The second variation of the Hodge norm and higher Prym representations

Let chi is an element of H1(sigma h,Q)$\chi \in H<^>1(\Sigma _h,\mathbb {Q})$ denote a rational cohomology class, and let H chi$\operatorname{H}_\chi$ denote its Hodge norm. We recover the result that H chi$\operatorname{H}_\chi$ is a plurisubharmonic function on the Teichmuller space Th${\mathcal {T}}_h$, and characterize complex directions along which the complex Hessian of H chi$\operatorname{H}_\chi$ vanishes. Moreover, we find examples of chi is an element of H1(sigma h,Q)$\chi \in H<^>1(\Sigma _{h},\mathbb {Q})$ such that H chi$\operatorname{H}_\chi$ is not strictly plurisubharmonic. As part of this construction, we find an unbranched covering pi:sigma h ->sigma 2$\pi:\Sigma _{h}\rightarrow \Sigma _2$ such that the subgroup of H1(sigma h,Q)$H_1(\Sigma _{h},\mathbb {Q})$ generated by lifts of simple curves from sigma 2$\Sigma _2$ is strictly contained in H1(sigma h,Q)$H_1(\Sigma _{h},\mathbb {Q})$. Finally, combining the characterization theorem with the Riemann-Roch, and the Li-Yau [Invent. Math. 69 (1982), no. 2, 269-291] gonality estimate, we show that geometrically uniform covers of sigma g$\Sigma _g$ satisfy the Putman-Wieland Conjecture about the induced Higher Prym representations.