The monodromy of meromorphic projective structures

Abstract

We study projective structures on a surface having poles of prescribed orders. We obtain a monodromy map from a complex manifold parameterising such structures to the stack of framed ${\mathrm{PGL}}_2(\mathbb{C})$ local systems on the associated marked bordered surface. We prove that the image of this map is contained in the union of the domains of the cluster charts. We discuss a number of open questions concerning this monodromy map.

arXiv comment54 pages. Version 2: Incorporated suggestions from referee

References in corpus (11)

• Four-dimensional wall-crossing via three-dimensional field theory
• Riemann-Hilbert problems from Donaldson-Thomas theory
• Twisted wild character varieties
• Voros symbols as cluster coordinates
• Painlevé monodromy manifolds, decorated character varieties and cluster algebras
• Projective structures and projective bundles over compact Riemann surfaces
• Stability conditions, cluster varieties, and Riemann-Hilbert problems from surfaces
• Geometry and braiding of Stokes data; Fission and wild character varieties
• Exact WKB analysis and cluster algebras
• G-bundles, isomonodromy and quantum Weyl groups
• Stability conditions and cluster varieties from quivers of type $A$

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• Toric degenerations of cluster varieties and cluster duality
• Monodromy groups of $\mathbb{C}{\mathrm{P}}^1$-structures on punctured surfaces
• Classification of algebraic solutions of irregular Garnier systems
• Naturality of ${\mathrm{SL}}_3$ quantum trace maps for surfaces
• A characterization of pseudo-Anosov mapping classes via tropical cluster transformations
• Dominating surface-group representations into ${\mathrm{PSL}}_2(\mathbb{C})$ in the relative representation variety
• Stability conditions and Teichmüller space
• Quantum Analytic Langlands Correspondence
• Tame and relatively elliptic ${\mathbb{CP}}^1$-structures on the thrice-punctured sphere
• Monodromy of Schwarzian equations with regular singularities
• Nilpotent Higgs bundles and families of flat connections
• Neck-Pinching of $C{P}^1$-structures in the $PSL(2,C)$-character variety
• On the relative opers in dimension one
• Meromorphic Projective Structures, Opers and Monodromy
• Parabolic SL(r)-opers
• Parabolic opers and differential operators
• On Generalized WKB Expansion of Monodromy Generating Function