
Here 
denotes the Tate twist. A chosen rational polarization
of 
converts this into a specified map 
. Algebraicity requires
a codimension-2 cycle on 
inducing that map, with the prescribed normalization.
