The Fano normal function

Abstract

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 $V$, $F-F^-$ in not algebraically equivalent to zero in $J(V)$ (already proved by van der Geer-Kouvidakis) and, moreover, there is no a divisor in $JV$ containing both $F$ and $F^-$ and such that these surfaces are homologically equivalent in the divisor. Our study of the infinitesimal variation of Hodge structure for $V$ produces intrinsically a threefold $\Xi (V)$ in $\mathbb G$ the Grasmannian of lines in $\mathbb P^4.$ We show that the infinitesimal invariant at $V$ attached to the normal function gives a section for 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 $V$.