Intermediate Jacobian
In mathematics, the intermediate Jacobian of a compact Kähler manifold or Hodge structure is a complex torus that is a common generalization of the Jacobian variety of a curve and the  Picard variety and the Albanese variety. It is obtained by putting a complex structure on the torus  for n odd. There are several different natural ways to put a complex structure on this torus,  giving several different sorts of intermediate Jacobians, including one due to  and one due to . The ones constructed by Weil have natural polarizations if M is projective, and so are abelian varieties, while the ones constructed by Griffiths behave well under holomorphic deformations. A complex structure on a real vector space is given by an automorphism I with square . The complex structures on  are defined using the Hodge decomposition

On  the Weil complex structure  is multiplication by , while the Griffiths complex structure  is multiplication by  if  and  if . Both these complex structures map  into itself and so defined complex structures on it. For  the intermediate Jacobian is the Picard variety, and for  it is the Albanese variety. In these two extreme cases the constructions of Weil and Griffiths are equivalent. used intermediate Jacobians to show that non-singular cubic threefolds are not rational, even though they are unirational. See also 

 Deligne cohomology

References

Hodge theory