Intermediate Jacobian

Template:Distinguish 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 $H^n(M,ℝ)/H^n(M,\mathbb{Z})$ 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 Template:Harvs and one due to Template:Harvs. 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.

Template:Harvtxt 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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