Projective test

Algorithm

Stack (abstract data type)

Engineering

Pure mathematics

Computer science

Surface (topology)

Mechanical engineering

Mathematics

Image (mathematics)

Artificial intelligence

Geometry

Programming language

Meromorphic function

Manifold (fluid mechanics)

Monodromy

Algebra over a field

Transactions of the American Mathematical Society

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.

The monodromy of meromorphic projective structures