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) [15] 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. Our study of the infinitesimal variation of Hodge structure for V produces intrinsically a threefold Ξ(V) in the Grassmannian of lines G in P4. We show that the infinitesimal invariant at V attached to the normal function gives a section of a natural bundle on Ξ(V) and more specifically that this section vanishes exactly on Ξ∩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.