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Yasuhiro Wakabayashi

Publications and source records attributed to Yasuhiro Wakabayashi.

At least 19 recordsLinked to original sources
arXiv2026

Complete flags on flat vector bundles in positive characteristic

Let $X$ be a connected, smooth, and projective curve of genus $g$ over an algebraically closed field of characteristic $p >0$. This paper investigates a characteristic-$p$ analogue of a well-known fact concerning flat vector bundles in characteristic $0$. That is to say, we prove that the inequality $g \leq 1$ holds if and only if any flat vector bundle on $X$ admits a complete flag. We also explore a generalization of this result in a broader setting.

math.AG↗
arXiv2026

Projective and affine structures in positive characteristic I: Chern class formulas and Characterizations of projective spaces

This paper aims to develop a theory of projective and affine structures on higher-dimensional varieties in positive characteristic. This theory deals with Frobenius-projective and Frobenius-affine structures, which have been previously investigated in the case where the underlying space is a curve. We first provide a description of such structures in terms of Berthelot's higher-level differential operators. That description leads us to obtain a positive characteristic version of Gunning's formulas, which give necessary conditions on Chern classes for the existence of Frobenius-projective and Frobenius-affine structures, respectively. Finally, we establish some characterizations of projective spaces using Frobenius-projective structures.

math.AG↗
arXiv2026

Genus formulas for dormant modular curves and asymptotic behavior of their function fields

Towers of algebraic function fields over finite fields play a fundamental role in arithmetic geometry and coding theory. Classical examples arising from modular and Drinfeld modular curves exhibit asymptotically good behavior. In this paper, we introduce an analogous construction derived from the moduli spaces of higher-level dormant $\mathrm{PGL}_2$-opers of prescribed radii on $4$-pointed stable curves of genus $0$. These spaces, which we refer to as dormant modular curves, form projective systems under level reduction. Building on previous results in the moduli theory of dormant opers, we establish an explicit formula for computing the genera of these curves. This formula allows us to study the asymptotic behavior of the corresponding towers of function fields and to compare them with the classical modular and Drinfeld modular cases.

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arXiv2026

Duality for dormant opers of classical types B and C

A $\mathfrak{g}$-oper for a simple Lie algebra $\mathfrak{g}$ is a specific type of flat principal bundle on an algebraic curve. When the base field is of prime characteristic $p$, those with vanishing $p$-curvature are called dormant $\mathfrak{g}$-opers, and they form finite and geometrically meaningful moduli spaces. In earlier work, a canonical duality was established between dormant $\mathfrak{sl}_n$-opers and dormant $\mathfrak{sl}_{p-n}$-opers. This duality has provided effective tools for the study of higher-rank cases, as well as for the computation and structural understanding of the associated enumerative invariants. The main result of this paper extends this duality phenomenon to classical Lie algebras of type B and C. More precisely, under the numerical condition $p-1 = 2 (\ell +m)$, we construct a canonical isomorphism between the moduli spaces of dormant $\mathfrak{so}_{2\ell +1}$-opers and dormant and $\mathfrak{sp}_{2m}$-opers with prescribed symmetric radii.

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arXiv2025

Arithmetic liftings and 2d TQFT for dormant opers of higher level

This manuscript represents an advance in the enumerative geometry of opers that takes the subject beyond our previous work. Motivated by a counting problem of linear differential equations in positive characteristic, we investigate the moduli space of opers from arithmetic and combinatorial points of view. We construct a compactified moduli space classifying dormant $\mathrm{PGL}_n^{(N)}$-opers (i.e., dormant $\mathrm{PGL}_n$-opers of level $N$) on pointed stable curves in characteristic $p>0$. One of the key results is the generic étaleness of that space for $n=2$, which is proved by obtaining a detailed understanding of relevant deformation spaces. This fact induces a certain arithmetic lifting of each dormant $\mathrm{PGL}_2^{(N)}$-oper on a general curve to characteristic $p^N$; this lifting is called the canonical diagonal lifting. On the other hand, the generic étaleness also implies that the degree function for the moduli spaces in the rank $2$ case satisfies factorization properties determined by various gluing morphisms of the underlying curves. That is to say, the degree function forms a $2$d TQFT (= a $2$-dimensional topological quantum field theory); it leads us to describe dormant $\mathrm{PGL}_2^{(N)}$-opers in terms of edge numberings on trivalent graphs, as well as lattice points inside generalized rational polytopes. These results yield an effective way of computing the numbers of such objects and $2$nd order differential equations in characteristic $p^N$ with a full set of solutions.

