arXiv subjects

Yasuhiro Wakabayashi

Publications and source records attributed to Yasuhiro Wakabayashi.

34 records · Page 2Linked to original sources

arXiv2022

Differential modules and dormant opers of higher level

In the first half of the present paper, we study higher-level generalizations of differential modules in positive characteristic. These objects may be regarded as ring-theoretic counterparts of vector bundles on a curve equipped with an action of the ring of (logarithmic) differential operators of finite level introduced by P. Berthelot (and C. Montagnon). The existence assertion for a cyclic vector of a differential module is generalized to higher level under mild conditions. In the second half, we introduce (dormant) opers of level $N > 0$ on a pointed smooth curve whose structure group is either $\mathrm{GL}_n$ or $\mathrm{PGL}_n$. Some of the results on higher-level differential modules are applied to prove a duality theorem between dormant $\mathrm{PGL}_n$-opers of level $N$ and dormant $\mathrm{PGL}_{p^N-n}$-opers of level $N$. Finally, in the case where the underlying curve is a $3$-pointed projective line, we establish a bijective correspondence between dormant $\mathrm{PGL}_2$-opers of level $N$ and certain tamely ramified coverings.

math.AG↗

arXiv2022

On the category of structure species

The purpose of the present paper is to make a mathematical study of the differences and relations among possible structures inherent in an object, as well as of the whole structure constituted by them (i.e., the structure of structures), against the background of the structuralism by Claude Lévi-Strauss and others. Our discussion focuses on Blanchard's categorical reformulation of the notion of structure species introduced originally by Bourbaki. The main result of the present paper asserts that a category can be reconstructed, up to a certain slight indeterminacy, from the category of structure species on it. This result is partially motivated by various reconstruction theorems that have been shown in the context of anabelian geometry.

math.CT↗

arXiv2021

Cyclic étale coverings of generic curves and ordinariness of dormant opers

The ordinariness of elliptic curves is essential in proving various expected properties of elliptic curves in positive characteristic and can be extended to algebraic curves of arbitrary genus. The present paper deals with another kind of extension, i.e., ordinariness of dormant $\mathfrak{sl}_2$-opers, or more generally, dormant $\mathfrak{sl}_n$-opers. We prove that, for dormant $\mathfrak{sl}_2$-opers on elliptic curves, this notion is essentially equivalent to the classical ordinariness. Moreover, the main result of the present paper asserts that the pull-back of an ordinary dormant $\mathfrak{sl}_n$-oper on a general curve by a cyclic covering is ordinary whenever the order of its Galois group is prime to the characteristic of the base field. This result may be regarded as an analogue of a result by S. Nakajima for ordinary algebraic curves.

math.AG↗

arXiv2021

Frobenius-Ehresmann structures and Cartan geometries in positive characteristic

The aim of the present paper is to lay the foundation for a theory of Ehresmann structures in positive characteristic, generalizing the Frobenius-projective and Frobenius-affine structures defined in the previous work. This theory deals with atlases of étale coordinate charts on varieties modeled on homogenous spaces of algebraic groups, which we call Frobenius-Ehresmann structures. These structures are compared with Cartan geometries in positive characteristic, as well as with higher-dimensional generalizations of dormant indigenous bundles. In particular, we investigate the conditions under which these geometric structures are equivalent to each other. Also, we consider the classification problem of Frobenius-Ehresmann structures on algebraic curves. The latter half of the present paper discusses the deformation theory of indigenous bundles in the algebraic setting. The tangent and obstruction spaces of various deformation functors are computed in terms of the hypercohomology groups of certain complexes. As a consequence, we formulate and prove the Ehresmann-Weil-Thurston principle for Frobenius-Ehresmann structures. This fact asserts that deformations of a variety equipped with a Frobenius-Ehresmann structure are completely determined by their monodromy crystals.

math.AG↗

arXiv2021

The symplectic nature of the space of dormant indigenous bundles on algebraic curves

We study the symplectic nature of the moduli stack classifying dormant curves over a field $K$ of positive characteristic, i.e., proper hyperbolic curves over $K$ equipped with a dormant indigenous bundle. The central objects of the present paper are the following two Deligne-Mumford stacks. One is the cotangent bundle ${^\circledcirc T^{\vee ^\mathrm{Zzz...}}_{g,K}}$ of the moduli stack ${^\circledcirc \mathfrak{M}^{^\mathrm{Zzz...}}_{g,K}}$ classifying ordinary dormant curves over $K$ of genus $g$. The other is the moduli stack ${^\circledcirc \mathfrak{S}^{^\mathrm{Zzz...}}_{g,K}}$ classifying ordinary dormant curves over $K$ equipped with an indigenous bundle. These Deligne-Mumford stacks admit canonical symplectic structures respectively. The main result of the present paper asserts that a canonical isomorphism ${^\circledcirc T^{\vee ^\mathrm{Zzz...}}_{g,K}} \rightarrow {^\circledcirc \mathfrak{S}^{^\mathrm{Zzz...}}_{g,K}}$ preserves the symplectic structure. This result may be thought of as a positive characteristic analogue of the works of S. Kawai (in the paper entitled "The symplectic nature of the space of projective connections on Riemann surfaces"), P. Arés-Gastesi, I. Biswas, and B. Loustau. Finally, as its application, we construct a Frobenius-constant quantization on the moduli stack ${^\circledcirc \mathfrak{S}^{^\mathrm{Zzz...}}_{g,K}}$.

