Deformation of moduli spaces of meromorphic $G$-connections on $\mathbb{P}^{1}$ via unfolding of irregular singularities
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866908671745720320 |
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| author | Hiroe, Kazuki |
| author_facet | Hiroe, Kazuki |
| contents | Unfolding singular points in linear differential equations is a classical technique for studying the properties of irregular singularities by relating them to regular singularities. In this paper, we propose a general framework for unfolding unramified irregular singularities of meromorphic connections on the trivial principal $G$-bundle over $\mathbb{P}^{1}$.
One of our main results is the description of the unfolding of singularities in terms of deformations of their moduli spaces. We show that every moduli space of irreducible meromorphic $G$-connections with unramified irregular singularities on $\mathbb{P}^{1}$ can be deformed into a moduli space of irreducible Fuchsian $G$-connections on $\mathbb{P}^{1}$.
Furthermore, we study the unfolding of additive Deligne-Simpson problems, in which the unfolding of irregular singularities naturally generates a family of such problems. As an application of our main result, we prove that a Deligne-Simpson problem for $G$-connections with unramified irregular singularities admits a solution if and only if every unfolded Deligne-Simpson problem in the family admits a simultaneous solution.
We also provide a combinatorial and diagrammatic framework of the unfolding process in terms of spectral types and unfolding diagrams. Finally, we address a conjecture proposed by Oshima concerning the existence of irreducible $G$-connections that realize prescribed spectral types and their unfoldings. Our main result gives an affirmative answer to this conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_20486 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Deformation of moduli spaces of meromorphic $G$-connections on $\mathbb{P}^{1}$ via unfolding of irregular singularities Hiroe, Kazuki Algebraic Geometry Classical Analysis and ODEs Representation Theory Unfolding singular points in linear differential equations is a classical technique for studying the properties of irregular singularities by relating them to regular singularities. In this paper, we propose a general framework for unfolding unramified irregular singularities of meromorphic connections on the trivial principal $G$-bundle over $\mathbb{P}^{1}$. One of our main results is the description of the unfolding of singularities in terms of deformations of their moduli spaces. We show that every moduli space of irreducible meromorphic $G$-connections with unramified irregular singularities on $\mathbb{P}^{1}$ can be deformed into a moduli space of irreducible Fuchsian $G$-connections on $\mathbb{P}^{1}$. Furthermore, we study the unfolding of additive Deligne-Simpson problems, in which the unfolding of irregular singularities naturally generates a family of such problems. As an application of our main result, we prove that a Deligne-Simpson problem for $G$-connections with unramified irregular singularities admits a solution if and only if every unfolded Deligne-Simpson problem in the family admits a simultaneous solution. We also provide a combinatorial and diagrammatic framework of the unfolding process in terms of spectral types and unfolding diagrams. Finally, we address a conjecture proposed by Oshima concerning the existence of irreducible $G$-connections that realize prescribed spectral types and their unfoldings. Our main result gives an affirmative answer to this conjecture. |
| title | Deformation of moduli spaces of meromorphic $G$-connections on $\mathbb{P}^{1}$ via unfolding of irregular singularities |
| topic | Algebraic Geometry Classical Analysis and ODEs Representation Theory |
| url | https://arxiv.org/abs/2407.20486 |