My Works on Moduli Spaces

Since 1991

 ${\huge{\color{Brown}{ \mathbb{I.\ Algebraic\ Structures\ of\ Punctured\ Riemann\ Surfaces}}}}$  ${\bigodot\ \Large{\overline{\pi_{g,N}}\,:\ \overline {{\mathcal C}_{g,N}}\,\to\,\overline{{\mathcal M}_{g,N}} \,:= \ \mathrm{universal\ curve \ of\ hyperbolic\ punctured\ Riemann\ surfaces\ of\ signature}\ (g,N)}}$  $\bigodot\ \Large{\mathbb P_i\ (i=1,2,\dots,N):\ \mathrm{sections\ corresponding\ to\ punctures} }$  $\bigodot\ \Large{\Delta_{\mathrm{bdy}}:\ \mathrm{the\ boundary\ divisor\ of}\ \overline{{\mathcal M}_{g,N}} }$  $\huge{\heartsuit\ \color{red}{\mathrm{Weil-Petersson\ Line\ Bundle:}}}$  $\boxed{\displaystyle{\Large{\Delta_{\mathrm{WP}}\,:=\,\langle\, K_{\overline{\pi_{g,N}}} (\mathbb P_1+\mathbb P_2+\cdots+\mathbb P_N),\,K_{\overline{\pi_{g,N}}} (\mathbb P_1+\mathbb P_2+\cdots+\mathbb P_N)\,\rangle }}}$  $\huge{\heartsuit\ \color{red}{\mathrm{Takhtajan-Zograf\ \, Line\ \,Bundle:}}}$  $\boxed{\Large{\begin{cases}~&\Delta_{\mathrm{TZ},i}\,:=\,\langle\, K_{\overline{\pi_{g,N}}} ,\,\mathbb P_i\,\rangle,\qquad i\,=\,1,\,2,\,\dots,\,N\\ ~&~\\ ~&\Delta_{\mathrm{TZ}\ }\,:=\langle\,K_{\overline{\pi_{g,N}}} ,\,\mathbb P_1+\mathbb P_2+\cdots+\mathbb P_N)\,\rangle\end{cases} }}$  $\huge{\heartsuit\ \color{red}{\mathrm{Grothendieck-Mumford\ Determinant\ Line\ Bundles:}}}$  $\Large{\boxed{\lambda_m=\begin{cases} \lambda\Big(mK_{\overline{\pi_{g,N}}}+(m-1)(\mathbb P_1+\mathbb P_2+\cdots+\mathbb P_N)\Big),&m\geq 1\\ &\\ \lambda\Big(((K_{\overline{\pi_{g,N}}}(\mathbb P_1+\mathbb P_2+\cdots+\mathbb P_N))^\vee\Big)^{\otimes -m}), &m\leq 0\end{cases}}}$  $\huge{\clubsuit\ \color{Blue}{\mathrm{Fundemantal\ Relation\ 0:}}}$  $\Large{\boxed{\displaystyle{\lambda_m\simeq \lambda_{1-m},\qquad\forall m\leq 0 }}}$   $\huge{\clubsuit\ \color{Blue}{\mathrm{Fundamental\ Relation\ I:}}\ (\mathrm{Deligne-Mumford}\ N=0,\ \mathrm{Weng}\ N>0)}$  $\Large{\boxed{\displaystyle{\lambda_m^{\otimes 12}\simeq \Delta_{\mathrm{WP}}^{\otimes(6m^2-6m+1)}\otimes \Delta_{TZ}^{\otimes -1}\otimes\Delta_{\mathrm{bdy}} }}}$   $\huge{\clubsuit\ \color{Blue}{\mathrm{Fundamental\ Relation\ II:}}}$  $\Large{\boxed{\displaystyle{\Delta_{\mathrm{WP}}^{\otimes N^2}\,\leq\, \Delta_{\mathrm{TZ}}^{\otimes(2g-2+N)^2} }}}$   $\huge{\clubsuit\ \color{Blue}{\mathrm{Fundamental\ Relation\ III:}\ (i)}\ (\mathrm{Xiao}\ \&\ \mathrm{Cornallba-Harris}) \ N=0: }$  $\Large{\boxed{\displaystyle{\Big(8+\frac{4}{g}\Big)\lambda_1\,\geq\,\Delta_{\mathrm{bdy}} }}}$   $\huge{\clubsuit\ \color{Blue}{\mathrm{Fundamental\ Relation\ III:}\ (ii)}\ N\geq 1:}$  $\Large{\boxed{\displaystyle{\Big(8+\frac{2N}{g-1+N}\Big)\lambda_1+\Delta_{\mathrm{TZ}}\,\geq\,\Delta_{\mathrm{bdy}} }}}$  $**********************************************************************************************$    ${\huge{\color{Brown}{ \mathbb{II.\ Arithmetic\ Structures\ of\ Punctured\ Riemann\ Surfaces}}}}$   $\bigodot\ \Large{\omega_{\mathrm{WP}}:\ \mathrm{the\ Kaehler\ form\ corresponding\ to\ the\ Weil-Petersson\ metric} }$  $\bigodot\ \Large{\omega_{\mathrm{TZ}}:\ \mathrm{the\ Kaehler\ form\ corresponding\ to\ the\ Takhtajan-Zograf\ metric} }$   $\huge{\clubsuit\ \color{Blue}{\mathrm{Fundamental\ Relation\ IV:}}\ (\mathrm{Wolpert,\ Weng})}$  $\Large{\boxed{\displaystyle{c_1\Big(\underline{\Delta_{\mathrm{WP}}}\Big) =\frac{\omega_{\mathrm{WP}}}{\pi^2} }}}$  $\huge{\clubsuit\ \color{Blue}{\mathrm{Fundamental\ Relation\ V:}}}$ $\Large{\boxed{\displaystyle{c_1\Big(\underline{\Delta_{\mathrm{TZ}}}\Big) =\frac{4}{3}\omega_{\mathrm{TZ}} }}}$   $\huge{\clubsuit\ \color{Blue}{\mathrm{Fundamental\ Relation\ VI:}}\ (\mathrm{Deligne}\ N=0,\ \mathrm{Weng}\ N\geq 1)}$  $\Large{\boxed{\displaystyle{\underline{\lambda_m}^{\otimes 12}\simeq\underline{\Delta_{\mathrm{WP}}}^{\otimes(6m^2-6m+1)} \otimes\underline{\Delta_{\mathrm{TZ}}}^{\otimes -1} }}}$   $\huge{\clubsuit\ \color{Blue}{\mathrm{Fundamental\ Relation\ VI}':}\ (\mathrm{Takhtajan-Zograf})}$  $\Large{\boxed{\displaystyle{c_1\Big(\lambda_m,h_Q(m)\Big)=\frac{6m^2-6m+1}{12}\cdot \frac{\omega_{\mathrm{WP}}}{\pi^2} -\frac{1}{9}\cdot \omega_{\mathrm{TZ}} }}}$