math.AG↗

arXiv2021

A theory of dormant opers on pointed stable curves -- a proof of Joshi's conjecture

This manuscript presents a detailed and original account of the theory of opers defined on pointed stable curves in arbitrary characteristic and their moduli. In particular, it includes the development of the study of dormant opers, which are opers of a certain sort in positive characteristic. The theory of dormant opers (or more generally, opers in positive characteristic) on pointed stable curves, which has proved to be rather rich and deep, was born in the work of S. Mochizuki, who developed the theory for $\mathfrak{sl}_2$-opers and used it to establish $p$-adic Teichmüller theory. Some parts of Mochizuki's work were later extended in the case of proper smooth curves by K. Joshi, C. Pauly, and other mathematicians. This manuscript represents an advance in the theory of opers that takes the subject beyond the work of Mochizuki, Joshi, and Pauly. In particular, we provide general unified formulations and the basics of principal bundles and connections defined on families of pointed stable curves. The notion of an oper is accordingly introduced in the context of logarithmic algebraic geometry. Some of the results can be regarded as generalizations of results obtained in the fundamental work on the geometric Langlands program developed by A. Beilinson and V. Drinfeld. We also describe various properties and assertions about (dormant) opers, such as duality, comparison with differential operators, and compactification of the moduli space. Our goal is to give an explicit formula, conjectured by Joshi, for the generic number of dormant $\mathfrak{sl}_n$-opers. We do so by obtaining a detailed understanding of the moduli space of dormant opers and computing the Gromov-Witten invariants for Quot-schemes in characteristic zero. This formula reveals an interaction between studies in $p$-adic Teichmüller theory and certain areas of mathematics, including Gromov-Witten theory.

math.AG↗

arXiv2020

Moduli of Tango structures and dormant Miura opers

The purpose of the present paper is to develop the theory of (pre-)Tango structures and (dormant generic) Miura $\mathfrak{g}$-opers (for a semisimple Lie algebra $\mathfrak{g}$) defined on pointed stable curves in positive characteristic. A (pre-)Tango structure is a certain line bundle of an algebraic curve in positive characteristic, which gives some pathological (relative to zero characteristic) phenomena. In the present paper, we construct the moduli spaces of (pre-)Tango structures and (dormant generic) Miura $\mathfrak{g}$-opers respectively, and prove certain properties of them. One of the main results of the present paper states that there exists a bijective correspondence between the (pre-)Tango structures (of prescribed monodromy) and the dormant generic Miura $\mathfrak{s} \mathfrak{l}_2$-opers (of prescribed exponent). By means of this correspondence, we achieve a detailed understanding of the moduli stack of (pre-)Tango structures. As an application, we construct a family of algebraic surfaces in positive characteristic parametrized by a higher dimensional base space whose fibers are pairwise non-isomorphic and violate the Kodaira vanishing theorem.

math.AG↗

arXiv2020

An effective version of Belyi's theorem in positive characteristic

The purpose of the present paper is to give an effective version of the noncritical $p$-tame Belyi theorem. That is to say, we compute explicitly an upper bound of the minimal degree of tamely ramified Belyi maps in positive characteristic which are unramified at a prescribed finite set of points.

math.AG↗

arXiv2020

Quantization on Algebraic Curves with Frobenius-Projective Structure

In the present paper, we study the relationship between deformation quantizations and Frobenius-projective structures defined on an algebraic curve in positive characteristic. A Frobenius-projective structure is an analogue of a complex projective structure on a Riemann surface, which was introduced by Y. Hoshi. Such an additional structure has some equivalent objects, e.g., a dormant $\mathrm{PGL}_2$-oper and a projective connection having a full set of solutions. The main result of the present paper provides a canonical construction of a Frobenius-constant quantization on the cotangent space minus the zero section on an algebraic curve by means of a Frobenius-projective structure. It may be thought of as a positive characteristic analogue of a result by D. Ben-Zvi and I. Biswas. Finally, we give a higher-dimensional variant of this result, as proved by I. Biswas in the complex case.

math.AG↗

arXiv2019

A combinatorial description of the dormant Miura transformation

A dormant generic Miura $\mathfrak{sl}_2$-oper is a flat $\mathrm{PGL}_2$-bundle over an algebraic curve in positive characteristic equipped with some additional data. In the present paper, we give a combinatorial description of dormant generic Miura $\mathfrak{sl}_2$-opers on a totally degenerate curve. The combinatorial objects that we use are certain branch numberings of $3$-regular graphs. Our description may be thought of as an analogue of the combinatorial description of dormant $\mathfrak{sl}_2$-opers given by S. Mochizuki, F. Liu, and B. Osserman. It allows us to think of the Miura transformation in terms of combinatorics. As an application, we identify the dormant generic Miura $\mathfrak{sl}_2$-opers on totally degenerate curves of genus $>0$.

