Relations on the period mapping giving extensions of mixed hodge structures on compact Riemann surfaces

Abstract

We consider a period map Ψ from Teichmüller space to $\mathcal{H}om(\mathcal{K}_2 {\text{,}}\mathcal{H}^1 {\text{)}}_\mathbb{R}$, which is a real vector bundle over the Siegel upper half space. This map lifts the Torelli map. We study the action of the mapping class group on this period map. We show that the period map from Teichmüller space modulo the Johnson kernel is generically injective. We derive relations that the quadratic periods must satisfy. These identities are generalizations of the symmetry of the Riemann period matrix. Using these higher bilinear relations, we show that the period map factors through a translation of the subbundle $( \wedge ^3 \mathcal{H}_{\text{1}} )_\mathbb{R}$ and is completely determined by the purely holomorphic quadratic periods. We apply this result 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 we compute the cocycle that exhibits the mapping class group modulo the Johnson kernel as an extension of the group SP _g (ℤ) by the group $( \wedge ^3 \mathcal{H}_{\text{1}} )_\mathbb{Z}$.