Moduli of Tango Structures and Dormant Miura Opers

Wakabayashi, Yasuhiro

doi:10.17323/1609-4514-2020-20-3-575-636

Cited by 8 publications

(28 citation statements)

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“…If N = 1, then these objects are equivalent to what we call dormant (generic) Miura PGL 2 -opers or Tango structures (cf. [41], [52], [55]).…”

Section: Case (A)mentioning

confidence: 99%

“…Frobenius-projective structures have provided a rich and deep story under the identifications with various equivalent realizations (in the case of N = 1), e.g., dormant indigenous bundles (= dormant PGL 2 -opers) and projective connections with a full set of solutions. See [40], [52], and [54] for the study of these objects and their moduli space in the context of p-adic Teichmüller theory developed by S. Mochizuki. Also, in [53], the author generalized Frobenius-projective structures to higher-dimensional varieties (i.e., generalized the local model to the n-dimensional projective space P n equipped with the natural PGL n+1 -action for an arbitrary n) including the case of infinite level. The main subject of loc.…”

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confidence: 99%

“…Cases (a) and (b) were discussed in the previous studies (cf. [23], [24], [25], [52], and [53]). By further examining the other cases, we will classify Frobenius-Ehresmann structures on curves (cf.…”

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confidence: 99%

“…Also, a detailed understanding of the moduli stack of dormant indigenous (Aff 1 , Aff ⊚ 1 )-bundles on curves was obtained by investigating its local structure (cf. [52]). As an application, we constructed a family of pathological algebraic surfaces (e.g., violating the Kodaira vanishing theorem) in characteristic p parametrized by a higher dimensional base space (cf.…”

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confidence: 99%

“…As an application, we constructed a family of pathological algebraic surfaces (e.g., violating the Kodaira vanishing theorem) in characteristic p parametrized by a higher dimensional base space (cf. [52,Theorem C]).…”

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confidence: 99%

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Frobenius-Ehresmann structures and Cartan geometries in positive characteristic

Wakabayashi

2021

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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. Contents1. Introduction 1 2. Frobenius-Ehresmann structures 11 3. Cartan geometries in positive characteristic 16 4. Indigenous bundles on smooth schemes 34 5. Classification for algebraic curves 54 6. Deformation theory of (dormant) indigenous bundles 57 7. Appendix: Connections and PD stratifications on a G-bundle 79 References 93

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“…If N = 1, then these objects are equivalent to what we call dormant (generic) Miura PGL 2 -opers or Tango structures (cf. [41], [52], [55]).…”

Section: Case (A)mentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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confidence: 99%

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confidence: 99%