摘要翻译：
本文讨论了复几何与可积系统之间的一种相互作用。第一节回顾了关于可积系统的经典结果。已发现的可积系统的新例子是基于运动方程的Lax表示。这些系统可以实现为在所谓的光谱曲线的雅可比变型上的直线运动。在第二节中，我们研究了导致可积系统的李代数理论方法，并将该方法应用于几个问题。在第三节中，我们讨论了代数完全可积性（A.C.I.）的概念哈密顿系统。代数可积性是指系统在由环面构成的相空间中是完全可积的，而环面又是复代数环面的实部（阿贝尔变体）。该方法用于说明如何确定交流阻抗。并应用于一些例子。最后，在第4节中，我们研究了一个A.C.I.在广义意义上，它表现为A.C.I.的覆盖。系统。复流的流形不变量是阿贝尔变换的覆盖。
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英文标题：
《Integrable systems and complex geometry》
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作者：
A. Lesfari
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最新提交年份：
2007
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分类信息：

一级分类：Mathematics        数学
二级分类：Dynamical Systems        动力系统
分类描述：Dynamics of differential equations and flows, mechanics, classical few-body problems, iterations, complex dynamics, delayed differential equations
微分方程和流动的动力学，力学，经典的少体问题，迭代，复杂动力学，延迟微分方程
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一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
  In this paper, we discuss an interaction between complex geometry and integrable systems. Section 1 reviews the classical results on integrable systems. New examples of integrable systems, which have been discovered, are based on the Lax representation of the equations of motion. These systems can be realized as straight line motions on a Jacobi variety of a so-called spectral curve. In section 2, we study a Lie algebra theoretical method leading to integrable systems and we apply the method to several problems. In section 3, we discuss the concept of the algebraic complete integrability (a.c.i.) of hamiltonian systems. Algebraic integrability means that the system is completely integrable in the sens of the phase space being folited by tori, which in addition are real parts of a complex algebraic tori (abelian varieties). The method is devoted to illustrate how to decide about the a.c.i. of hamiltonian systems and is applied to some examples. Finally, in section 4 we study an a.c.i. in the generalized sense which appears as covering of a.c.i. system. The manifold invariant by the complex flow is covering of abelian variety.
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PDF链接：
https://arxiv.org/pdf/0706.1579