Hyperbolic 2-spheres with conical singularities, accessory parameters and Kähler metrics on M_0,n

We show that the real-valued function S_α on the moduli space M_0,n of pointed rational curves, defined as the critical value of the Liouville action functional on a hyperbolic 2-sphere with n ≥ 3 conical singularities of arbitrary orders α = {α₁,..., α_n}, generates accessory parameters of the associated Fuchsian differential equation as their common antiderivative. We introduce a family of Kähler metrics on M_0,n parameterized by the set of orders a, explicitly relate accessory parameters to these metrics, and prove that the functions S_α are their Kähler potentials.