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The Monodromy of Meromorphic Projective Structures

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THE MONODROMY OF MEROMORPHIC PROJECTIVE STRUCTURES DYLAN G.L. ALLEGRETTI AND 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 PGL2(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. 1. Introduction This paper is concerned with the monodromy of projective structures on Riemann surfaces. A projective structure can be viewed as a global generalization of a differential equation of the form y00(z) − '(z) · y(z) = 0; where primes denote differentiation with respect to z. In another language it is an oper for the group PGL2(C). Our main focus will be on the case when the potential '(z) has poles, but we begin by recalling some of the classical results on holomorphic projective structures. For excellent surveys on this material we recommend [17, 36, 43]. 1.1. Holomorphic projective structures. A projective structure P on a Riemann surface S is an atlas of holomorphic charts 1 [ fi : Ui ! P ;S = Ui; i2I −1 with the property that each transition function gij = fi ◦ fj is the restriction of an element of 1 G = Aut(P ) = PGL2(C): Projective structures are closely related to quadratic differentials. If P is a projective structure on a Riemann surface S, with a local chart z : U ! P1, and φ(z) = '(z) dz⊗2 (1) is a quadratic differential on S, written in terms of the co-ordinate z, then one obtains a chart in a new projective structure P0 = P + φ on the surface S by considering the ratio of two independent 1 2 DYLAN G.L. ALLEGRETTI AND TOM BRIDGELAND solutions to the differential equation y00(z) − '(z) · y(z) = 0: (2) This construction gives the set of projective structures on a fixed compact Riemann surface S of genus g = g(S) the structure of an affine space for the vector space of holomorphic quadratic differentials 0 ⊗2 ∼ 3g−3 H (S; !S ) = C : (3) Let us fix a closed, oriented surface S of genus g = g(S). A marked projective structure is then defined to be a triple (S; P; θ), where S is a Riemann surface equipped with a projective structure P, and θ is a marking, that is, an isotopy class of orientation-preserving diffeomorphisms θ : S ! S. Two such triples (Si; Pi; θi) will be considered to be equivalent if there is a biholomorphism f : S1 ! S2 which preserves the projective structures and the markings in the obvious way. The set P(S) of equivalence classes of marked projective structures has the natural structure of a complex manifold of dimension 6g − 6. There is an obvious forgetful map p: P(S) ! T(S) (4) to the Teichm¨ullerspace T(S), viewed as the moduli space of Riemann surfaces S equipped with a marking θ : S ! S. A relative version of the construction described above shows that this map (4) is an affine bundle for the vector bundle q : Q(S) ! T(S); whose fibre over a marked Riemann surface (S; θ) is the vector space (3). 1.2. Monodromy of projective structures. A projective structure P on a Riemann surface S has an associated developing map: a holomorphic map 1 f : S~ ! P ; where π : S~ ! S is the universal covering surface, such that any injective locally-defined map of the form f ◦ π−1 is a chart in P. Such a developing map gives rise to a monodromy representation ρ: π1(S) ! G; (5) defined by the the condition f(γ · x) = ρ(γ) · f(x). The developing map f is unique up to post-composition with an element of the group G, and the monodromy representation is therefore well-defined up to overall conjugation by an element of G. More abstractly, we can think in terms of a G local system naturally associated to the projective structure. THE MONODROMY OF MEROMORPHIC PROJECTIVE STRUCTURES 3 Let us fix a closed, oriented surface S as above and consider the quotient stack X(S) = Hom(π1(S);G)=G (6) parameterising representations (5) up to overall conjugation, or equivalently, isomorphism classes of G local systems on S. The monodromy of a marked projective structure on S defines a point of this stack in the obvious way. Gunning [26] proved that when g = g(S) > 1, this point lies in the open substack X∗(S) consisting of representations with non-commutative image. This substack is a (possibly non-Hausdorff) complex manifold, and there is a holomorphic map ∗ F : P(S) ! X (S) (7) sending a marked projective structure to its monodromy representation. The map F has been studied for over a century. Let us briefly recall some of the better known results. A result of Poincar´e[39] shows that F is injective when restricted to each fibre of the forgetful map (4). Hejhal [27] proved that F is a local homeomorphism, and Earle [18] and Hubbard [28] showed that F is moreover a local biholomorphism. It is also known that F has infinite fibres and is not a covering map of its image, see e.g. [17]. Finally, a famous theorem of Gallo, Kapovich and Marden [25] characterises the image of F . 