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

:$H^(M,) \otimes = H^(M)\oplus\cdots\oplus H^(M).$

On $H^$ the Weil complex structure $I_W$ is multiplication by $i^$, 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

References

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*{{Citation , last1=Weil , first1=André , author1-link=André Weil , title=On Picard varieties , mr=0050330 , year=1952 , journal=American Journal of Mathematics , issn=0002-9327 , volume=74 , pages=865–894 , doi=10.2307/2372230 , issue=4 , jstor=2372230

Hodge theory