Potentials and Chern forms for Weil-Petersson and Takhtajan-Zograf metrics on moduli spaces

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• 作者：Jinsung Park^a ; ^{jinsung@kias.re.kr" class="auth_mail" title="E-mail the corresponding author} ; Leon A. Takhtajan^b ; ^c ; ^{leontak@math.sunysb.edu" class="auth_mail" title="E-mail the corresponding author} ; Lee-Peng Teo^d ; ^{lpteo@xmu.edu.my" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 14H60 ; 32G15 ; secondary ; 53C80
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：10 January 2017
• 年：2017
• 卷：305
• 期：Complete
• 页码：856-894
• 全文大小：631 K

文摘

For the TZ metric on the moduli space M_0,n${\mathcal{M}}_{0,n}$ of n  -pointed rational curves, we construct a Kähler potential in terms of the Fourier coefficients of the Klein's Hauptmodul. We define the space S_g,n${\mathfrak{S}}_{g,n}$ as holomorphic fibration S_g,n→S_g${\mathfrak{S}}_{g,n}\to {\mathfrak{S}}_g$ over the Schottky space S_g${\mathfrak{S}}_g$ of compact Riemann surfaces of genus g, where the fibers are configuration spaces of n   points. For the tautological line bundles L_i${\mathcal{L}}_i$ over S_g,n${\mathfrak{S}}_{g,n}$, we define Hermitian metrics h_i$h_i$ in terms of Fourier coefficients of a covering map J of the Schottky domain. We define the regularized classical Liouville action S   and show that exp⁡{S/π}$\exp \{S/\pi \}$ is a Hermitian metric in the line bundle $\mathcal{L}={\otimes }_{i=1}^n{\mathcal{L}}_i$ over S_g,n${\mathfrak{S}}_{g,n}$. We explicitly compute the Chern forms of these Hermitian line bundles We prove that a smooth real-valued function $\&<font\; color="red">min</font>us;\mathcal{S}=\&<font\; color="red">min</font>us;S+\pi {\sum }_{i=1}^n\log h_i$ on S_g,n${\mathfrak{S}}_{g,n}$, a potential for this special difference of WP and TZ metrics, coincides with the renormalized hyperbolic volume of a corresponding Schottky 3-manifold. We extend these results to the quasi-Fuchsian groups of type (g,n)$(g,n)$.