论文总字数：28296字
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\fancyhead[l]{\kaishu{~~~东~南~大~学~本~科~毕~业~论~文}}
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\begin{center}{\kaishu \zihao{2}{复代数几何的基本概念}}\end{center}
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\begin{center}{\kaishu\zihao{4} 摘\ \ \ \ 要}
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{\kaishu{\quad 黎曼曲面是一个历史悠久的课题，充满数学的独特魅力。复代数几何，在复分析，抽象代数以及拓扑的基础上，引入更多的定义与概念去解决黎曼曲面即一维复流形上的问题。本文首先简略地介绍了课题背景以及黎曼曲面课题的开拓者波恩哈德·黎曼及其生平。接着我们以黎曼曲面为切入点，介绍了黎曼曲面以及代数曲线的概念及它们之间的联系，提出了黎曼球面和复环面两种形象的例子，并介绍了拓扑概念上的亏格。然后我们在黎曼曲面的基础上定义了全纯函数和半纯函数，简要地介绍了一些定义性质。再通过全纯（半纯）函数我们定义了全纯（半纯）微分，得到了全纯微分的Stokes定理和留数定理。在此基础上，我们定义了微分形式。之后我们研究了平面代数曲线的奇点定义了$k$重点。之后定义了因子以及分歧因子的概念。同时我们了解了亏格公式及其意义。在因子全纯微分等概念的基础上我们做了一些必要的说明和定义以为黎曼-罗赫定理的叙述与证明做铺垫。最后我们成功叙述黎曼-罗赫定理并在黎曼球面的条件下完成证明。}}
\vskip 1cm \noindent{\kaishu 关键词: \ \ 黎曼面,\ 复分析,\ 抽象代数,\ 几何}
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\begin{center}{\heiti \large Basic concepts of complex algebraic geometry}\end{center}
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\begin{center}{\rm\zihao{4} Abstract}
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The Riemann surface is a long-standing subject, full of the unique charm of mathematics. Complex algebraic geometry, based on complex analysis, abstract algebra and topology, introduces more definitions and concepts to solve the problem of Riemann surfaces, ie one-dimensional complex manifolds. This article first briefly introduces the background of the subject and the pioneer of the Riemann surface project, Bonnhard Riemann and his life. Then we use the Riemann surface as the entry point, introduce the concept of Riemann surface and algebraic curve and the relationship between them, and put forward two examples of Riemann spherical and complex torus, and introduce the loss of topology concept. grid. Then we define the holomorphic function and the semi-pure function based on the Riemann surface, and briefly introduce some definition properties. Then we define the holomorphic (semi-pure) differential by the purely pure (semi-pure) function, and obtain the Stokes theorem and the residue theorem of the all-pure differential. On this basis, we define the differential form. Later we studied the singularity of the plane algebraic curve to define the k-emphasis. The concept of factors and divergence factors are then defined. At the same time, we understand the genus formula and its significance. Based on the concept of factor-pure differential, we have made some necessary explanations and definitions to pave the way for the narrative and proof of Riemann-Roche theorem. Finally, we successfully describe the Riemann-Roche theorem and complete the proof under the conditions of Riemann's spherical surface.
\vskip 0.8cm \noindent{\rm Key Words:\ Riemann surface,\ Complex analysis,\ Abstract algebra,\ geometry}
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黎曼曲面是一个历史悠久的理论课题，由黎曼，克莱因和魏尔（1851-1913）提出。黎曼曲面同时也是一维复流形和代数曲线。从多复变函数论的角度看，黎曼曲面被认为是研究解析函数的整体行为的自然选择，特别是像平方根和自然对数这样的多值函数。
另一种看法是将黎曼曲面研究看为二维实流形，因为高斯（1822）已经考虑了在欧几里德空间中取一块光滑定向表面并将其共形地嵌入复平面的问题。一个实二维流形当且仅当是可定向的时，它可以被看做一个黎曼曲面(通常有几种不同的方式)。
第四个观点来自克莱因，庞加莱和科贝（1882-1907）的均匀化理论，他们表明每个黎曼面（根据定义，它是一个配有复分析结构的连通表面）也允许黎曼度量。
紧黎曼曲面是近代数学的一个重要研究对象，它无论在代数几何、自守函数、或微分几何中，都存在于一个非常重要的地位。与此同时，在多复变函数论研究中，紧黎曼曲面理论中的一些技巧和想法也存在着重要的影响。
黎曼曲面理论在整个现代数学的发展历史中扮演了一个举足轻重角色，它有重要历史意义，它和20世纪一大批全新的数学研究领域有着密切的关系，所以说它举足轻重。受到影响的现代数学分支学科有很多，例如：数论、代数几何、多复分析、代数拓扑、偏微分方程和整体微分几何等等。
举个例子，在代数几何中，在代数簇的研究中，通过黎曼曲面的方法，首次引入了解析以及拓扑的工具。反过来，这种方法同时也促成了多复分析理论、整体微分几何和代数拓扑的诞生与发展。特别的，从黎曼曲面的概念当中，慢慢地微分流形的精确定义被数学家们引申出来，由此２０世纪几何与拓扑的研究建立起了一个巨大的平台。
很显然Riemann-Roch(黎曼-洛赫)定理是紧黎曼面的理论中最重要的结果。它是代数几何理论中最为重要的几个定理之一，最开始这个定理是建立在代数曲线理论上的，后来有很多数学家都考虑过将它推广到高维的情形下。
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\chapter{波恩哈德·黎曼 }
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