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 PGL 2 (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.

Voros symbols as cluster coordinates

“…In [4], we considered a moduli space parametrizing projective structures with poles of prescribed orders. To define such a projective structure, let us fix an ordinary projective structure P 0 on S. Then a meromorphic projective structure is defined as a collection of charts w : U → CP 1 given by ratios of solutions of the equation (2) as above, where now the quadratic differential φ is allowed to have poles.…”

“…of degree 2 N where N is the number of marked points in the interior of S. In terms of the space Proj * (S, M), the main result of [4] was the following.…”

“…First, we restrict attention to the open subset Proj • (S, M) ⊆ Proj(S, M), whose complement is the codimension 2 locus of projective structures with apparent singularities. We do this because the monodromy map of [4] is not defined at projective structures with apparent singularities. Second, we consider a finite cover…”

“…Then, for each puncture p, we can define a loop δ p ∈ π 1 (S, x) which travels from x to U (p) along β p , travels along a small loop around p in the counterclockwise direction, and then returns to x along β p . Lemma 6.2 [4,Lemma 4.2]. There is a bijection between the set of isomorphism classes of rigidified framed P GL 2 (C)-local systems on (S, M) and the set of points of the complex projective variety…”

“…The framing at p is given by an eigenline of the monodromy, and hence for a generic framed local system, there is a unique way of modifying the framing at p to get a different framed local system. Lemma 6.5 [4,Lemma 9.4]. There is a birational action of the group (Z/2Z) P on the stack X (S, M) of framed local systems in which the nontrivial generator corresponding to p ∈ P acts by fixing the underlying local system and exchanging the two generically possible choices of framing at p.…”