1.3. Meromorphic projective structures. In this paper we study a monodromy map analogous to (7) but for meromorphic projective structures. The notion of a meromorphic projective structure has meaning because of the above-mentioned fact that the set of projective structures on a fixed Riemann surface S is an affine space for the space of quadratic differentials. Concretely, if we fix a holomorphic projective structure on S as above, the local charts in a meromorphic projective structure are obtained by taking ratios of solutions to the equation (2), where the quadratic differential (1) is now allowed to have poles. Note that when '(z) has a pole p of order m > 2 the equation (2) has an irregular singularity, and one should expect generalised monodromy in the form of Stokes data to enter the picture. A solution y(z) to the equation (2) defined in a sectorial neighbourhood centered at p is called subdominant if y(z) ! 0 as z ! p. Standard results in the theory of differential equations show that there are m − 2 distinguished sectors centered at p, known as Stokes sectors, in which there exist unique-up-to-scale subdominant solutions to (2). The rays in the centres of these distinguished sectors are called Stokes directions and coincide with the asymptotic directions at p of the horizontal foliation defined by the quadratic differential φ(z). 4 DYLAN G.L. ALLEGRETTI AND TOM BRIDGELAND Our results depend on a choice of genus g ≥ 0 and a non-empty collection of positive integers giving the orders of poles of the projective structures. It is more convenient to represent this data in the form of a marked bordered surface (S; M). This is a compact, connected, oriented surface S with (possibly empty) boundary, equipped with a non-empty collection of marked points M ⊂ S, such that each boundary component of S contains at least one point of M. We denote by P ⊂ M the set of internal marked points, which we also refer to as punctures. A meromorphic projective structure P on a compact Riemann surface S naturally determines a marked bordered surface (S; M). The surface S is the real oriented blow-up of S at the poles of P of order > 2, which are precisely the irregular singularities of the differential equation (2), and the boundary marked points correspond to the Stokes directions. The internal marked points are the poles of P of order ≤ 2, which are precisely the regular singularities of the equation (2). Let us fix a marked bordered surface (S; M). If g(S) = 0 we assume that jMj ≥ 3. In Section8 we introduce a complex manifold P(S; M) parameterising marked meromorphic projective structures in much the same way as before. The points of P(S; M) are equivalence-classes of triples (S; P; θ), where S is a Riemann surface equipped with a meromorphic projective structure P, and θ is an isotopy class of orientation-preserving diffeomorphisms between the marked bordered surface canonically associated to (S; P) and the fixed surface (S; M). It will be convenient to introduce two modifications to the manifold P(S; M). Firstly, we define a dense open subset ◦ P (S; M) ⊂ P(S; M); whose complement consists of projective structures with apparent singularities: regular singularities for which the corresponding monodromy transformation is trivial as an element of G = PGL2(C). We do this because the monodromy map of Theorem 1.1 is not well-defined for such projective structures. Secondly, we introduce a branched cover ∗ ◦ π : P (S; M) ! P (S; M) (8) of degree 2jPj, whose points parameterise marked projective structures equipped with a signing: a choice of eigenline for the monodromy around each regular singularity. The main point of this is to obtain a monodromy map taking values in the stack of framed local systems, which as we explain below, is rational and carries interesting birational co-ordinate systems. An analogous cover also turns out to be very natural in the context of moduli spaces of meromorphic quadratic differentials [14, Section 6.2]. THE MONODROMY OF MEROMORPHIC PROJECTIVE STRUCTURES 5 1.4. Monodromy and framed local systems. Let us now turn to the analogue of the character stack (6) in the meromorphic situation.
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