Moscow Mathematical Journal

Volume 20, Issue 3, July–September 2020 pp. 575–636.

Moduli of Tango Structures and Dormant Miura Opers

Authors: Yasuhiro Wakabayashi (1)
Author institution:(1) Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan

Summary:

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 monodromies) and the dormant generic Miura $\mathfrak{sl}_2$ -opers (of prescribed exponents). By using 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.

2010 Math. Subj. Class. Primary: 14H10; Secondary: 14H60

Keywords: Oper, dormant oper, Miura oper, Miura transformation, Tango structure, Raynaud surface, pathology,

$p$ -curvature

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