The Infinitesimal Invariant of C ⁺ -C^-

Abstract

Our aim is to compute the infinitesimal invariant of a normal function associated to a very natural cycle: the curve in its Jacobian. We have to solve an exercise along some lines explained on these Cime lectures. This “exercise” is a joint work with Alberto Collino [CP] with a big hint given by F. Bardelli and S. Muller Stach that we will explain below.

The normal function constructed from the basic cycle C ⁺ -C^- is a natural object and appears in different areas. An example of this is given for instance by the work about the harmonic integral (cf. [H1], [H2], [Hain1] and [Pu]). An interesting connection between the cycle and the work of Johnson (cf. [J1] and [J2]) on the mapping class group has been recently found by R. Hain (cf. [Hain2] and see (1.11) below) and therefore some possible relations with the syzygies of the canonical curve was pointed out to us by C. Voisin.

The Griffiths infinitesimal invariant was introduced as a tool (cf. [Gr1]) to decide when a normal function is locally constant. The work of M. Green (cf. [G]) and C. Voisin (cf. [V1]) made possible, in many cases, to compute it, so the infinitesimal invariant has become a main tool for the study of the normal functions.

We have found that our infinitesimal invariant carries important information and, in genus 3, it determines the curve (see (2.6)). In [Gr1] a similar Torelli theorem is proved. Griffiths considered the normal function defined by the differente of the two g₃¹ on the Jacobian of a curve of genus 4. This work and in particular the treatment of the Schiffer variations was our model.

Our computations allows us to improve Ceresa’s theorem by making possible to study the algebraic dependence of C ⁺ and C^- when C varies in subvarieties of the moduli space of curves. In this way we obtain some answers to a question asked by H.Clemens (see section 1 below).

In section 1 we recall some definitions, results and problems on the basic cycle. We introduce next, in section 2, the infinitesimal invariant and state our Torelli theorem. Sketches of the proof of it and of the main formula (3.4) are given in section 3.