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 $V, F-F^{-}$is not algebraically equivalent to zero in $J(V)$ (proved also by van der Geer and Kouvidakis (2010) [15] with different methods) and, moreover, that there is no divisor in $J V$ 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 the Grassmannian of lines $\mathbb{G}$ in $\mathbb{P}^4$. 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 $V$.