Generic global generation for unitary local systems on curves (parabolic case)
Show that for g≥3 and a unitary representation \rho: \pi_1(\Sigma_{g,n})\to U(r), for a generic complex structure X with compactification \overline{X} and boundary D, the associated parabolic bundle \widehat{\mathscr{E}_0\otimes \omega_{\overline{X}}(D)} is generically generated by global sections.
References
Conjecture [ggg conjecture] Let $g\geq 3$, and let $$\rho: \pi_1(\Sigma_{g,n})\to U(r)$$ be a unitary representation. Fix a generic complex structure $X$ on $\Sigma_{g,n}$, with $\overline{X}$ the corresponding compact Riemann surface, and $D=\overline{X}\setminus X$, and let $\mathscr{E}\star$ be the corresponding parabolic bundle. Then $\widehat{\mathscr{E}_0\otimes \omega{\overline{X}}(D)}$ is generically generated by global sections.