The monodromy of meromorphic projective structures
Dylan G. L. Allegretti, Tom Bridgeland
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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P upper G upper L 2 left-parenthesis double-struck upper C right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>PGL</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo> </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {PGL}_2(\mathbb {C})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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.