# 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 Weil (1952) and one due to Griffiths (1968, 1968b); 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

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.

Clemens & Griffiths (1972) used intermediate Jacobians to show that non-singular cubic threefolds are not rational, even though they are unirational.

## References[edit]

- Clemens, C. Herbert; Griffiths, Phillip A. (1972), "The intermediate Jacobian of the cubic threefold",
*Annals of Mathematics*, Second Series,**95**(2): 281–356, CiteSeerX 10.1.1.401.4550, doi:10.2307/1970801, ISSN 0003-486X, JSTOR 1970801, MR 0302652 - Griffiths, Phillip A. (1968), "Periods of integrals on algebraic manifolds. I. Construction and properties of the modular varieties",
*American Journal of Mathematics*,**90**(2): 568–626, doi:10.2307/2373545, ISSN 0002-9327, JSTOR 2373545, MR 0229641 - Griffiths, Phillip A. (1968b), "Periods of integrals on algebraic manifolds. II. Local study of the period mapping",
*American Journal of Mathematics*,**90**(3): 805–865, doi:10.2307/2373485, ISSN 0002-9327, JSTOR 2373485, MR 0233825 - Griffiths, Phillip; Harris, Joseph (1994),
*Principles of algebraic geometry*, Wiley Classics Library, New York: John Wiley & Sons, doi:10.1002/9781118032527, ISBN 978-0-471-05059-9, MR 1288523 - Kulikov, Vik.S. (2001) [1994], "i/i051870", in Hazewinkel, Michiel (ed.),
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Weil, André (1952), "On Picard varieties",
*American Journal of Mathematics*,**74**(4): 865–894, doi:10.2307/2372230, ISSN 0002-9327, JSTOR 2372230, MR 0050330