arXiv subjects

Shunya Adachi

Publications and source records attributed to Shunya Adachi.

5 recordsLinked to original sources

arXiv2026

On the Riemann-Hilbert problem for hyperplane arrangements with a good line

We study a variant of the Riemann-Hilbert problem on the complements of hyperplane arrangements. This problem asks whether a given local system on the complement can be realized as the solution sheaf of a logarithmic Pfaffian system with constant coefficients. In this paper, we generalize Katz's middle convolution as a functor for local systems on hyperplane complements and show that it preserves the solvability of this problem.

math.AG↗

arXiv2025

Middle Laplace transform and middle convolution for linear Pfaffian systems with irregular singularities

We introduce a transformation of linear Pfaffian systems, which we call the middle Laplace transform, as a formulation of the Laplace transform from the perspective of Katz theory. While the definition of the middle Laplace transform is purely algebraic, its categorical interpretation is also provided. We then show the fundamental properties (invertibility, irreducibility) of the middle Laplace transform. As an application of the middle Laplace transform, we define the middle convolution for linear Pfaffian systems with irregular singularities. This gives a generalization of Haraoka's middle convolution, which was defined for linear Pfaffian systems with logarithmic singularities. The fundamental properties (additivity, irreducibility) of the middle convolution follow from the properties of the middle Laplace transform. Some examples related to hypergeometric functions with two variables are also given.

math.CA↗

arXiv2023

Unitary monodromies of rank two Fuchsian systems with $(n+1)$ singularities

We study the unitarity of monodromies of rank two Fuchsian systems of SL type with $(n+1)$ regular singularities on the Riemann sphere, namely, we give a sufficient and necessary condition for the monodromy group to be conjugate to a subgroup of a special unitary group $\mathrm{SU}(p,q)$. When $n\ge 3$, the moduli space of irreducible monodromies can be realized as an affine algebraic set in $\mathbb{C}^m$ for some $m \in \mathbb{N}$. In this paper, we give a characterization and construction of unitary monodromies in terms of this affine algebraic set. The signatures of unitary monodromies are also classified.

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arXiv2020

On a Connection Problem for the Generalized Hypergeometric Equation

We study a connection problem between the fundamental systems of solutions at singular points $0$ and $1$ for the generalized hypergeometric equation which is satisfied by the generalized hypergeometric series ${}_nF_{n-1}$. In general, the local solution space around $x=1$ consists of one dimensional singular solution space and $n-1$ dimensional holomorphic solution space. Therefore in the case of $n\ge3$, the expression of connection matrix depends on the choice of the fundamental system of solutions at $x=1$. On the connection problem for ordinary differential equations, Schäfke and Schmidt (LNM 810, Springer, 1980) gave an impressive idea which focuses on the series expansion of fundamental system of solutions. We apply their idea to solve the connection problem for the generalized hypergeometric equation and derive the connection matrix.

math.CA↗

arXiv2019

The $q$-Borel Sum of Divergent Basic Hypergeometric Series ${}_rφ_s(a;b;q,x)$

We study the divergent basic hypergeometric series which is a $q$-analog of divergent hypergeometric series. This series formally satisfies the linear $q$-difference equation. In this paper, for that equation, we give an actual solution which admits basic hypergeometric series as a $q$-Gevrey asymptotic expansion. Such an actual solution is obtained by using $q$-Borel summability, which is a $q$-analog of Borel summability. Our result shows a $q$-analog of the Stokes phenomenon. Additionally, we show that letting $q\to1$ in our result gives the Borel sum of classical hypergeometric series. The same problem was already considered by Dreyfus, but we note that our result is remarkably different from his one.

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