Intermediate Jacobian

# Intermediate Jacobian

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Encyclopedia
In mathematics, the intermediate Jacobian of a compact Kähler manifold
Kähler manifold
In mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...

or Hodge structure
Hodge structure
In mathematics, a Hodge structure, named after W. V. D. Hodge, is an algebraic structure at the level of linear algebra, similar to the one that Hodge theory gives to the cohomology groups of a smooth and compact Kähler manifold...

is a complex torus
Complex torus
In mathematics, a complex torus is a particular kind of complex manifold M whose underlying smooth manifold is a torus in the usual sense...

that is a common generalization of the Jacobian variety
Jacobian variety
In mathematics, the Jacobian variety J of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles...

of a curve and the Picard variety and the Albanese variety
Albanese variety
In mathematics, the Albanese variety A, named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve, and is the abelian variety generated by a variety V. In other words there is a morphism from the variety V to its Albanese variety A, such that any morphism from V to an...

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

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
Albanese variety
In mathematics, the Albanese variety A, named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve, and is the abelian variety generated by a variety V. In other words there is a morphism from the variety V to its Albanese variety A, such that any morphism from V to an...

. In these two extreme cases the constructions of Weil and Griffiths are equivalent.

used intermediate Jacobians to show that non-singular cubic threefold
Cubic threefold
In algebraic geometry, a cubic threefold is a hypersurface of degree 3 in 4-dimensional projective space. Cubic threefolds are all unirational, but used intermediate Jacobians to show that non-singular cubic threefolds are not rational. The space of lines on a non-singular cubic 3-fold is a...

s are not rational
Rational variety
In mathematics, a rational variety is an algebraic variety, over a given field K, which is birationally equivalent to projective space of some dimension over K...

, even though they are unirational.