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 $$H^n(M,\R)/H^n(M,\Z)$$ 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 decomposition


 * $$ H^{n}(M,{\R}) \otimes {\C} = H^{n,0}(M)\oplus\cdots\oplus H^{0,n}(M).$$

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.