# 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 Hn(M,R)/Hn(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 Hn(M,R) are defined using the Hodge decomposition

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

On Hp,q the Weil complex structure IW is multiplication by ipq, while the Griffiths complex structure IG is multiplication by i if p > q and −i if p < q. Both these complex structures map Hn(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

• 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