math.AG↗

arXiv2019

Symplectic geometry of $p$-adic Teichmüller uniformization for ordinary nilpotent indigenous bundles

The aim of the present paper is to provide a new aspect of the $p$-adic Teichmüller theory established by S. Mochizuki. We study the symplectic geometry of the $p$-adic formal stacks $\widehat{\mathcal{M}}_{g, \mathbb{Z}_p}$ (= the moduli classifying $p$-adic formal curves of fixed genus $g>1$) and $\widehat{\mathcal{S}}_{g, \mathbb{Z}_p}$ (= the moduli classifying $p$-adic formal curves of genus $g$ equipped with an indigenous bundle). A major achievement in the (classical) $p$-adic Teichmüller theory is the construction of the locus $\widehat{\mathcal{N}}_{g, \mathbb{Z}_p}^{\mathrm{ord}}$ in $\widehat{\mathcal{S}}_{g, \mathbb{Z}_p}$ classifying $p$-adic canonical liftings of ordinary nilpotent indigenous bundles. The formal stack $\widehat{\mathcal{N}}_{g, \mathbb{Z}_p}^{\mathrm{ord}}$ embodies a $p$-adic analogue of uniformization of hyperbolic Riemann surfaces, as well as a hyperbolic analogue of Serre-Tate theory of ordinary abelian varieties. In the present paper, the canonical symplectic structure on the cotangent bundle $T^\vee_{\mathbb{Z}_p} \widehat{\mathcal{M}}_{g, \mathbb{Z}_p}$ of $\widehat{\mathcal{M}}_{g, \mathbb{Z}_p}$ is compared to Goldman's symplectic structure defined on $\widehat{\mathcal{S}}_{g, \mathbb{Z}_p}$ after base-change by the projection $\widehat{\mathcal{N}}_{g, \mathbb{Z}_p}^{\mathrm{ord}} \rightarrow \widehat{\mathcal{M}}_{g, \mathbb{Z}_p}$. We can think of this comparison as a $p$-adic analogue of certain results in the theory of projective structures on Riemann surfaces proved by S. Kawai and other mathematicians.

math.AG↗

arXiv2017

Duality for dormant opers

In the present paper, we prove that on a fixed pointed stable curve of characteristic $p>0$, there exists a duality between dormant $\mathfrak{sl}_n$-opers ($1 < n 0$.

math.AG↗

arXiv2016

Moduli of log twisted $\mathcal{N} =1$ SUSY curves

The goal of the present paper is to construct a smooth compactification of the moduli superstack classifying pointed $\mathcal{N} =1$ SUSY (= $\text{SUSY}_1$) curves. This construction is based on the Abramovich-Jarvis-Chiodo compactification of the moduli stack classifying spin curves. First, we give a general framework of a theory of log superschemes (or more generally, log superstacks). Then, we introduce the notion of a pointed (stable) log twisted $\text{SUSY}_1$ curve; it may be thought of as a logarithmic and twisted generalization of the classical notion of a pointed $\text{SUSY}_1$ curve, as well as a supersymmetric analogue of the notion of a pointed (log) twisted curve. The main result of the present paper asserts that the moduli superstack classifying pointed stable log twisted $\text{SUSY}_1$ curves may be represented by a log superstack whose underlying superstack is a superproper and supersmooth Deligne-Mumford superstack. Consequently, this moduli superstack forms a smooth compactification different from the compactification proposed by P. Deligne.

math.AG↗

arXiv2016

Categorical representation of superschemes

In the present paper, we prove that a locally noetherian superscheme $X^\circledS$ may be reconstructed (up to certain equivalence) category-theoretically from the category of noetherian superschemes over $X^\circledS$. This result is a supergeometric generalization of the result proved by Shinichi Mochizuki concerning categorical reconstruction of schemes.

math.AG↗

arXiv2015

On the cuspidalization problem for hyperbolic curves over finite fields

In this paper, we study some group-theoretic constructions associated to arithmetic fundamental groups of hyperbolic curves over finite fields. One of the main results of this paper asserts that any Frobenius-preserving isomorphism between the geometrically pro-$l$ fundamental groups of hyperbolic curves with one given point removed induces an isomorphism between the geometrically pro-$l$ fundamental groups of the hyperbolic curves obtained by removing other points. Finally, we apply this result to obtain results concerning certain cuspidalization problems for fundamental groups of (not necessarily proper) hyperbolic curves over finite fields.

math.AG↗

arXiv2014

An explicit formula for the generic number of dormant indigenous bundles

A dormant indigenous bundle is an integrable projective line bundle on a proper hyperbolic curve of positive characteristic satisfying certain conditions. Dormant indigenous bundles were introduced and studied in the $p$-adic Teichmuller theory developed by S. Mochizuki. Kirti Joshi proposed a conjecture concerning an explicit formula for the degree over the moduli stack of curves of the moduli stack classifying dormant indigenous bundles. In this paper, we give a proof for this conjecture of Joshi.

math.AG↗