Lagrangian fibration in duality on moduli space of rank two logarithmic connections over the projective line

Frank Loray (1) , Masa-Hiko Saito (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 Kobe - Department of Mathematics, Graduate School of Science

We study the moduli space of logarithmic connections of rank 2 on the Riemann sphere minus n points with fixed spectral data. There are two natural Lagrangian maps: one towards apparent singularities of the associated fuchsian scalar equation, and another one towards moduli of parabolic bundles. We show that these are transversal and dual to each other. In case n=5, we recover the beautiful geometry of Del Pezzo surfaces of degree 4.

Mots clés: Painlevé VI; Garnier systems; Logarithmic connection; Parabolic structure; Projective line; apparent singularity; Okamoto-Painlevé pair; Del Pezzo surface

Domaines: Géométrie algébrique [math.AG]

Frank Loray, Masa-Hiko Saito. Lagrangian fibration in duality on moduli space of rank two logarithmic connections over the projective line. International Mathematics Research Notices, 2015, 4 (4), pp.995-1043. ⟨10.1093/imrn/rnt232⟩. ⟨hal-00789255v2⟩