ar X iv : m at h / 07 03 23 0 v 3 [ m at h . A G ] 1 6 Ja n 20 08 DEFORMATIONS OF FUCHSIAN EQUATIONS AND LOGARITHMIC CONNECTIONS

— We give a geometric proof to the classical fact that the dimension of the deformations of a given generic Fuchsian equation without changing the semi-simple conjugacy class of its local monodromies (“number of accessory parameters”) is equal to half the dimension of the moduli space of deformations of the associated local system. We do this by constructing a weight 1 Hodge structure on the infinitesimal deformations of logarithmic connections, such that deformations as an equation correspond to the (1, 0)-part. This answers a question of Nicholas Katz, who noticed the dimension doubling mentioned above. We then show that the Hitchin map restricted to deformations of the Fuchsian equation is a one-to-one étale map. Finally, we give a positive answer to a conjecture of Ohtsuki about the maximal number of apparent singularities for a Fuchsian equation with given semisimple monodromy, and define a Lagrangian foliation of the moduli space of connections whose leaves consist of logarithmic connections that can be realised as Fuchsian equations having apparent singularities in a prescribed finite set.