2007/03/08 by Szilárd Szabó, Szabo, Szilard
Mathematics · #14H60 #32G34 #34M35 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.math/0703230
openalex publication_date 2007/03/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 etale 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.