Boundary fibers of the holonomy map for complex projective structures

Abstract

We study the Morgan--Shalen boundary of holonomy representations of complex projective structures on a compact Riemann surface of genus at least two. We prove that every direction defined by a holomorphic quadratic differential with a multiple zero admits distinct projective marked length limits. Together with Dumas's uniqueness theorem for simple zeros, this gives a complete characterization of directions with unique limits. We also establish that no nonzero abelian length function can occur in the holonomy boundary of any such fixed surface, regardless of the limiting quadratic differential. Finally, we characterize all directional boundary fibers by a necessary-and-sufficient analytic condition involving valued representations and the holomorphic jet bundle underlying an oper.