因子化分支联络的模空间与广义同构单值变形
Moduli Space of Factorized Ramified Connections and Generalized Isomonodromic Deformation
摘要：
在光滑射影曲线上的一般分歧非正则奇异联络上引入了分解分歧结构的概念。利用具有因子化分歧结构联络的变形理论，在分歧联络的模空间上构造了一个标准2-形式。由于分解分歧结构提供了一个对偶的切空间的模空间，2-形式成为非退化。通过构造模空间的一个开折，证明了分歧联络的模空间上的2-形式是d-闭的。基于Stokes数据，我们引入了单位圆盘上一般非分歧非正则奇异联络的局部广义等单点形变的概念。将Jimbo-Miwa-Ueno理论应用于一般的非分歧联络，局部广义等单径形变等价于联络族可扩为可积联络。我们对分歧联络给出同样的陈述。基于Jimbo-Miwa-Ueno理论的这一原理，通过构造泛联络族的水平提升，在泛分歧联络的模空间上构造了一个全局广义等单点形变.作为整体广义等单径变形的结果，我们可以将模空间上的相对辛形式提升为全闭形式，称为广义等单径2-形式。
英文摘要：
We introduce the notion of factorized ramified structure on a generic ramified irregular singular connection on a smooth projective curve. By using the deformation theory of connections with factorized ramified structure, we construct a canonical 2-form on the moduli space of ramified connections. Since the factorized ramified structure provides a duality on the tangent space of the moduli space, the 2-form becomes nondegenerate. We prove that the 2-form on the moduli space of ramified connections is d-closed via constructing an unfolding of the moduli space. Based on the Stokes data, we introduce the notion of local generalized isomonodromic deformation for generic unramified irregular singular connections on a unit disk. Applying the Jimbo-Miwa-Ueno theory to generic unramified connections, the local generalized isomonodromic deformationis equivalent to the extendability of the family of connections to an integrable connection. We give the same statement for ramified connections. Based on this principle of Jimbo-Miwa-Ueno theory, we construct a global generalized isomonodromic deformation on the moduli space of generic ramified connections by constructing a horizontal lift of a universal family of connections. As a consequence of the global generalized isomonodromic deformation, we can lift the relative symplectic form on the moduli space to a total closed form, which is called a generalized isomonodromic 2-form.
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