Hamiltonian structures of isomonodromic deformations on moduli spaces of parabolic connections

Komyo, Arata

doi:10.48550/arxiv.1611.03601

Cited by 3 publications

(5 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For the proof of Proposition 3.2 the reader is referred to [Ko,Proposition 3.8] (Proposition 3.4 of the arxiv version of [Ko]), [Ch2, p. 1417, Proposition 5.1], [Ch1], [In] and [BS].…”

Section: Letmentioning

confidence: 99%

On the monodromy map for the logarithmic differential systems

Aprodu¹,

Biswas²,

Dumitrescu³

et al. 2022

Preprint

View full text Add to dashboard Cite

We study the monodromy map for logarithmic g-differential systems over an oriented surface S 0 of genus g, with g being the Lie algebra of a complex reductive affine algebraic group G. These logarithmic g-differential systems are triples of the form (X, D, Φ), where (X, D) ∈ T g,d is an element of the Teichmüller space of complex structures on S 0 with d ≥ 1 ordered marked points D ⊂ S 0 = X and Φ is a logarithmic connection on the trivial holomorphic principal G-bundle X × G over X whose polar part is contained in the divisor D. We prove that the monodromy map from the space of logarithmic g-differential systems to the character variety of G-representations of the fundamental group of S 0 \ D is an immersion at the generic point, in the following two cases:(1) g ≥ 2, d ≥ 1, and dim C G ≥ d + 2;(2) g = 1 and dim C G ≥ d. The above monodromy map is nowhere an immersion in the following two cases:(1) g = 0 and d ≥ 4;(2) g ≥ 1 and dim C G < d+3g−3 g . This extends to the logarithmic case the main results in [CDHL], [BD] dealing with nonsingular holomorphic g-differential systems (which corresponds to the case of d = 0).

show abstract

“…For the proof of Proposition 3.2 the reader is referred to [Ko,Proposition 3.8] (Proposition 3.4 of the arxiv version of [Ko]), [Ch2, p. 1417, Proposition 5.1], [Ch1], [In] and [BS].…”

Section: Letmentioning

confidence: 99%

On the monodromy map for the logarithmic differential systems

Aprodu¹,

Biswas²,

Dumitrescu³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…In this section, we study the moduli space of parabolic connections with a quadratic differential, which is generalization of the moduli space of parabolic connections studied by Inaba-Iwasaki-Saito [16] and Inaba [15]. In Section 3.2, we describe the (algebraic) tangent sheaf of this moduli space in terms of the hypercohomology of a certain complex by generalization of the description of the tangent sheaf of the moduli space of parabolic connections in [15,16,18]. Moreover, we describe the analytic tangent sheaf in terms of the hypercohomology of a certain analytic complex as in [15,Section 7].…”

Section: Moduli Scheme Of Parabolic Connections With a Quadratic Diff...mentioning

confidence: 99%

“…This description is more simple than the algebraic one. In Section 3.3, we recall the description of the vector fields associated to the isomonodromic deformations in terms of the description of the (algebraic) tangent sheaf as in [14, Section 6] and [18,Section 3.3]. In Section 3.4, we show that the moduli space of parabolic connections with a quadratic differential is endowed with a symplectic structures.…”

Section: Moduli Scheme Of Parabolic Connections With a Quadratic Diff...mentioning

confidence: 99%

See 1 more Smart Citation

The Moduli Spaces of Parabolic Connections with a Quadratic Differential and Isomonodromic Deformations

Komyo¹

2018

SIGMA

Self Cite

View full text Add to dashboard Cite

In this paper, we study the moduli spaces of parabolic connections with a quadratic differential. We endow these moduli spaces with symplectic structures by using the fundamental 2-forms on the moduli spaces of parabolic connections (which are phase spaces of isomonodromic deformation systems). Moreover, we see that the moduli spaces of parabolic connections with a quadratic differential are equipped with structures of twisted cotangent bundles.

show abstract

“…The formulation of isomonodromic deformation in a higher genus case requires an appropriate setting of the moduli space of connections, which is done in the work with K. Iwasaki and M.-H. Saito in [10] and in [11]. A cohomological description of the isomonodromic deformation on the moduli space is also established by I. Biswas, V. Heu, J. Hurtubise and A. Komyo in [2], [3] and [16]. Conceptually, the isomonodromic deformation is obtained by pulling back, via the Riemann-Hilbert morphism, the local trivial foliation on the family of character varieties.…”

Section: Introductionmentioning

confidence: 99%

Moduli space of factorized ramified connections and generalized isomonodromic deformation

Inaba¹

2021

Preprint

View full text Add to dashboard Cite

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.

show abstract

Hamiltonian structures of isomonodromic deformations on moduli spaces of parabolic connections

Abstract: In this paper, we treat moduli spaces of parabolic connections. We take affine open coverings of the moduli spaces, and we construct a Hamiltonian structure of an algebraic vector field determined by the isomonodromic deformation on each affine open set of the coverings.

Cited by 3 publications

References 21 publications

On the monodromy map for the logarithmic differential systems

On the monodromy map for the logarithmic differential systems

The Moduli Spaces of Parabolic Connections with a Quadratic Differential and Isomonodromic Deformations

Moduli space of factorized ramified connections and generalized isomonodromic deformation

Contact Info

Product

Resources

About