A Twisted bidifferential, second fundamental form of Prym map and projective structures

Abstract

Given a point $[(\widetilde{C},C)]$ of the moduli space ${\mathcal R}_g$, where $\widetilde{C}$ is an étale double cover of a smooth curve $C$ of genus $g \,\geq\, 7$, we construct a twisted meromorphic bidifferential $\widehatη_P$ on $C\times C$ with a pole on the diagonal. It is shown that the second fundamental form $ρ_P$ of the Prym map from ${\mathcal R}_g$ to ${\mathcal A}_{g-1}$ at $[(\widetilde{C},C)]$ is given by this $\widehatη_P$. This allows us to compare $ρ_P$ with the second fundamental forms of the two Torelli maps, to ${\mathcal A}_{2g-1}$ and ${\mathcal A}_{g}$, at the points $[\widetilde{C}]$ and $[C]$ respectively. It is shown that $\widehatη_P$ induces a projective structure on $C$ that depends on the double cover $\widetilde{C}\, \longrightarrow\, C$; these projective structures together give a ${\mathcal C}^{\infty}$ section $β_P$ of the bundle of projective structures on the moduli space ${\mathcal R}_g$. Finally, it is proved that the $(0,\,1)$-component of the differential of the section $β_P$ is the pullback of the Kähler form of the Siegel metric on ${\mathcal A}_{g-1}$ via the Prym map. This is a key ingredient in the proof that this section $β_P$ is not the pullback of a section of projective structures on ${\mathcal M}_{g}$ and, in addition, that it differs from the known projective structures on ${\mathcal R}_g$.