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

TL;DR

This paper explores the duality and Lagrangian fibrations in the moduli space of rank 2 logarithmic connections on the Riemann sphere, revealing geometric structures and dualities, especially for five punctures.

Contribution

It demonstrates the transversality and duality of two natural Lagrangian maps in the moduli space of logarithmic connections, connecting to Del Pezzo surface geometry.

Findings

Lagrangian maps are transversal and dual

Connection to Del Pezzo surfaces for five points

Geometric structures of moduli space elucidated

Abstract

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.