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 ${\displaystyle H^{n}(M,\mathbb {R} )/H^{n}(M,\mathbb {Z} )}$ 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 André Weil (1952) and one due to Phillip 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.

Not to be confused with generalized Jacobian.

A complex structure on a real vector space is given by an automorphism I with square ${\displaystyle -1}$. The complex structures on ${\displaystyle H^{n}(M,\mathbb {R} )}$ are defined using the Hodge decomposition

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

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

For ${\displaystyle n=1}$ the intermediate Jacobian is the Picard variety, and for ${\displaystyle n=2\dim(M)-1}$ 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], "Intermediate Jacobian", Encyclopedia of Mathematics, EMS Press
• 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