Moduli of Tango structures and dormant Miura opers

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.