math.AG↗
arXiv2025

Frobenius pull-back of parabolic bundles and dormant opers

We study parabolic bundles on an algebraic curve in positive characteristic. Our motivation is to properly formulate Frobenius pull-backs of parabolic bundles in a way that extends various previous facts and arguments for the usual non-parabolic Frobenius pull-backs. After defining that operation, we generalize a classical result by Cartier concerning Frobenius descent, that is, we establish a bijective correspondence (including the version using higher-level $\mathcal{D}$-modules) between parabolic flat bundles with vanishing $p$-curvature on a pointed curve and parabolic bundles on its Frobenius twist. This correspondence gives a description of maximally Frobenius-destabilized parabolic bundles in terms of dormant opers admitting logarithmic poles. As an application of that description together with a previous result in the enumerative geometry of dormant opers, we obtain an explicit formula for computing the number of such parabolic bundles of rank $2$ under certain assumptions.

math.AG↗
arXiv2025

The irreducibility of the moduli space of pointed stable curves with dormant $\mathrm{PGL}_2^{(N)}$-oper

A $\mathrm{PGL}_n^{(N)}$-oper is a specific type of flat $\mathrm{PGL}_n$-bundle on an algebraic curve in prime characteristic $p$ enhanced by an action of the sheaf of differential operators of level $N-1$. In this paper, we introduce and study a higher-level generalization of the Hitchin-Mochizuki morphism on the moduli space of $\mathrm{PGL}_n^{(N)}$-opers, defined via the characteristic polynomials of their $p^N$-curvatures. As an application, we prove the irreducibility of the moduli space classifying pointed stable curves equipped with dormant $\mathrm{PGL}_2^{(N)}$-opers, i.e., $\mathrm{PGL}_2^{(N)}$-opers with vanishing $p^N$-curvature.

math.AG↗
arXiv2025

Explicit computation of the generic degree of the generalized Verschiebung in rank two

The purpose of this paper is to apply previous work on dormant opers to the study of the moduli space of stable bundles in positive characteristic. We affirmatively resolve the rank $2$ case of a conjecture proposed by the second author, which predicts a direct relationship between the number of higher-level dormant $\mathrm{PGL}_2$-opers and the generic degree of the generalized Verschiebung map for rank $2$ stable bundles induced by Frobenius pull-back. As a consequence, we obtain a procedure for explicitly determining these generic degrees in the previously unexplored range of genera by counting certain combinatorial objects.

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arXiv2025

Generalized hypergeometric equations and $2$d TQFT for dormant opers in characteristic $\leq 7$

This note studies $\mathrm{PGL}_n$-opers arising from generalized hypergeometric differential equations in prime characteristic $p$. We prove that these opers are rigid within the class of dormant opers. By combining this rigidity result with previous work in the enumerative geometry of dormant opers, we obtain a complete and explicit description of the $2$d TQFTs that compute the number of dormant $\mathrm{PGL}_n$-opers for primes $p \leq 7$.

math.AG↗
arXiv2025

The moduli space of dormant opers on elliptic curves

A dormant oper is a specific type of principal bundle with a flat connection, defined on an algebraic curve in positive characteristic. The moduli spaces of dormant opers and their variants, known as dormant Miura opers, have been studied in various contexts. This paper focuses on the case where the underlying spaces are (possibly nodal) elliptic curves and provides a detailed examination of the geometric structures of their moduli spaces. In particular, we explicitly describe dormant (generic Miura) opers in terms of regular elements in an associated Lie algebra and establish the connectedness of these moduli spaces. We also explore generalizations to higher level and prime-power characteristic.

math.AG↗
arXiv2024

Gaudin model modulo $p$, Tango structures, and dormant Miura opers

In the present paper, we study the Bethe ansatz equations for Gaudin model and Miura opers in characteristic $p>0$. Our study is based on a work by E. Frenkel, in which solutions to the Bethe ansatz equations are described in terms of Miura opers on the complex projective line. The main result of the present paper provides a positive characteristic analogue of this description. We pay particular attention to the case of Miura $\mathrm{PGL}_2$-opers because dormant generic Miura $\mathrm{PGL}_2$-opers correspond bijectively to Tango structures, which bring various sorts of exotic phenomena in positive characteristic, e.g., counter-examples to the Kodaira vanishing theorem. As a consequence, we construct new examples of Tango structures by means of solutions to the Bethe ansatz equations modulo $p$.

math.AG↗
arXiv2024

The generic étaleness of the moduli space of dormant $\mathfrak{so}_{2\ell}$-opers

The generic étaleness is an important property on the moduli space of dormant $\mathfrak{g}$-opers (for a simple Lie algebra $\mathfrak{g}$) in the context of enumerative geometry. In the previous study, this property has been verified under the assumption that $\mathfrak{g}$ is either $\mathfrak{sl}_\ell$, $\mathfrak{so}_{2\ell -1}$, or $\mathfrak{sp}_{2\ell}$ for any sufficiently small positive integer $\ell$. The purpose of the present paper is to prove the generic étaleness for one of the remaining cases, i.e., $\mathfrak{g} = \mathfrak{so}_{2\ell}$. As an application of this result, we obtain a factorization formula for computing the generic degree induced from pull-back along various clutching morphisms between moduli spaces of pointed stable curves.

