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

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 ${}^{⊚}{T}_{g,K}^{{\vee }^{\mathrm{Z}\mathrm{z}\mathrm{z}\mathrm{.}\mathrm{.}\mathrm{.}}}$ of the moduli stack ${}^{⊚}{\mathfrak{M}}_{g,K}^{{}^{\mathrm{Z}\mathrm{z}\mathrm{z}\mathrm{.}\mathrm{.}\mathrm{.}}}$ classifying ordinary dormant curves over $K$ of genus $g$. The other is the moduli stack ${}^{⊚}{\mathfrak{S}}_{g,K}^{{}^{\mathrm{Z}\mathrm{z}\mathrm{z}\mathrm{.}\mathrm{.}\mathrm{.}}}$ 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 ${^\circledcirc T^{\vee ^$\mathrm{Z}\mathrm{z}\mathrm{z}\mathrm{.}\mathrm{.}\mathrm{.}$}{g,K}} \rightarrow {^\circledcirc \mathfrak{S}^{^$\mathrm{Z}\mathrm{z}\mathrm{z}\mathrm{.}\mathrm{.}\mathrm{.}$}{g,K}}$ 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. Ar'{e}s-Gastesi, I. Biswas, and B. Loustau. Finally, as its application, we construct a Frobenius-constant quantization on the moduli stack ${^\circledcirc \mathfrak{S}^{^$\mathrm{Z}\mathrm{z}\mathrm{z}\mathrm{.}\mathrm{.}\mathrm{.}$}_{g,K}}$.