Let $X$ be a Fano threefold over $\mathbb{C}$ (e.g. a cubic threefold in $\mathbb{P}^4$). Then one may define the intermediate Jacobian $J(X)$ of $X$ via the Hodge decomposition on $H^3(X,\mathbb{C})$. A priori this is just a complex torus, however the vanishing of $H^{3,0}(X)$ implies that this is in fact an abelian variety with

$$\dim J(X) = 1/2 \cdot \dim H^3(X,\mathbb{C}).$$

Note that we constructed $J(X)$ complex analytically, but it turned out that it was algebraic. This is much the same as with the Jacobian of algebraic curves, but it turns out that there is an algebraic definition for the Jacobians of curves which works over any field (namely Pic^0). My question is whether something similar happens here.

Does there exist an algebraic definition for the intermediate Jacobian?

More specifically, let $X$ be a Fano threefold over a field $k$ (not necessarily of charactertistic zero). May one define an intermediate Jacobian $J(X)$ here over $k$, which in the case where $k=\mathbb{C}$ recovers the above definition?

More generally, I would like to be able to define the intermediate Jacobian for a family of Fano threefolds parametrised by suitable schemes (any algebraic definition should also hopefully give this).

I have a vague idea how one might go about this for cubic threefolds, but it seems a little ad hoc. Namely, Clemens and Griffiths showed that the Fano variety of lines $F(X)$ in a cubic threefold over $\mathbb{C}$ is a surface of general type, and that the Abel-Jacobian map $F(X) \to J(X)$ induces an isomorphism of abelian varieties $Alb(F(X)) \to J(X)$ (Here $Alb$ means take the Albanese variety). Therefore over general fields, it seems possible that one could simply *define* $J(X) := Alb(F(X))$.

There are a few reasons I don't like this. Namely:

- I don't know anything about $F(X)$ over general fields (e.g. is it smooth in char. $p$?).
- It's not clear that this will work or make sense for general Fano threefolds (I'm not sure that $Alb(F(X)) \to J(X)$ need be an isomorphism in general.)
- Ultimately I want a Torelli type theorem to hold, which I am not sure this approach gives.
- It is very ad hoc.