Moduli space of connections on rational irregular curves

1. Problem Statement

The paper addresses the construction of a compactification of the moduli space of rank-two meromorphic connections on the Riemann sphere ($\mathbb{P}^1$) with a specific polar divisor configuration: one double pole and two simple poles. These connections are referred to as Painlevé V (PV) type connections (or confluent Heun equations).

While the moduli space for generic parameters is well-understood (related to the Okamoto spaces of initial conditions for the Painlevé V equation), a complete algebraic compactification that includes the boundary cases (where singularities coalesce or parameters degenerate) was lacking. Specifically, the author aims to:

1. Construct a compactified moduli space $\mathfrak{Con}^V_\Theta$ that extends the known Okamoto spaces.
2. Describe the geometry of the fibers over the boundary points ($t=0$ and $t=\infty$) of the time parameter.
3. Define the concept of irregular stable nodal curves to rigorously describe the degeneration of the underlying geometric data.

2. Methodology

The author employs a combination of algebraic geometry, the theory of isomonodromic deformations, and birational geometry. The methodology proceeds in three main stages:

A. Normal Form and Parameter Identification

Using the normal form established by Diarra and Loray, the author identifies that any PV connection (up to meromorphic gauge equivalence) is uniquely determined by three complex parameters:

• $q$: The location of an "apparent singularity" (a point where the connection matrix has integer eigenvalues but trivial monodromy).
• $t$: A parameter associated with the irregular singularity (acting as the time variable in the Painlevé V equation).
• $\hat{p}$: A parameter related to the eigenvector of the residual matrix at the apparent singularity.

The author interprets these parameters geometrically:

• $q$ and $t$ define a rational irregular curve (a sphere with four marked points and a marked tangent vector at one point).
• $\hat{p}$ acts as a fiber coordinate.
  Thus, the moduli space is viewed as a family of line bundles over the moduli space of these irregular curves.

B. Compactification of the Base (Irregular Curves)

To compactify the moduli space, the author first compactifies the base space $\mathcal{M}$ (the space of rational irregular curves).

• Inspiration: The technique follows Kapranov's compactification of $\mathcal{M}_{0,n}$ using marked conics in $\mathbb{P}^2$.
• Construction: The author establishes an isomorphism between $\mathcal{M}$ and the moduli space of smooth marked conics in $\mathbb{P}^2$ passing through three fixed points with a prescribed tangent direction.
• Result: The compactification $\overline{\mathcal{M}}$ is obtained by contracting a specific $(-2)$-curve in a weak Del Pezzo surface of degree 5. This results in a surface with an $A_1$-type singularity.

C. Extension of the Line Bundle and Boundary Analysis

The author studies how the trivial line bundle $\mathcal{M} \times \mathbb{C}$ (parameterized by $\hat{p}$) extends over the compactified base $\overline{\mathcal{M}}$.

• Divisor Calculation: By analyzing the behavior of the section $\hat{p}=1$ near the boundary components (where $q \to 0, 1, \infty$ or $t \to 0, \infty$), the author determines the divisor $D$ required to extend the bundle.
• Irregular Stable Nodal Curves: Inspired by Deligne-Mumford, the author defines "irregular stable nodal curves" to describe the limit curves when parameters degenerate. These are curves with two smooth components connected at a node, encoding the confluence of singularities.
• Connection Limits: The paper explicitly computes the limit of the connection matrix as parameters approach the boundary. It is shown that for the limit to exist, the point $(q, \hat{p})$ must follow specific invariant trajectories (Taylor expansions) of the Riccati foliation.
• Blow-ups: The compactification requires blowing up specific sections in the boundary divisors to resolve the connections that appear only on specific sub-loci (e.g., specific slopes of convergence).

3. Key Contributions and Results

Theorem A: Compactification of the Base

The compactification $\overline{\mathcal{M}}$ of the moduli space of rational irregular curves is isomorphic to the singular surface obtained by contracting the unique $(-2)$-curve in a weak Del Pezzo surface of degree 5. The resulting singularity is of $A_1$-type.

Theorem B: Extension of the Line Bundle

The trivial line bundle over the open set extends to the smooth resolution $\widetilde{\mathcal{M}}$ as the line bundle $\mathcal{O}_{\widetilde{\mathcal{M}}}(D)$, where the divisor $D$ is given by:
$$D = A_0 + 2Q_1 + 2B^1_0 - B^0_\infty - B^\infty_\infty$$
Here, $A_i, Q_j, B^i_j$ represent specific boundary components corresponding to the degeneration of parameters $t$ and $q$.

Theorem C: Construction of the Compactified Moduli Space

The compactified moduli space $\mathfrak{Con}^V_\Theta$ is constructed as a specific open subset of a projective bundle over the blown-up space. Specifically, it is obtained by:

1. Taking the projective bundle $X = \mathbb{P}(\mathcal{O} \oplus \mathcal{O}_{\widetilde{\mathcal{M}}}(D))$.
2. Blowing up special sections in the boundary components $\mathcal{Q}_i$.
3. Removing the strict transforms of the boundary divisors and the infinity section.

Theorem D: Geometry of the Fibers

The space $\mathfrak{Con}^V_\Theta$ admits a fibration $\pi: \mathfrak{Con}^V_\Theta \to \mathbb{P}^1$ (parameterized by $t$) with the following fiber structure:

• Generic Fiber ($t \in \mathbb{C}^*$): The fiber $\pi^{-1}(t)$ is an Okamoto space, which is the 8-point blow-up of the second Hirzebruch surface $\mathbb{F}_2$. This recovers the known initial condition spaces for Painlevé V.
• Boundary Fiber ($t=0$): The fiber is the union of two surfaces: $\mathcal{A}_0 \cong \mathbb{F}_1$ and $\mathcal{B}^1_0$ (a double blow-up of $\mathbb{F}_1$).
• Boundary Fiber ($t=\infty$): The fiber is a union of four surfaces: $\mathcal{B}^0_\infty, \mathcal{B}^\infty_\infty$ (both double blow-ups of $\mathbb{F}_1$) and $\mathcal{F}^\pm_\infty$ (blow-ups of specific sections).

Symmetries

The paper identifies a group of order 16 acting on the moduli space, generated by involutions that exchange the residual spectral data ($\kappa^+ \leftrightarrow \kappa^-$) and swap the roles of the singularities (e.g., $0 \leftrightarrow \infty$). These symmetries are well-defined on the compactified space.

4. Significance

• Algebraic Framework for Painlevé V: The work provides a rigorous algebraic geometric framework for the Painlevé V equation, extending the analytic results of Okamoto and the symplectic results of Boalch.
• Boundary Geometry: It explicitly describes the geometry of the moduli space at the "infinity" of the parameter space, which is crucial for understanding the global behavior of isomonodromic deformations and the Riemann-Hilbert correspondence.
• Irregular Stable Curves: The introduction of "irregular stable nodal curves" generalizes the Deligne-Mumford compactification to the context of irregular singularities, offering a new tool for studying degenerating connections.
• Future Applications: The author notes that this compactification serves as a foundation for studying the isomonodromic foliation on the boundary components, which is essential for understanding the global monodromy and Stokes phenomena of the system.

In summary, Morbello successfully constructs a quasi-projective variety that compactifies the moduli space of PV connections, bridging the gap between the smooth Okamoto spaces and their singular limits using advanced techniques in birational geometry and the theory of irregular connections.