My Works on Moduli Spaces
Since 1991
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\[{\huge{\color{Brown}{
\mathbb{I.\ Algebraic\ Structures\ of\ Punctured\ Riemann\ Surfaces}}}}\]
\[\]
\({\bigodot\ \Large{\overline{\pi_{g,N}}\,:\ \overline {{\mathcal C}_{g,N}}\,\to\,\overline{{\mathcal M}_{g,N}}
\,:=
\ \mathrm{universal\ curve
\ of\ hyperbolic\ punctured\ Riemann\ surfaces\ of\ signature}\ (g,N)}}
\)
\[\]
\(\bigodot\ \Large{\mathbb P_i\ (i=1,2,\dots,N):\ \mathrm{sections\ corresponding\ to\ punctures}
}\)
\[\]
\(\bigodot\ \Large{\Delta_{\mathrm{bdy}}:\ \mathrm{the\ boundary\ divisor\ of}\ \overline{{\mathcal M}_{g,N}}
}\)
\[\]
\(\huge{\heartsuit\ \color{red}{\mathrm{Weil-Petersson\ Line\ Bundle:}}}\)
\[\]
\[\boxed{\displaystyle{\Large{\Delta_{\mathrm{WP}}\,:=\,\langle\,
K_{\overline{\pi_{g,N}}}
(\mathbb P_1+\mathbb P_2+\cdots+\mathbb P_N),\,K_{\overline{\pi_{g,N}}}
(\mathbb P_1+\mathbb P_2+\cdots+\mathbb P_N)\,\rangle
}}}\]
\[\]
\(\huge{\heartsuit\ \color{red}{\mathrm{Takhtajan-Zograf\ \, Line\ \,Bundle:}}}\)
\[\]
\[\boxed{\Large{\begin{cases}~&\Delta_{\mathrm{TZ},i}\,:=\,\langle\, K_{\overline{\pi_{g,N}}}
,\,\mathbb P_i\,\rangle,\qquad i\,=\,1,\,2,\,\dots,\,N\\
~&~\\
~&\Delta_{\mathrm{TZ}\ }\,:=\langle\,K_{\overline{\pi_{g,N}}}
,\,\mathbb P_1+\mathbb P_2+\cdots+\mathbb P_N)\,\rangle\end{cases}
}}\]
\[\]
\(\huge{\heartsuit\ \color{red}{\mathrm{Grothendieck-Mumford\ Determinant\ Line\ Bundles:}}}\)
\[\]
\[\Large{\boxed{\lambda_m=\begin{cases}
\lambda\Big(mK_{\overline{\pi_{g,N}}}+(m-1)(\mathbb P_1+\mathbb P_2+\cdots+\mathbb P_N)\Big),&m\geq 1\\
&\\
\lambda\Big(((K_{\overline{\pi_{g,N}}}(\mathbb P_1+\mathbb P_2+\cdots+\mathbb P_N))^\vee\Big)^{\otimes -m}),
&m\leq 0\end{cases}}}\]
\[\]
\(\huge{\clubsuit\ \color{Blue}{\mathrm{Fundemantal\ Relation\ 0:}}}\)
\[\]
\[\Large{\boxed{\displaystyle{\lambda_m\simeq \lambda_{1-m},\qquad\forall m\leq 0
}}}\]
\[\]
\[\]
\(\huge{\clubsuit\ \color{Blue}{\mathrm{Fundamental\ Relation\ I:}}\ (\mathrm{Deligne-Mumford}\ N=0,\ \mathrm{Weng}\ N>0)}\)
\[\]
\[\Large{\boxed{\displaystyle{\lambda_m^{\otimes 12}\simeq \Delta_{\mathrm{WP}}^{\otimes(6m^2-6m+1)}\otimes
\Delta_{TZ}^{\otimes -1}\otimes\Delta_{\mathrm{bdy}}
}}}\]
\[\]
\[\]
\(\huge{\clubsuit\ \color{Blue}{\mathrm{Fundamental\ Relation\ II:}}}\)
\[\]
\[\Large{\boxed{\displaystyle{\Delta_{\mathrm{WP}}^{\otimes N^2}\,\leq\, \Delta_{\mathrm{TZ}}^{\otimes(2g-2+N)^2}
}}}\]
\[\]
\[\]
\(\huge{\clubsuit\ \color{Blue}{\mathrm{Fundamental\ Relation\ III:}\ (i)}\ (\mathrm{Xiao}\ \&\ \mathrm{Cornallba-Harris})
\ N=0:
}\)
\[\]
\[\Large{\boxed{\displaystyle{\Big(8+\frac{4}{g}\Big)\lambda_1\,\geq\,\Delta_{\mathrm{bdy}}
}}}\]
\[\]
\[\]
\(\huge{\clubsuit\ \color{Blue}{\mathrm{Fundamental\ Relation\ III:}\ (ii)}\ N\geq 1:}\)
\[\]
\[\Large{\boxed{\displaystyle{\Big(8+\frac{2N}{g-1+N}\Big)\lambda_1+\Delta_{\mathrm{TZ}}\,\geq\,\Delta_{\mathrm{bdy}}
}}}\]
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\[**********************************************************************************************\]
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\[{\huge{\color{Brown}{
\mathbb{II.\ Arithmetic\ Structures\ of\ Punctured\ Riemann\ Surfaces}}}}\]
\[\]
\[\]
\(\bigodot\ \Large{\omega_{\mathrm{WP}}:\ \mathrm{the\ Kaehler\ form\ corresponding\ to\ the\
Weil-Petersson\ metric}
}\)
\[\]
\(\bigodot\ \Large{\omega_{\mathrm{TZ}}:\ \mathrm{the\ Kaehler\ form\ corresponding\ to\
the\ Takhtajan-Zograf\ metric}
}\)
\[\]
\[\]
\(\huge{\clubsuit\ \color{Blue}{\mathrm{Fundamental\ Relation\ IV:}}\ (\mathrm{Wolpert,\ Weng})}\)
\[\]
\[\Large{\boxed{\displaystyle{c_1\Big(\underline{\Delta_{\mathrm{WP}}}\Big)
=\frac{\omega_{\mathrm{WP}}}{\pi^2}
}}}\]
\[\]
\(\huge{\clubsuit\ \color{Blue}{\mathrm{Fundamental\ Relation\ V:}}}\)
\[\Large{\boxed{\displaystyle{c_1\Big(\underline{\Delta_{\mathrm{TZ}}}\Big)
=\frac{4}{3}\omega_{\mathrm{TZ}}
}}}\]
\[\]
\[\]
\(\huge{\clubsuit\ \color{Blue}{\mathrm{Fundamental\ Relation\ VI:}}\ (\mathrm{Deligne}\ N=0,\ \mathrm{Weng}\ N\geq 1)}\)
\[\]
\[\Large{\boxed{\displaystyle{\underline{\lambda_m}^{\otimes 12}\simeq\underline{\Delta_{\mathrm{WP}}}^{\otimes(6m^2-6m+1)}
\otimes\underline{\Delta_{\mathrm{TZ}}}^{\otimes -1}
}}}\]
\[\]
\[\]
\(\huge{\clubsuit\ \color{Blue}{\mathrm{Fundamental\ Relation\ VI}':}\ (\mathrm{Takhtajan-Zograf})}\)
\[\]
\[\Large{\boxed{\displaystyle{c_1\Big(\lambda_m,h_Q(m)\Big)=\frac{6m^2-6m+1}{12}\cdot \frac{\omega_{\mathrm{WP}}}{\pi^2}
-\frac{1}{9}\cdot \omega_{\mathrm{TZ}}
}}}\]
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