Let $\chi \in H^1({\mathrm{\Sigma }}_h,\mathbb{Q})$ 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 ${\mathcal{T}}_h$, and characterize complex directions along which the complex Hessian of $H_{\chi }$ vanishes. Moreover, we find examples of $\chi \in H^1({\mathrm{\Sigma }}_h,\mathbb{Q})$ such that $H_{\chi }$ is not strictly plurisubharmonic. As part of this construction, we find an unbranched covering $\pi :{\mathrm{\Sigma }}_h\to {\mathrm{\Sigma }}_2$ such that the subgroup of $H_1({\mathrm{\Sigma }}_h,\mathbb{Q})$ generated by lifts of simple curves from ${\mathrm{\Sigma }}_2$ is strictly contained in $H_1({\mathrm{\Sigma }}_h,\mathbb{Q})$. 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 ${\mathrm{\Sigma }}_g$ satisfy the Putman–Wieland Conjecture about the induced Higher Prym representations.

中文翻译：

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Hodge 范数的第二个变体和更高的 Prym 表示

让$\chi \epsilon H^1（{\mathrm{\Sigma }}_H,\mathbb{问}）$表示有理上同调类，并令$H_{\chi }$表示其霍奇范数。我们恢复的结果是$H_{\chi }$是 Teichmüller 空间上的多次谐波函数${\mathcal{时间}}_H$，并描述复数 Hessian 矩阵所沿的复数方向$H_{\chi }$消失。此外，我们还发现了这样的例子$\chi \epsilon H^1（{\mathrm{\Sigma }}_H,\mathbb{问}）$这样$H_{\chi }$不是严格的多次谐波。作为这个结构的一部分，我们发现了一个无分支的覆盖物$\pi ：{\mathrm{\Sigma }}_H\to {\mathrm{\Sigma }}_2$使得子群$H_1（{\mathrm{\Sigma }}_H,\mathbb{问}）$由简单曲线的升力产生${\mathrm{\Sigma }}_2$严格包含在$H_1（{\mathrm{\Sigma }}_H,\mathbb{问}）$。最后，将表征定理与黎曼-罗赫和李-丘[发明。数学。 69（1982），没有。 2, 269–291] gonality 估计，我们表明几何均匀覆盖${\mathrm{\Sigma }}_G$满足关于诱导的 High Prym 表示的 Putman-Wieland 猜想。