On a smooth complex projective variety of $\dim X=n$, we have $n$ complex tori associated to it via $J^k(X)=F^kH^{2k-1}(X,\mathbb{C})/H\_k(X,\mathbb{Z})$ (assuming I've got all the indices right) called the $k$th intermediate Jacobian.


If $k=1$, we have $J^1(X)=H^{1,0}/H\_1$, and so $J^1(X)\cong Jac(X)$ is an abelian variety (the bilinear form is a polarization because it has to be definite on each piece of the Hodge decomposition (I think) ) and is in fact isomorphic as PPAV's to the Jacobian of the variety.


If $k=n$, we have $H^{2n-1,1}/H\_{2n-1}$, which is also a PPAV, and is in fact the Albanese of $X$.


The ones in the middle, however, the "true" intermediate Jacobians, are generally only complex tori. One example of an application is that Clemens and Griffiths proved that cubic threefolds are unirational but not rational using $J^2(X)$ for $X$ a cubic threefold.


So, what information do the intermediate Jacobians contain? I've been told that we don't really know much about that, but what is known, beyond Clemens/Griffiths?



What do intermediate Jacobians do?
Probably you know this, but just to be sure: they receive cycle class maps from
codimension $k$ cycles. More precisely, if $Z$ is a cycle on $X$ of codimension $k$
that is cohomologically trivial, then it gives an element in $J^k(X)$. This map,
from cohomologically trivial cycles to $J^k(X)$, is a (generalized) Abel-Jacobia map.


If we let $CH^k(X)$ be the Chow group of codimension $k$ cycles on $X$, then we get
a filtration on $CH^k(X)$, namely $CH^k(X) \supset \text{ cohomologically trivial cycles}
\supset \text{ Albanese trivial cycles}.$ (Albanese trivial means "has trivial image
in $J^k(X)$.) It is conjectured that this filtration can be further extended; this is
part of the Bloch--Beilinson series of conjectures on Chow groups and motives.


On the other other hand, if $X$ is defined over a number field $K$, and we look just 
at cycles defined over $K$, then it is conjectured that Albanese trivial implies rationally
trivial (i.e. any cohomologically trivial cycle that is also is in the kernel of themap to $J^k(X)$ (and  *is defined over a number field* ) is in fact rationally equivalent to zero). I believe this conjecture is extremely wide open at the moment.


