TFPT scientific source excerpt
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000113 | \begin{theorem}[Rigid admissible family holonomy class]
000114 | \label{thm:rigid-family-local-system}
000115 | On the rigid four-punctured family section
000116 | \[
000117 | X_f^\circ\cong \mathbb P^1\setminus\mu_4,
000118 | \]
000119 | fix the determinant line, the local monodromy conjugacy classes at the four punctures, and the
000120 | $D_4$ symmetry inherited from the seam winding theorem. Then there exists a unique admissible
000121 | flat Hermitian family connection
000122 | \[
000123 | \nabla_F^\star
000124 | \]
000125 | on the flat Hermitian family bundle $L_F\to X_f^\circ$ with fiber
000126 | $F\cong H^1(X_f^\circ,\mathbb C)\cong\mathbb C^3$ whose monodromy representation is the explicit
000127 | holonomy datum $\rho_F$ of
000128 | \TFPTxref{def:canonical-cycle-holonomy,thm:d4-monodromy-rigidity}. Hence its $SU(3)_F$
000129 | conjugacy class $[\nabla_F^\star]$ is unique.
000130 | \end{theorem}
000131 | 
000132 | \begin{proof}
000133 | By \TFPTxref{thm:topological-family-closure}, the family section is the rigid four-punctured sphere
000134 | \[
000135 | X_f^\circ\cong\mathbb P^1\setminus\mu_4,
000136 | \]
000137 | and it carries the full $D_4$ symmetry inherited from the seam winding data. The determinant line
000138 | is fixed to be trivial, and the local puncture conjugacy classes are fixed by the admissible cusp
000139 | labels. Therefore \TFPTxref{thm:d4-monodromy-rigidity} determines one monodromy representation
000140 | \[
000141 | \rho_F:\pi_1(X_f^\circ)\to SU(3)_F
000142 | \]
000143 | uniquely up to global $SU(3)_F$ conjugation.
000144 | 
000145 | Construct the flat Hermitian bundle explicitly as
000146 | \[
000147 | L_F:=(\widetilde X_f^\circ\times\mathbb C^3)/\pi_1(X_f^\circ),
000148 | \]
000149 | where $\pi_1(X_f^\circ)$ acts on $\mathbb C^3$ through $\rho_F$. Because the image of $\rho_F$ lies
000150 | in $SU(3)_F$, the standard Hermitian form on $\mathbb C^3$ descends to a Hermitian metric on
000151 | $L_F$, and the trivial connection on $\widetilde X_f^\circ\times\mathbb C^3$ descends to a flat
000152 | Hermitian connection realizing the same monodromy representation.
000153 | 
000154 | Conversely, any admissible flat Hermitian family connection with the same puncture data has the same
000155 | monodromy representation up to global $SU(3)_F$ conjugation. The unitary Riemann--Hilbert
000156 | correspondence for flat Hermitian bundles then identifies it with the descended bundle above.
000157 | Hence there is exactly one admissible flat Hermitian connection class realizing the rigid puncture
000158 | data. We denote this distinguished class by
000159 | \[
000160 | [\nabla_F^\star].
000161 | \]
000162 | \end{proof}
000163 |