Relations on the period mapping giving extensions of mixed Hodge structures on compact Riemann surfaces
The goal of this paper is to produce `higher bilinear period relations' for a compact Riemann surface from a point of view which combines that of \textit{R. C. Gunning} [Probl. Analysis, Princeton Univ. 1969, 239-247 (1970; Zbl 0211.10502) and \textit{E. Jablow} [Duke Math. J. 53, 221-232 (1986; Zbl 0607.30043)] on quadratic periods with that of \textit{R. Hain} [Proc. Sympos. Pure Math. 46, 247-282 (1987; Zbl 0654.14006)] and \textit{M. Pulte} [Duke Math. J. 57, No. 3, 721-760 (1988; Zbl 0678.14005)] on variations of mixed Hodge structure. The authors consider a period map \(\Psi\) from Teichmüller space to \(\Hom ({\mathcal K}_2, {\mathcal H} {\mathfrak s}p1)_\mathbb{R}\), which is a real vector bundle over the Siegel upper half space. This map lifts the Torelli map. The period map from Teichmüller space modulo the Johnson kernel is generically injective. The relations that the quadratic periods must satisfy are derived. These identities are generalizations of the symmetry of the Riemann period matrix. The period map factors through a translation of the subbundle \((\Lambda {\mathfrak s} p3{\mathcal H}_1)_\mathbb{R}\) and is completely determined by the purely holomorphic quadratic periods. This result is applied to strengthen some theorems in the literature. One application is that the quadratic periods, along with the abelian periods, determine a generic marked compact Riemann surface up to an element of the kernel of Johnson's homomorphism. Another application is that the cocycle that exhibits the mapping class group modulo the Johnson kernel as an extension of the group \(SP_g (\mathbb{Z})\) by the group \((\Lambda {\mathfrak s} p3 H_1)_\mathbb{Z}\) is computed.
scientific article; zbMATH DE number 843615