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We study the symplectic nature of the moduli stack classifying dormant curvesover a field $K$ of positive characteristic, i.e., proper hyperbolic curvesover $K$ equipped with a dormant indigenous bundle. The central objects of thepresent paper are the following two Deligne-Mumford stacks. One is thecotangent bundle ${^\circledcirc T^{\vee ^\mathrm{Zzz...}}_{g,K}}$ of themoduli stack ${^\circledcirc \mathfrak{M}^{^\mathrm{Zzz...}}_{g,K}}$classifying ordinary dormant curves over $K$ of genus $g$. The other is themoduli stack ${^\circledcirc \mathfrak{S}^{^\mathrm{Zzz...}}_{g,K}}$classifying ordinary dormant curves over $K$ equipped with an indigenousbundle. These Deligne-Mumford stacks admit canonical symplectic structuresrespectively. The main result of the present paper asserts that a canonicalisomorphism ${^\circledcirc T^{\vee ^\mathrm{Zzz...}}_{g,K}} \rightarrow{^\circledcirc \mathfrak{S}^{^\mathrm{Zzz...}}_{g,K}}$ preserves the symplecticstructure. This result may be thought of as a positive characteristic analogueof the works of S. Kawai (in the paper entitled "The symplectic nature of thespace of projective connections on Riemann surfaces"), P. Ar\'{e}s-Gastesi, I.Biswas, and B. Loustau. Finally, as its application, we construct aFrobenius-constant quantization on the moduli stack ${^\circledcirc\mathfrak{S}^{^\mathrm{Zzz...}}_{g,K}}$.

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