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arXiv2023

Infinitesimal deformations of opers in positive characteristic and the de Rham cohomology of symmetric products

The Eichler-Shimura isomorphism describes a certain cohomology group with coefficients in a space of polynomials by using holomorphic modular/cusp forms. It determines a canonical decomposition of the corresponding de Rham cohomology group associated to a specific oper on a Riemann surface. One purpose of the present paper is to establish its analogue for opers in positive characteristic. We first discuss some basic properties on the (parabolic) de Rham cohomology groups and deformation spaces of $G$-opers (where $G$ is a semisimple algebraic group of adjoint type) in a general formulation. In particular, it is shown that the deformation space of a $G$-oper induced from an $\mathrm{SL}_2$-oper decomposes into a direct sum of the (parabolic) de Rham cohomology groups of its symmetric products. As a consequence, we obtain an Eichler-Shimura-type decomposition for dormant opers on general pointed stable curves by considering a transversal intersection of related spaces in the de Rham moduli space.

math.AG↗
arXiv2023

Holonomic $\mathcal{D}$-modules of arithmetic type and middle convolution

The aim of the present paper is to study arithmetic properties of $\mathcal{D}$-modules on an algebraic variety over the field of algebraic numbers. We first provide a framework for extending a class of $G$-connections (resp., globally nilpotent connections; resp., almost everywhere nilpotent connections) to holonomic $\mathcal{D}$-modules. It is shown that the derived category of $\mathcal{D}$-modules in each of such extended classes carries a Grothendieck six-functor formalism. This fact leads us to obtain the stability of the middle convolution for $G$-connections with respect to the global inverse radii. As a consequence of our study of middle convolution, we prove equivalences between various arithmetic properties on rigid Fuchsian systems. This result gives, for such systems of differential equations, an affirmative answer to a conjecture described in a paper written by Y. André and F. Baldassarri.

math.AG↗
arXiv2023

Opers with real monodromy and Eichler-Shimura isomorphism

The purpose of the present paper is to investigate $G$-opers on pointed Riemann surfaces (for a simple algebraic group $G$ of adjoint type) and their monodromy maps. In the first part, we review some general facts on $G$-opers, or more generally, principal $G$-bundles with holomorphic connection having simple poles along marked points, including the correspondence with $G$-representations of the fundamental group. One of the main results, proved in the second part, asserts that the space of certain $G$-opers with real monodromy forms a discrete set. This fact generalizes the discreteness theorem for real projective structures, already proved by G. Faltings. As an application, we establish the Eichler-Shimura isomorphism for each $\mathrm{PSL}_2$-oper with real monodromy. The resulting decomposition of the (parabolic) de Rham cohomology group of its symmetric product defines a polarized real Hodge structure.

math.CV↗
arXiv2022

Dormant opers and Gauss maps in positive characteristic

The Gauss map of a given projective variety is the rational map that sends a smooth point to the tangent space at that point, considered as a point of the Grassmann variety. The present paper aims to generalize a result by H. Kaji on Gauss maps in positive characteristic and establish an interaction with the study of dormant opers, as well as Frobenius-projective structures. We first prove a correspondence between dormant opers on a smooth projective variety $X$ and closed immersions from $X$ into a projective space with purely inseparable Gauss map. By using this, we determine the subfields of the function field of a smooth curve in positive characteristic induced by Gauss maps. Moreover, the correspondence gives us a Frobenius-projective structure on a Fermat hypersurface. This example embodies an exotic phenomenon of algebraic geometry in positive characteristic.

math.AG↗
arXiv2022

Topological quantum field theory for dormant opers

The purpose of the present paper is to develop the enumerative geometry of dormant $G$-opers for a semisimple algebraic group $G$. In the present paper, we construct a compact moduli stack admitting a perfect obstruction theory by introducing the notion of a dormant faithful twisted $G$-oper (or a "$G$-do'per", for short). The resulting virtual fundamental class induces a semisimple $2$d TQFT (= $2$-dimensional topological quantum field theory) counting the number of $G$-do'pers. This $2$d TQFT gives an analogue of the Witten-Kontsevich theorem describing the intersection numbers of psi classes on the moduli stack of $G$-do'pers.

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