Local deformations of branched projective structures: Schiffer variations and the Teichmüller map

Abstract

We study a class of continuous deformations of branched complex projective structures on closed surfaces of genus $g\ge 2$, which preserve the holonomy representation of the structure and the order of the branch points. In the case of non-elementary holonomy we show that when the underlying complex structure is infinitesimally preserved the branch points are necessarily arranged on a canonical divisor, and we establish a partial converse for hyperelliptic structures.

arXiv comment29 pages, 2 figures; v2: funding information updated

References in corpus (7)

• Branched projective structures with quasi-Fuchsian holonomy
• 2 π-grafting and complex projective structures, I
• The oriented graph of multi-graftings in the Fuchsian case
• Bubbling complex projective structures with quasi-Fuchsian holonomy
• Holonomy map fibers of $\mathbb{C}{\mathrm{P}}^1$-structures in moduli space
• Multi(de)grafting quasi-Fuchsian complex projective structures via bubbles
• Complex projective structures with maximal number of Möbius transformations

Cited by in corpus (1)

• Tame and relatively elliptic ${\mathbb{CP}}^1$-structures on the thrice-punctured sphere