[Paper Review] The symplectic nature of the space of dormant indigenous bundles on algebraic curves

Yasuhiro Wakabayashi|arXiv (Cornell University)|Nov 5, 2014

Algebraic Geometry and Number Theory31 references3 citations

TL;DR

This paper establishes a canonical symplectic isomorphism between the cotangent stack of ordinary dormant curves and the moduli stack of dormant indigenous bundles on algebraic curves over a positive characteristic field. The isomorphism preserves the symplectic structures, generalizing classical results of Kawai and others in complex geometry, and leads to a construction of a Frobenius-constant quantization on the moduli stack of dormant indigenous bundles.

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 ${^\circledcirc T^{\vee ^\mathrm{Zzz...}}_{g,K}}$ of the moduli stack ${^\circledcirc \mathfrak{M}^{^\mathrm{Zzz...}}_{g,K}}$ classifying ordinary dormant curves over $K$ of genus $g$. The other is the moduli stack ${^\circledcirc \mathfrak{S}^{^\mathrm{Zzz...}}_{g,K}}$ 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 ${^\circledcirc T^{\vee ^\mathrm{Zzz...}}_{g,K}} ightarrow {^\circledcirc \mathfrak{S}^{^\mathrm{Zzz...}}_{g,K}}$ 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. Arés-Gastesi, I. Biswas, and B. Loustau. Finally, as its application, we construct a Frobenius-constant quantization on the moduli stack ${^\circledcirc \mathfrak{S}^{^\mathrm{Zzz...}}_{g,K}}$.

Motivation & Objective

• To investigate the symplectic geometry of moduli stacks of dormant indigenous bundles in positive characteristic.
• To establish a canonical symplectic structure on the moduli stack of ordinary dormant curves equipped with an indigenous bundle.
• To prove that the natural map from the cotangent stack of ordinary dormant curves to the moduli stack of such bundles is a symplectomorphism.
• To apply the symplectic isomorphism to construct a Frobenius-constant quantization on the moduli stack of dormant indigenous bundles.
• To provide a positive characteristic analogue of classical symplectic results on projective structures and Teichmüller spaces.

Proposed method

• Define the moduli stack ${{}^{ ext{ extcircledcirc}}}Τ^{Σ}_{g,K}$ classifying ordinary dormant curves of genus $g$ over a field $K$ of positive characteristic.
• Construct the cotangent stack ${{}^{ ext{ extcircledcirc}}}T^{ ext{ extdollar}∓ ext{ extsc{Zzz...}}}_{g,K}$ of this moduli stack and equip it with a canonical symplectic structure.
• Define the moduli stack ${{}^{ ext{ extcircledcirc}}}Τ^{Σ}_{g,K}$ classifying ordinary dormant curves equipped with an indigenous bundle, and equip it with a canonical symplectic structure.
• Establish a canonical isomorphism $\Psi_{g,K}: {{ }^{ ext{ extcircledcirc}}}T^{ ext{ extdollar}∓ ext{ extsc{Zzz...}}}_{g,K} \xrightarrow{\sim} {{ }^{ ext{ extcircledcirc}}}Τ^{Σ}_{g,K}$ as Deligne-Mumford stacks.
• Prove that this isomorphism preserves the symplectic structures, i.e., it is a symplectomorphism.
• Use the symplectomorphism to transport a restricted structure and a Frobenius-constant quantization from the cotangent stack to the moduli stack of dormant indigenous bundles.

Experimental results

Research questions

• RQ1 Is there a canonical symplectic structure on the moduli stack of ordinary dormant curves equipped with an indigenous bundle in positive characteristic?
• RQ2 Does the natural map from the cotangent stack of ordinary dormant curves to the moduli stack of such bundles preserve the symplectic structure?
• RQ3 Can the symplectic isomorphism between these stacks be used to construct a Frobenius-constant quantization on the moduli stack of dormant indigenous bundles?
• RQ4 How does this construction relate to classical symplectic geometry of projective structures on Riemann surfaces?
• RQ5 What is the role of the $p$-curvature and $p$-adic Teichmüller theory in this symplectic framework?

Key findings

• The moduli stack ${{}^{ ext{ extcircledcirc}}}Τ^{Σ}_{g,K}$ of ordinary dormant curves with an indigenous bundle admits a canonical symplectic structure.
• The cotangent stack ${{}^{ ext{ extcircledcirc}}}T^{ ext{ extdollar}∓ ext{ extsc{Zzz...}}}_{g,K}$ of ordinary dormant curves also admits a canonical symplectic structure.
• There exists a canonical isomorphism $\Psi_{g,K}: {{ }^{ ext{ extcircledcirc}}}T^{ ext{ extdollar}∓ ext{ extsc{Zzz...}}}_{g,K} \xrightarrow{\sim} {{ }^{ ext{ extcircledcirc}}}Τ^{Σ}_{g,K}$ of Deligne-Mumford stacks.
• This isomorphism $\Psi_{g,K}$ preserves the symplectic structures, making it a symplectomorphism.
• The symplectomorphism allows the transfer of a restricted structure and a Frobenius-constant quantization from the cotangent stack to the moduli stack of dormant indigenous bundles.
• As a consequence, the moduli stack ${{}^{ ext{ extcircledcirc}}}Τ^{Σ}_{g,K}$ admits a canonical Frobenius-constant quantization.