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

Abstract

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 ${}^{⊚}{T}_{g,K}^{{\vee }^{Zzz...}}$ of the moduli stack ${}^{⊚}{\mathfrak{M}}_{g,K}^{{}^{Zzz...}}$ classifying ordinary dormant curves over $K$ of genus $g$. The other is the moduli stack ${}^{⊚}{\mathfrak{S}}_{g,K}^{{}^{Zzz...}}$ 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 ${}^{⊚}{T}_{g,K}^{{\vee }^{Zzz...}}\to {}^{⊚}{\mathfrak{S}}_{g,K}^{{}^{Zzz...}}$ 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 ${}^{⊚}{\mathfrak{S}}_{g,K}^{{}^{Zzz...}}$.

35 pages, revised version

References in corpus (1)

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

Cited by in corpus (2)

• A theory of dormant opers on pointed stable curves -- a proof of Joshi's conjecture
• Symplectic geometry of $p$-adic Teichmüller uniformization for ordinary nilpotent indigenous bundles