TITLE: What do intermediate Jacobians do?
QUESTION [14 upvotes]: 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?

REPLY [11 votes]: Recently I learned from a talk of Nick Addington one beautiful  classical example where intermediate Jacobians contain all information about the variety. Namely, if we consider an intersection of two quadrics in $\mathbb CP^n$ then to such a variety we can associate a hyperelliptic curve.  To do this we consider the corresponding pencil of quadrics. Singular quadrics in the pencil define a finite number of points on $CP^1$ and we take the double cover of $CP^1$ with the ramification at these points. The Jacobian of the hyperelliptic curve is isomorphic to the intermediate Jacobian of the intersection of to quadrics (this is the PhD of Miles Reid). The consruction was first considered by Weil. All this is very well explained in the introduction of the article of Nick http://arxiv.org/abs/0904.1764. 
You can reconstruct the hyperelliptic curve from its Jacobian and you can reconstructs the intersection of two quadrics from the curve.
Intermediate Jacobians of the intersection of 3 quadrics were considered by Turin. He proved the corresponding Torelli theorem: On intersections of quadrics. Russ. Math. Surv., 30, 51–105, 1975.