The symplectic nature of the space of dormant indigenous bundles on algebraic curves
The symplectic nature of the space of dormant indigenous bundles on algebraic curves
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代数曲线上休眠本征丛空间的辛性质
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发表时间：
2014
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通讯作者：
Y. Wakabayashi
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作者：
Y. Wakabayashi
研究了具有正特征域$K$上的模堆分类休眠曲线的辛性质，即$K$上的固有双曲曲线配备了一个休眠的固有束。本论文的中心对象是以下两个Deligne-Mumford堆栈。一个是余切束${^\circledcirc T^{\vee ^\ mathm {Zzz…}} _ {g、K}} $模栈的$ {^ \ circledcirc \ mathfrak {M} ^ {^ \ mathrm{打鼾声……}}_{g,K}}$分类$g$属$K$上的普通休眠曲线。另一个是模栈${^\circledcirc \mathfrak{S}^{^\mathrm{Zzz…}}_{g,K}}$对具有固有束的K$上的普通休眠曲线进行分类。这些delign - mumford堆栈分别承认正则辛结构。本文的主要结果证明了正则同构${^\circledcirc T^{\vee ^\ mathm {Zzz…}} _ {g、K}} \ rightarrow {^ \ circledcirc \ mathfrak{年代}^ {^ \ mathrm{打鼾声……}}_{g,K}}$保留辛结构。这个结果可以被认为是S. Kawai（在题为“Riemann曲面上投影连接空间的辛性质”的论文中），P. Ar\ {e} - gastesi， I. Biswas和B. Loustau的工作的一个正特征模拟。最后，作为其应用，我们在模栈${^\circledcirc \mathfrak{S}^{^\ maththrm {Zzz…}}_{g,K}}$上构造了一个frobenius常数量化。
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 ${^\circledcirc T^{\vee ^\mathrm{Zzz...}}_{g,K}} \rightarrow {^\circledcirc \mathfrak{S}^{^\mathrm{Zzz...}}_{g,K}}$ 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. Ar\'{e}s-Gastesi, I. Biswas, and B. Loustau. Finally, as its application, we construct a Frobenius-constant quantization on the moduli stack ${^\circledcirc \mathfrak{S}^{^\mathrm{Zzz...}}_{g,K}}$.