Description of generalized isomonodromic deformations of rank two linear differential equations using apparent singularities

Komyo, Arata

doi:10.48550/arxiv.2003.08045

Cited by 3 publications

(8 citation statements)

References 17 publications

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Section: Background and Motivationmentioning

confidence: 99%

“…On the other hand, there is a study of explicit description of isomonodromic deformations of second order linear differential equation with irregular singularities by using apparent singularities in [32]. So the purpose of this paper is to give the explicit irregular Garnier system corresponding to the remained algebraic solution by using the study in [32]. In consequence, we will give an explicit form of the algebraic solution by using the fixed system on P 1 and the algebraic family of ramified covers P 1 → P 1 which are found by Diarra-Loray [14, the first line of Table 2].…”

Section: Background and Motivationmentioning

confidence: 99%

“…When we fix the time variables (that is, local formal data), we call the moduli space the moduli space with fixed time variables. When the time variables may vary, we call the moduli space the extended moduli space (for details, for example, see [32,Section 2.4]). The isomonodromic deformations mean vector fields on the extended moduli space.…”

Section: The Irregular Garnier Systemmentioning

confidence: 99%

“…We will use the coordinate system given by apparent singularities (for details, see [15, Section 1] and [32, Section 2]). By using the coordinate system, we have a birational map from this moduli space with fixed time variables to Sym 3 (C 2 ) (see [15,Theorem 1.1] and [32,Section 2.4]). Here Sym 3 (C 2 ) is the 3-fold symmetric product of C 2 .…”

Section: The Irregular Garnier Systemmentioning

confidence: 99%

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A nonclassical algebraic solution of a 3-variable irregular Garnier system

Komyo¹

2022

Preprint

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In this paper, a nonclassical algebraic solution of a 3-variable irregular Garnier system is constructed. Diarra-Loray have studied classification of algebraic solutions of irregular Garnier systems. There are two type of the algebraic solutions: classical type and pull-back type. They have shown that there are exactly three nonclassical algebraic solutions for N -variables irregular Garnier systems with N > 1. Explicit forms of two of the three solutions are already given. The solution constructed in the present paper is the remained algebraic solution.

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Section: Background and Motivationmentioning

confidence: 99%

Section: Background and Motivationmentioning

confidence: 99%

Section: The Irregular Garnier Systemmentioning

confidence: 99%

Section: The Irregular Garnier Systemmentioning

confidence: 99%

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A nonclassical algebraic solution of a 3-variable irregular Garnier system

Komyo¹

2022

Preprint

Self Cite

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“…Such a process is called a shearing transformation method in a certain context. Its explicit description is given by K. Diara, F. Loray and A. Komyo in [9] and [17] for rank 2 ramified connections on P 1 . On the other hand, we give a brief idea of producing the Stokes data corresponding to the unramified connection on the local analytic ramified cover.…”

Section: Introductionmentioning

confidence: 99%

Moduli space of factorized ramified connections and generalized isomonodromic deformation

Inaba¹

2021

Preprint

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We introduce the notion of factorized ramified structure on a generic ramified connection on a smooth projective curve. We construct a canonical 2-form on the moduli space of ramified connections by the deformation theory of connections with factorized ramified structure. Since the factorized ramified structure provides a duality on the tangent space of the moduli space, the 2-from becomes nondegenerate. We prove that the 2-form on the moduli space of ramified connections is d-closed via constructing an unfolding of the moduli space. Based on the Stokes data, we introduce the notion of local generalized isomonodromic deformation for generic unramified connections on a unit disk. Applying the Jimbo-Miwa-Ueno theory to generic unramified connections, the local generalized isomonodromic deformation is equivalent to the extendability of the family of connections to an integrable connection. We give the same statement for ramified connections. Based on this principle of Jimbo-Miwa-Ueno theory, we construct a global generalized isomonodromic deformation on the moduli space of generic ramified connections by constructing a horizontal lift of a universal family of connections.

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Moduli Space of Factorized Ramified Connections and Generalized Isomonodromic Deformation

Inaba¹

2023

SIGMA

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We introduce the notion of factorized ramified structure on a generic ramified irregular singular connection on a smooth projective curve. By using the deformation theory of connections with factorized ramified structure, we construct a canonical 2-form on the moduli space of ramified connections. Since the factorized ramified structure provides a duality on the tangent space of the moduli space, the 2-form becomes nondegenerate. We prove that the 2-form on the moduli space of ramified connections is d -closed via constructing an unfolding of the moduli space. Based on the Stokes data, we introduce the notion of local generalized isomonodromic deformation for generic unramified irregular singular connections on a unit disk. Applying the Jimbo-Miwa-Ueno theory to generic unramified connections, the local generalized isomonodromic deformationis equivalent to the extendability of the family of connections to an integrable connection. We give the same statement for ramified connections. Based on this principle of Jimbo-Miwa-Ueno theory, we construct a global generalized isomonodromic deformation on the moduli space of generic ramified connections by constructing a horizontal lift of a universal family of connections. As a consequence of the global generalized isomonodromic deformation, we can lift the relative symplectic form on the moduli space to a total closed form, which is called a generalized isomonodromic 2-form.

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Description of generalized isomonodromic deformations of rank two linear differential equations using apparent singularities

Cited by 3 publications

References 17 publications

A nonclassical algebraic solution of a 3-variable irregular Garnier system

A nonclassical algebraic solution of a 3-variable irregular Garnier system

Moduli space of factorized ramified connections and generalized isomonodromic deformation

Moduli Space of Factorized Ramified Connections and Generalized Isomonodromic Deformation

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