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

Given a point $[(\tilde{C},C)]$ of the moduli space ${\mathcal{R}}_g$, where $\tilde{C}$ is an étale double cover of a smooth curve $C$ of genus $g\ge 7$, we construct a twisted meromorphic bidifferential ${\widehat{\eta }}_P$ on $C\times C$ with a pole on the diagonal. It is shown that the second fundamental form ${\rho }_P$ of the Prym map from ${\mathcal{R}}_g$ to ${\mathcal{A}}_{g-1}$ at $[(\tilde{C},C)]$ is given by this ${\widehat{\eta }}_P$. This allows us to compare ${\rho }_P$ with the second fundamental forms of the two Torelli maps, to ${\mathcal{A}}_{2g-1}$ and ${\mathcal{A}}_g$, at the points $[\tilde{C}]$ and $[C]$ respectively. It is shown that ${\widehat{\eta }}_P$ induces a projective structure on $C$ that depends on the double cover $\tilde{C}⟶C$; these projective structures together give a ${\mathcal{C}}^{\mathrm{\infty }}$ section ${\beta }_P$ 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 ${\beta }_P$ 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 ${\beta }_P$ 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 ${\mathcal{R}}_g$.