Moduli of Tango structures and dormant Miura opers

TL;DR

This paper develops the theory of Tango structures and dormant Miura opers on pointed stable curves in positive characteristic, constructing their moduli spaces and revealing a bijective correspondence between them.

Contribution

It introduces the moduli spaces of (pre-)Tango structures and dormant Miura $rak{g}$-opers, establishing a key correspondence for $rak{sl}_2$ and analyzing their properties.

Findings

Established a bijective correspondence between Tango structures and dormant Miura $rak{sl}_2$-opers.

Constructed moduli spaces for these structures and studied their properties.

Produced algebraic surfaces in positive characteristic violating Kodaira vanishing.

Abstract

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.