The Fano normal function

The Fano surface F of lines in the cubic threefold V is naturally embedded in the intermediate Jacobian $J(V)$, we call ``Fano cycle'' the difference $F - F^{ - }$, this is homologous to 0 in $J(V)$. We study the normal function on the moduli space which computes the Abel - Jacobi image of the Fano cycle. By means of the related infinitesimal invariant we can prove that the primitive part of the normal function is not of torsion. As a consequence we get that, for a general $V,F - F^{ - }$ is not algebraically equivalent to zero in $J(V)$ (proved also by van der Geer and Kouvidakis ( (2010) with different methods) and, moreover, that there is no divisor in JV containing both F and $F^{ - }$ and such that these surfaces are homologically equivalent in the divisor.\par Our study of the infinitesimal variation of Hodge structure for V produces intrinsically a threefold $\Xi (V)$ in the Grassmannian of lines $\Bbb G$ in $\Bbb P^{4}$. We show that the infinitesimal invariant at V attached to the normal function gives a section of a natural bundle on $\Xi (V)$ and more specifically that this section vanishes exactly on $\Xi \cap F$, which turns out to be the curve in F parameterizing the ``double lines'' in the threefold. We prove that this curve reconstructs V and hence we get a Torelli-like result: the infinitesimal invariant for the Fano cycle determines V.