亚纯微分模空间上的Baker-Akhiezer旋量核与τ-函数
Baker-Akhiezer spinor kernel and tau-functions on moduli spaces of meromorphic differentials
摘要：
本文研究了黎曼曲面上亚纯微分的模空间上的Baker-Akhiezer旋量核。我们引入了Baker-Akhiezer τ-函数，它与Bergman τ-函数（以前在Hurwitz空间、全纯微分空间和二次微分空间中研究过）和KP τ-函数都有关系。特别地，我们在亚纯微分的模空间上导出了具有指定奇点的Rauch-Ahlfors型变分公式：我们使用由亚纯微分的绝对周期和相对周期组成的同调坐标系，并说明如何改变与黎曼曲面相关的基本对象（$b$ -周期矩阵、归一化阿贝尔微分、Bergman双微分、Szegö核和Baker-Akhiezer旋量核）。变分公式编码依赖于黎曼曲面的模和亚纯微分的选择（变化的亚纯微分，同时保持黎曼曲面固定对应于流动的KP型）。通过分析Bergman和Baker-Akhiezer τ-函数的整体性质，建立了模空间上各种除子类之间的关系.
英文摘要：
In this paper we study Baker-Akhiezer spinor kernel on moduli spaces of meromorphic differentials on Riemann surfaces. We introduce the Baker-Akhiezer tau-function which is related to both Bergman tau-function (which was studied before in the context of Hurwitz spaces and spaces of holomorphic and quadratic differentials) and KP tau-function on such spaces. In particular, we derive variational formulas of Rauch-Ahlfors type on moduli spaces of meromorphic differentials with prescribed singularities: we use the system of homological coordinates, consisting of absolute and relative periods of the meromorphic differential, and show how to vary the fundamental objects associated to a Riemann surface (the matrix of $b$-periods, normalized Abelian differentials, the Bergman bidifferential, the Szegö kernel and the Baker-Akhiezer spinor kernel) with respect to these coordinates. The variational formulas encode dependence both on the moduli of the Riemann surface and on the choice of meromorphic differential (variation of the meromorphic differential while keeping the Riemann surface fixed corresponds to flows of KP type). Analyzing the global properties of the Bergman and Baker-Akhiezer tau-functions we establish relationships between various divisor classes on the moduli spaces.
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