[Paper Review] Normal forms for rank two linear irregular differential equations and moduli spaces

Karamoko Diarra, Frank Loray|arXiv (Cornell University)|Jul 17, 2019

Nonlinear Waves and Solitons23 references4 citations

TL;DR

This paper establishes a unique normal form for rank two irregular meromorphic connections on the Riemann sphere by introducing apparent singular points and fixing the Birkhoff-Grothendieck decomposition. It identifies an open subset of the moduli space of such connections with an open subset of a Hilbert scheme of points on the blow-up of a Hirzebruch surface, generalizing previous logarithmic results and providing a symplectic structure compatible with isomonodromic systems.

ABSTRACT

We provide a unique normal form for rank two irregular connections on the Riemann sphere.In fact, we provide a birational model where we introduce apparent singular points and where the bundlehas a fixed Birkhoff-Grothendieck decomposition. The essential poles and the apparent poles provide twoparabolic structures. The first one only depend on the formal type of the singular points. The latter one determine the connection (accessory parameters). As a consequence, an open set of the corresponding moduli space of connections is canonically identified with an open set of some Hilbert scheme of points on the explicit blow-up of some Hirzebruch surface. This generalizes to the irregular case a description dueto Oblezin, and Saito-Szabo in the logarithmic case. This approach is also very close to the work of Dubrovin-Mazzocco with the cyclic vector.

Motivation & Objective

• To construct an explicit birational model for the moduli space of rank two irregular connections on P^1.
• To identify the moduli space of such connections with an open subset of a Hilbert scheme of points on a blow-up of a Hirzebruch surface.
• To establish a symplectic structure on the moduli space compatible with isomonodromic deformations.
• To generalize previous logarithmic results of Szabó and Oblezin to the irregular case.
• To provide a canonical normal form that separates formal invariants (essential poles) from accessory parameters (apparent poles).

Proposed method

• Introduce a birational transformation that resolves singularities by adding apparent singular points, fixing the Birkhoff-Grothendieck decomposition of the bundle.
• Define two parabolic structures: one from the formal type (essential poles), one from the connection (apparent poles), which encode the accessory parameters.
• Construct a birational map Φ from the moduli space M_μ^Λ to the Hilbert scheme Hilb^{n−3}(Ω_D), where Ω_D is the total space of the line bundle Ω^1_{P^1}(D).
• Use the Liouville symplectic form on the cotangent bundle Ω_0 to pull back a symplectic structure to an open set of Hilb^{n−3}(Ω_D), and show it pulls back to the natural symplectic form on M_μ^Λ.
• Apply a companion matrix normalization to transform the connection into a scalar second-order differential equation, enabling comparison with known symplectic forms in the literature.
• Verify that the symplectic form ω = ∑ dp_j ∧ dq_j on the moduli space matches known forms in the literature (e.g., [12], [4], [20]) via explicit coordinate comparison.

Experimental results

Research questions

• RQ1 Can a unique normal form be constructed for rank two irregular connections on P^1 that separates formal invariants from accessory parameters?
• RQ2 Is the moduli space of such connections birationally equivalent to a Hilbert scheme of points on a blow-up of a Hirzebruch surface?
• RQ3 Does the symplectic structure on the moduli space match the pullback of the Liouville form via the birational map Φ?
• RQ4 How does the normal form relate to known scalar equations and symplectic structures in the logarithmic case?
• RQ5 Can the normal form be used to construct isomonodromic Hamiltonian systems, as confirmed by Komyo’s subsequent work?

Key findings

• The moduli space M_μ^Λ of μ-stable Λ-connections admits a birational map Φ to the Hilbert scheme Hilb^{n−3}(Ω_D), where n = deg(D).
• The symplectic structure on M_μ^Λ pulls back via Φ to the symplectic form ∑_{j=1}^{n−3} dp_j ∧ dq_j, matching known forms in the literature.
• The map Φ is symplectic, as confirmed by Komyo [13], and verified explicitly in the Fuchsian case and up to 5 poles in the irregular case.
• The normal form separates the formal data (essential poles) from the accessory parameters (apparent poles), with the latter determining the connection uniquely.
• The construction generalizes previous results of Szabó and Oblezin from the logarithmic to the irregular setting.
• The scalar differential equation associated with the normal form matches known forms in Okamoto and Kimura’s work, confirming consistency of the symplectic structure.