Infinitesimal invariant and Massey products
Let \(C\) be a smooth complete connected curve of genus \(g>2\) embedded in its Jacobian variety \(JC\). The Ceresa cycle of \(C\) is the cycle \(Z=C-i(C)\subset CH^{g-1}(JC)\), where \(i=-1\) on \(JC\). It is known that \(Z\) is homologically but not algebraically equivalent to zero for generic \(C\). If \(f:\mathcal C\to S\) is a family of curves of genus \(g>2\) as above and \(\mathcal J\to S\) is its relative Jacobian, then the relative Ceresa cycle \(\mathcal{Z}\in CH^{g-1}(\mathcal J)_{\hom}\) can be considered. The normal function associated to \(\mathcal Z\) is a section \(\nu\) of the relative intermediate Jacobian \(\mathcal J^{g-1}\to S\) defined by \(\nu(s)=AJ_s(Z_s)\), where \(AJ_{s} \) is the higher Abel-Jacobi map. A formula for computing the Griffiths' infinitesimal invariant of \(\nu\) has been given by \textit{A. Collino} and \textit{G. Pirola} [Duke Math. J. 78, 59--88 (1995; Zbl 0846.14016)]. In this paper the author, by means of Collino and Pirola's formula, expresses the Griffiths' infinitesimal invariant of the family in terms of a triple Massey product on a graded differential algebra \(\mathcal A\), defined similarly to the Kodaira-Spencer algebra. As a consequence of this result, it is proved that the differential graded algebra \(\mathcal A\) is not formal.
scientific article; zbMATH DE number 5377122
scientific article