Moduli spaces for linear differential equations and the Painlevé equations

Details Moduli spaces for linear differential equations and the Painlevé equations Moduli spaces for linear differential equations and the Painlevé equations

2009-02-10

arxiv.org/abs/0902.1702v5

Mathematics

Algebraic Geometry

43 pages, (The signs for Hamiltonian systems in Section 4 and the equations (3)(PV) and (7)(degenerate PV) are corrected.)

Scientific paper

In this paper, we give a systematic construction of ten isomonodromic families of connections of rank two on $\mathbb{P}^1$ inducing Painlev\'e equations. The classification of ten families is given by considering the Riemann-Hilbert morphism from a moduli space of connections with certain type of regular and irregular singularities to a corresponding catetorical moduli space of analytic data (i.e., ordinary monodromy, Stokes matrices and links), which is called the monodromy space. Explicit equations of the monodromy spaces for ten families are calculated. Moreover, we obtain natural explicit families of connections for these ten cases and calculate isomonodromic equations which give Painlev\'e equations of all types.

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