The second variation of the Hodge norm and higher Prym representations

Let $\chi \in H^1({\Sigma }_h,Q)$ denote a rational cohomology class, and let $H_{\chi }$ denote its Hodge norm. We recover the result that $H_{\chi }$ is a plurisubharmonic function on the Teichmüller space $T_h$, and characterize complex directions along which the complex Hessian of $H_{\chi }$ vanishes. Moreover, we find examples of $\chi \in H^1({\Sigma }_h,Q)$ such that $H_{\chi }$ is not strictly plurisubharmonic. As part of this construction, we find an unbranched covering $\pi :{\Sigma }_h\to {\Sigma }_2$ such that the subgroup of $H_1({\Sigma }_h,Q)$ generated by lifts of simple curves from ${\Sigma }_2$ is strictly contained in $H_1({\Sigma }_h,Q)$. 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 ${\Sigma }_g$ satisfy the Putman–Wieland Conjecture about the induced Higher Prym representations.