Moduli spaces for linear differential equations and the Painlevé equations

Put, Marius van der; Saito, Misato

doi:10.5802/aif.2502

Cited by 78 publications

(108 citation statements)

References 24 publications

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“…As explained in [13], [20], and [21], in order to construct the moduli space of generalized monodromy data and define the Riemann-Hilbert correspondence, we need to fix a formal type of the parabolic connection (E, ∇, {l (i) j }) at each irregular or regular singular point t i . However the counterexample in Remark 1.2 shows that for a special ν, one cannot determine the formal type of a connection (E, ∇, {l (i) j }) ∈ M α D/C (r, d, (m i )) ν , that is, the reductions up to the order m i are not enough to determine the formal type for a special ν.…”

Section: By Proposition 11 There Exists a Filtration By C[[z I ]]-Smentioning

confidence: 99%

Moduli of unramified irregular singular parabolic connections on a smooth projective curve

Inaba¹,

Saito²

2013

Kyoto J. Math.

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In this paper we construct a coarse moduli scheme of stable unramified irregular singular parabolic connections on a smooth projective curve and prove that the constructed moduli space is smooth and has a symplectic structure. Moreover, we will construct the moduli space of generalized monodromy data coming from topological monodromies, formal monodromies, links, and Stokes data associated to the generic irregular connections. We will prove that for a generic choice of generalized local exponents, the generalized Riemann-Hilbert correspondence from the moduli space of the connections to the moduli space of the associated generalized monodromy data gives an analytic isomorphism. This shows that differential systems arising from (generalized) isomonodromic deformations of corresponding unramified irregular singular parabolic connections admit the geometric Painlevé property as in the regular singular cases proved generally. IntroductionLet m, l be positive integers, and let ν be an element of C dw/w lm−l+1 + · · · + C dw/w. We denote the C [[w]]-module C[[w]] ⊕r with the connectionHere e 1 , . . . , e r is the canonical basis of C[[w]] ⊕r and e 0 = 0.We have the following fundamental theorem. THEOREM 0.1 (HUKUHARA AND TURRITTIN) Let V be a free C[[z]]-module of rank r, and let ∇ : V → V ⊗ dz/z m be a connection. Then there are positive integers l, s, r 1 , .. . , r s such that for a variable w which is the family of moduli spaces of α-stable parabolic connections over a certain space T •,s νres of parameters including generic, simple exponents ν with the fixed residue part ν res (see (26)). The isomonodromic flows define an isomonodromic foliation or an isomonodromic differential system on the phase space, and its geometric Painlevé property follows easily from the definition based on Theorem 5.1. The geometric Painlevé property gives a complete and clear proof of the analytic Painlevé property for the isomonodromic differential systems with nonresonant and irreducible exponents ν res or p = e(ν res ).As explained in [9], it is important to construct the fibers of the phase space of the isomonodromic differential system as smooth algebraic schemes. One can use affine algebraic coordinates of the fibers over an open set of parameter spaces to write down the differential systems explicitly. Then the differential systems satisfy the analytic Painlevé property, which easily follows from the geometric Painlevé property.We should mention that Malgrange [14] and [15] and Miwa [16] gave proofs of the analytic Painlevé property for isomonodromic differential systems for irregular connections on P 1 . However, to give a complete proof of the geometric Painlevé property, we believe that our algebro-geometric construction of the family of the moduli spaces of connections is indispensable (see also [10], [11], [8] for the regular singular cases).Bremer and Sage [2] studied the moduli space of irregular singular connections on P 1 . They considered also the ramified case. However, they assumed that the bundle V is trivial, whi...

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Section: By Proposition 11 There Exists a Filtration By C[[z I ]]-Smentioning

confidence: 99%

Moduli of unramified irregular singular parabolic connections on a smooth projective curve

Inaba¹,

Saito²

2013

Kyoto J. Math.

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“…As shown by [20,Section 3], for each (θ 0 , θ x , θ ∞ ) ∈ C 3 , M 0 is identified with an affine cubic surface given by…”

Section: Almost-completeness Of Our Critical Behavioursmentioning

confidence: 99%

“…Although our definition of the Stokes multipliers is slightly different from that of [20], the equation of M 0 is essentially the same. If…”

Section: Almost-completeness Of Our Critical Behavioursmentioning

confidence: 99%

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Critical Behaviours of the Fifth Painlevé Transcendents and the Monodromy Data

Shimomura

2017

Kyushu J. Math.

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Abstract. For the fifth Painlevé equation, we present families of convergent series solutions near the origin and the corresponding monodromy data for the associated isomonodromy linear system. These solutions are of complex power type, of inverse logarithmic type and of Taylor series type. For generic parameters the total set of these critical behaviours is almost complete. For the complex power type of solutions in the generic case, we clarify the structure of the analytic continuation on the universal covering around the origin, and examine the distribution of zeros, poles and 1-points. It is shown that two kinds of spiral domains including a sector as a special case are alternately arrayed; the domains of one kind contain sequences both of zeros and of poles, and those of the other kind sequences of 1-points.

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“…As one expects, the reducibility of the monodromy is involved in the solubility, but more unexpectedly, ramification (or rather its absence) also is. Van der Put and Saito [2009] have already provided a counterexample to this version of the problem. However, it is not clear whether a bundle with minimal Poincaré ranks provides the most natural generalization of the logarithmic bundles required in the regular case.…”

Section: Introductionmentioning

confidence: 99%

Exponents of a meromorphic connection on a compact Riemann surface

Corel¹

2009

Pacific J. Math.

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We give a general definition of the exponents of a meromorphic connection ∇ on a holomorphic vector bundle Ᏹ of rank n over a compact Riemann surface X. We prove that they can be computed as invariants of a vector bundle Ᏹ L canonically attached to Ᏹ, which we construct and call the Levelt bundle of Ᏹ, and whose degree (equal to the sum of the exponents) we estimate by upper and lower bounds (Fuchs' relations). We use this definition to construct, for every linear differential equation on a compact Riemann surface (with regular or irregular singularities), the companion bundle of the equation, a vector bundle endowed with a meromorphic connection that is equivalent to the given equation and has precisely the same singularities and the same set of exponents.

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Moduli spaces for linear differential equations and the Painlevé equations

Cited by 78 publications

References 24 publications

Moduli of unramified irregular singular parabolic connections on a smooth projective curve

Moduli of unramified irregular singular parabolic connections on a smooth projective curve

Critical Behaviours of the Fifth Painlevé Transcendents and the Monodromy Data

Exponents of a meromorphic connection on a compact Riemann surface

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