标题： 代数德拉姆定理和贝克-阿基耶泽函数 显示英文标题

标题： Algebraic de Rham theorem and Baker-Akhiezer function

摘要： 对于代数曲线——紧致黎曼曲面的情况，证明了亏格为$g$的黎曼曲面$X$的德雷姆上同调群$H^{1}_{\mathrm{dR}}(X,\mathbb{C})$具有辛向量空间的自然结构。 每个非特殊有效除子$D$的选择，其在$X$上的次数为$g$，定义了$H^{1}_{\mathrm{dR}}(X,\mathbb{C})$的辛基，该基由全纯微分和在$D$上有极点的第二类微分组成。 这个结果，即代数德拉姆定理，用于用二类微分描述Picard和Jacobian簇$X$的切空间，并在Jacobian的$X$上定义自然的向量场，这些向量场移动除子$D$的点。 在代数曲线上的Lax形式主义中，这些向量场对应于可积系统理论中的Dubrovin方程，而Baker-Akhierzer函数通过沿积分曲线的积分自然得到。 显示英文摘要

摘要： For the case of algebraic curves - compact Riemann surfaces - it is shown that de Rham cohomology group $H^{1}_{\mathrm{dR}}(X,\mathbb{C})$ of a genus $g$ Riemann surface $X$ has a natural structure of a symplectic vector space. Every choice of a non-special effective divisor $D$ of degree $g$ on $X$ defines a symplectic basis of $H^{1}_{\mathrm{dR}}(X,\mathbb{C})$, consisting of holomorphic differentials and differentials of the second kind with poles on $D$. This result, the algebraic de Rham theorem, is used to describe the tangent space to Picard and Jacobian varieties of $X$ in terms of differentials of the second kind, and to define a natural vector fields on the Jacobian of $X$ that move points of the divisor $D$. In terms of the Lax formalism on algebraic curves, these vector fields correspond to the Dubrovin equations in the theory of integrable systems, and the Baker-Akhierzer function is naturally obtained by the integration along the integral curves.