Intermediate Jacobian Explained

Intermediate Jacobian should not be confused with generalized 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

H ^{n(M,\R)/H} ^{n(M,\Z)} 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

-1 . The complex structures on H ^{n(M,\R)} are defined using the Hodge decompositionH ^{n} (M,{\R}) ⊗ {\C} = H ^{n,0} (M) ⊕ … ⊕ H ^{0,n} (M).

On

H ^{p,q} the Weil complex structure I _{W} is multiplication by i ^{p-q} , while the Griffiths complex structure I _{G} is multiplication by i if p > q and - i if p < q . Both these complex structures map H ^{n(M,\R)} into itself and so defined complex structures on it.

For

n =1 the intermediate Jacobian is the Picard variety, and for n =2 \dim (M) -1 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