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.



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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.



Algebraic definition for intermediate Jacobians
There are many answers, but none of them is completely satisfactory. First of all, there is the following book by Gerd Welters.


MR0633157 (84k:14035) Reviewed   

Welters, G. E.   

Abel-Jacobi isogenies for certain types of Fano threefolds.   

Mathematical Centre Tracts, 141. Mathematisch Centrum, Amsterdam, 1981. i+139 pp.  
 
ISBN: 90-6196-227-7   

14K30 (14J30) 


In this book, Welters proves that for many Fano threefolds $X$, for some "natural" irreducible component $M$ of the Hilbert scheme / Chow variety / Kontsevich space of effective curves on $X$, the induced Abel-Jacobi map $\alpha:\text{Alb}(M)\to J(X)$ is an isogeny. Thus, if you are willing to work up to isogeny, then you may take $\text{Alb}(M)$ as one algebraic definition. 


Of course that seems *ad hoc*: why use $M$ instead of some other irreducible component? This was one motivation for the study of the geometry of the different irreducible components $M\_\beta$ by myself, Harris, Roth, de Jong, Coskun, etc. Harris raised the question whether the Albanese varieties $\text{Alb}(M\_\beta)$ might stabilize as the curve class $\beta$ becomes more and more positive. Alas, in her thesis, Ana-Maria Castravet proved that this does not hold, at least for $(2,2)$ complete intersections in $\mathbb{P}^5$, and other Fano manifolds arising as moduli spaces of stable bundles on a curve. Castravet found that there are two different Abelian varieties that both occur infinitely often as $\text{Alb}(M\_\beta)$, one of which is $J(X)$, but the other of which is $J(X)\times J(X)$. 


MR2039210 (2005i:14038) Reviewed   

Castravet, Ana-Maria(1-TX)   

Rational families of vector bundles on curves. (English summary)   

Internat. J. Math. 15 (2004), no. 1, 13–45.   

14H60 (14J45 14M20)   



If you really only care about the intermediate Jacobian up to isogeny in a *strong* sense, i.e., you really only need the functor $T\mapsto J(X)(T)\otimes\_{\mathbb{Z}}\mathbb{Q}$ rather than $T\mapsto J(X)(T)$, then you can access this "motivically". Certainly the Bloch-Beilinson conjecture describes this as a certain subquotient of the Chow group of $X$. But I believe there are actually theorems that show how to describe $J(X)\otimes \mathbb{Q}$ purely algebraically, over the same (characteristic 0) base field as $X$. My recollection is that this is unknown in positive characteristic, but I will look into this and revise this post.


Finally, there is also a definition of Yi Zhu for $J(X)$ when $X$ admits a morphism $X\to C$ where $C$ is a smooth, projective curve and the fibers $Y$ of $X$ have sufficiently high "coniveau", i.e., something like $Y$ is rationally connected and $h^{2,1}(Y)=0$. Note, $X$ itself cannot be rationally connected, because $C$ is not rationally connected. However, this sort of fibration does arise in the study of rationally simply connected fibrations, for instance. Zhu constructs $J(X)$ as a certain moduli space of torsors on $C$ for the torus bundle over $C$ that is (Pontrjagin / Cartier) dual to the relative Picard of $X$ over $C$. 


Homogeneous Fibrations over Curves   

Yi Zhu   

<http://arxiv.org/abs/1111.2963>


Zhu's definition works over an arbitrary base field, even in positive characteristic.


