This is a review text file submitted electronically to MR. Reviewer: W.M.Goldman Reviewer number: 014521 Address:

Math Dept. University of Maryland,College Park, Maryland 20742 Author: Nigel Hitchin Short title: Geometrical aspects of Schlesinger’s equation Control number: 1 484 592 Primary classification: 32G34 Secondary classification(s): 53C 81R25 Review text:

This very interesting paper interprets the classical differential equation of Schlesinger in terms of a projective structure on a complex surface and relates its solutions to the symplectic geometry of the moduli space of representations of the fundamental group of a quadruply-punctured sphere S in SL(2, ℂ).

A projective structure on a manifold M can be defined as a projective equivalence class of affine connections. Two affine connections are said to be projectively equivalent if they possess the same unparametrized geodesics. Thus a projective structure gives a well-defined 1-dimensional foliation of the projectivized tangent bundle P T M. For a geodesically convex neighborhood U ⊂ M the leaf space U^{∗} is a manifold of dimension 2 dim(M) − 2 and possesses an interesting geometry of its own.

In dimension two, É. Cartan proved that the geodesics of a projective structure are defined by a differential equation of the form

\frac{d^{2}y}{dx^{2}} = F (\frac{dy}{dx})

where F (u) is a cubic polynomial in u whose coefficients are functions of local coordinates (x, y).

The present paper considers the case when M is a complex surface and the projective structure is holomorphic, that is, the affine connections are holomorphic affine connections on M. In that case, the points of U determine lines in U^{∗} of self-intersection number 1, generalizing the duality between points and lines in the projective plane. The geometry of U^{∗} with this family of lines determines the projective structure on U.

A significant feature of this correspondence is that although U has only a local rigid geometry (a projective structure), the geometry of its dual U^{∗} is locally trivial (U^{∗} is just a complex manifold) but its global structure (the collection of lines) reflects the geometry of U.

This paper discusses an important example of this construction. Start with a connection on the trivial rank two trivial bundle over ℂ × V where V ⊂ ℂ^{3} is an open subset. The connection is defined by a meromorphic matrix-valued 1-form

A = ∑_{k=1}^{3} A_{k} \frac{dz − dx_{k}}{z − x_{k}}

where z ∈ ℂ and (x_{1}, …, x_{3}) ∈ V and A_{k} : V ⟶ 𝔤l(2, ℂ) is holomorphic. For fixed (x_{1}, …, x_{3}) ∈ V, the 1-form A has simple poles at x_{1}, x_{2}, x_{3} with residues A_{1}(x), A_{2}(x), A_{3}(x) and ∞ with residue A_{4}(x). By the vanishing of the sum of the residues of a meromorphic 1-form on a closed Riemann surface,

A_{1} + A_{2} + A_{3} + A_{4} = 0

Then Schlesinger’s equation is the condition that the associated connection is flat: dA + \frac{1}{2}[A, A] = 0. Although this construction works in general, for any rank m and number of points n, one obtains a two-dimensional moduli space for m = 2 and n = 3.

To obtain a surface M with an affine connection (with a resulting projective structure), normalize the parameter x ∈ V to (x_{1}, …, x_{3}) = (0, 1, t) by applying an affine transformation of ℂ. Then A may be regarded a connection on a holomorphic rank two vector bundle over S = P^{1} −{0, 1, t, ∞}. Schlesinger’s equation reduces to the system

\frac{dA_{1}}{dt} = \frac{[A_{3}, A_{1}]}{t},
\frac{dA_{2}}{dt} = \frac{[A_{3}, A_{2}]}{t − 1},
\frac{dA_{3}}{dt} = \frac{[A_{1}, A_{3}]}{t} + \frac{[A_{2}, A_{3}]}{t − 1},
\frac{dA_{4}}{dt} = 0

By writing each of these equations in Lax form, one sees that each A_{k}(t) (for k = 1, 2, 3, 4) lies in a single conjugacy class O_{k} ∈ 𝔰𝔩(2, ℂ). The moduli space M is thus determined by four adjoint orbits

O_{1}, O_{2}, O_{3}, O_{4} ∈ SL(2, ℂ)

and the two-dimensional moduli space M is the quotient of

{(A_{1}, A_{2}, A_{3}, A_{4}) ∈ O_{1} × O_{2} × O_{3} × O_{4} | A_{1} + A_{2} + A_{3} + A_{4} = 0}

(which has dimension five) by SL(2, ℂ). Schlesinger’s equation then determines flows (varying with t) on M (assuming no A_{k} is zero). The functions

x = tr(A_{1}A_{3})
y = tr(A_{2}A_{3})

define local coordinates on M. By differentiating the above system, one obtains a system of equations on M equivalent to the geodesic equations for an affine connection on M. Thus (Proposition 1), Schlesinger’s equation defines a projective structure on M.

This equation is equivalent to Painlevé’s sixth equation and cannot be solved explicitly. However, U^{∗} possesses a Poisson geometry which can be used to study the geometry resulting from Schlesinger’s equation. The Wronskian of two solutions of the linearization of Cartan’s equation for y(x) (tangent vectors to U^{∗}) is seen to be constant, from which one obtains an exterior 2-form on U^{∗}. This exterior 2-form is singular along a divisor D ⊂ U^{∗} with three components. The paper explicitly determines this 2-form for the system arising from Schlesinger’s equation.

For a solution z(t) of Schlesinger’s equation, the monodromy matrices A_{k}(z_{t}) remain in a fixed conjugacy class, and hence solutions define isomonodromic representations of the fundamental group of a quadruply-punctured sphere. (A family of representations of π_{1}(S) is said to be isomonodromic if for each boundary component b_{k} ⊂ ∂S the representations of π_{1}(b_{k}) remain in a single conjugacy class.) Furthermore the braid group H(3) (which appears as a mapping class group of S) acts on the space M of representations whose restriction to each boundary component remains in a fixed conjugacy class C_{k} in SL(2, ℂ). In this way the complement U^{∗}\D identifies with a quotient of an open subset of M by H(3).

For fixed conjugacy classes C_{k} ∈ SL(2, ℂ), the space of equivalences classes of representations of π_{1}(S) taking π_{1}(b_{k}) to C_{k} has a natural symplectic structure. This structure induces a symplectic structure on M, which is shown to be equivalent to the Poisson structure on U^{∗} arising from the Wronskian constructed previously.

The paper concludes with some speculative remarks concerning degeneration. The projective structure on M gives, indirectly, an extension of the space of flat connections to a Poisson manifold U^{∗} containing rational curves and the author proposes a more direct construction using flat connections.