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Initial value space of the four dimensional Painlevé system with $(A_5+A_1)^{(1)}$ symmetry

Kazuya Matsugashita, Takao Suzuki

TL;DR

This work extends the Okamoto program to a four-dimensional Painlevé-type system with $(A_5+A_1)^{(1)}$ symmetry by constructing a holomorphic, symplectic initial value space. The authors build a Hirzebruch-type compactification $\,\Sigma_ta$ and identify accessible singularities, then resolve them through a sequence of blow-ups (for $C^0$) and formal power-series methods (for $P^0$), aided by the system’s birational symmetries. The resolution yields a smooth initial value space $E$, assembled from 33 coordinate charts, on which the dynamics is governed by a unique degree-5 polynomial Hamiltonian in $(q_1,q_2,p_1,p_2)$. This provides a geometric IVS for the four-dimensional Painlevé system, consistent with Sakai’s classification and extending known results about holomorphy and symmetries in higher dimensions.

Abstract

The initial value spaces of the Painlevé equations are proposed by Okamoto. They are symplectic manifolds in which the Painlevé equations are described as polynomial Hamiltonian systems on all coordinates. In this article, we construct an initial value space of the four dimensional Painlevé system with affine Weyl group symmetry of type $(A_5+A_1)^{(1)}$.

Initial value space of the four dimensional Painlevé system with $(A_5+A_1)^{(1)}$ symmetry

TL;DR

This work extends the Okamoto program to a four-dimensional Painlevé-type system with symmetry by constructing a holomorphic, symplectic initial value space. The authors build a Hirzebruch-type compactification and identify accessible singularities, then resolve them through a sequence of blow-ups (for ) and formal power-series methods (for ), aided by the system’s birational symmetries. The resolution yields a smooth initial value space , assembled from 33 coordinate charts, on which the dynamics is governed by a unique degree-5 polynomial Hamiltonian in . This provides a geometric IVS for the four-dimensional Painlevé system, consistent with Sakai’s classification and extending known results about holomorphy and symmetries in higher dimensions.

Abstract

The initial value spaces of the Painlevé equations are proposed by Okamoto. They are symplectic manifolds in which the Painlevé equations are described as polynomial Hamiltonian systems on all coordinates. In this article, we construct an initial value space of the four dimensional Painlevé system with affine Weyl group symmetry of type .

Paper Structure

This paper contains 14 sections, 11 theorems, 166 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Initial value space
  4. Four dimensional FST system
  5. Compactification of the initial phase space
  6. Accessible singularities
  7. Resolving the accessible singularities $C^0$
  8. Blowing up
  9. Formal power series
  10. Resolving the accessible singularities $P^0$
  11. Proof of Proposition \ref{['prop:resolve_P']}
  12. Main result
  13. Proof of Lemma \ref{['lem:as_list']}
  14. Proof of Lemma \ref{['lem:C_connect']}

Key Result

Proposition 3.2

The system $\mathcal{H}^0$ is described as the polynomial Hamiltonian system on any affine chart $W_{i0}$.

Figures (1)

  • Figure 1: Exceptional divisors

Theorems & Definitions (21)

  • Remark 2.1
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Remark 3.4
  • Proposition 3.5
  • proof
  • Lemma 4.1
  • ...and 11 more