Canonical coordinates for moduli spaces of rank two irregular connections on curves

TL;DR

This paper introduces a geometric method to identify canonical coordinates on moduli spaces of rank 2 irregular connections on curves, enabling separation of variables and explicit symplectic structures.

Contribution

It provides a new geometric framework for canonical coordinates on moduli spaces of rank 2 connections, including explicit formulas and the case of elliptic curves.

Findings

Established a symplectic map to Hilbert schemes of points

Derived Darboux coordinates for moduli spaces

Explicit symplectic structure for SL2-connections on Matsumoto's model

Abstract

In this paper, we study a geometric counterpart of the cyclic vector which allow us to put a rank 2 meromorphic connection on a curve into a ``companion'' normal form. This allow us to naturally identify an open set of the moduli space of $\mathrm{GL}_2$-connections (with fixed generic spectral data, i.e. unramified, non resonant) with some Hilbert scheme of points on the twisted cotangent bundle of the curve. We prove that this map is symplectic, therefore providing Darboux (or canonical) coordinates on the moduli space, i.e. separation of variables. On the other hand, for $\mathrm{SL}_2$-connections, we give an explicit formula for the symplectic structure for a birational model given by Matsumoto. We finally detail the case of an elliptic curve with a divisor of degree $2$.