Topological methods in algebraic geometry
Topological methods in algebraic geometry
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DOI：
10.1007/978-3-662-30697-0
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发表时间：
1966
期刊：
影响因子：
--
通讯作者：
F. Hirzebruch
中科院分区：
文献类型：
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作者：
F. Hirzebruch
它主要涉及证明黎曼-罗赫定理的显着推广，并阐述卡勒流形、滑轮、向量丛和配边理论的必要部分。这些理论中的大部分在当时还是相当新的，而赫策布鲁赫的书仍然是极少数的记载之一。陈省身在他的评论（Bull. American Math. Soc. 1958）中将其与 Lefschetz 的 L'Analysis situs et la ge'om&trie alge'brique（1924）和 Hodge 的调和积分的理论与应用（1941）列为代数簇拓扑和超越理论发展的里程碑。此版本是原文的英文翻译，并附有进一步的参考文献，两个附录和一些参考书目注释。第一个附录是一个实质性的补充，添加它是为了考虑继 Hirzenbruch 发现可微流形指数定理和代数流形上向量丛的黎曼-罗赫定理之后的一些发展，其中包括： 用于射影簇映射的黎曼-罗赫定理及其可微流形的类似物；微分算子理论中的指数定理和不动点公式。它主要从几何角度对这些结果以及主要应用进行了出色的总结：将霍奇指数定理和赫兹布鲁赫黎曼-罗赫定理扩展到紧复流形；完整性和非沉浸定理；自守形式的某些空间的维数的计算。第二个附录是 A. Borel 之前未发表的一篇论文，在第一版中提到，该论文涉及复丛 5 上同调的谱序列基础。论文给出了两个应用：复流形的xy-亏格的乘性性质（向量丛黎曼-罗赫定理的原始证明所需）； Calabi-Eckmann 流形（具有某些复杂结构的奇维球体的乘积）的 5-上同调公式。
It was mainly concerned with proving a remarkable generalization of the Riemann-Roch theorem and giving an exposition of the necessary parts of the theories of Kahler manifolds, sheaves, vector bundles and cobordism. Much of these theories was then quite new, and Hirzebruch's book is still one of the very few accounts. In his review (Bull. American Math. Soc. 1958), Chern ranked it with Lefschetz's L'Analysis situs et la ge'om&trie alge'brique (1924) and Hodge's The theory and applications of harmonic integrals (1941) as a landmark in the development of the topological and transcendental theory of algebraic varieties.This edition is a translation of the original text into English, with further references, two appendices and some bibliographical notes. The first appendix is a substantial addition, and has been added to take account of some of the developments that have followed Hirzenbruch's discoveries of the index theorem for differentiable manifolds and the Riemann-Roch theorem for vector bundles on algebraic manifolds, these include: the Riemann-Roch theorem for a mapping of projective varieties and its analogues for differentiable manifolds; the index theorems and fixed point formulas in the theory of differential operators. It gives excellent summaries of these results, mainly from a geometric point of view, and of the main applications: the extension of Hodge's index theorem and Hirzebruch's Riemann-Roch theorem to compact complex manifolds; integrality and non-immersion theorems; the computation of the dimensions of certain spaces of automorphic forms. The second appendix is a formerly unpublished paper of A. Borel's, referred to in the first edition, on a spectral sequence fundamental for the 5-cohomology of complex bundles. The paper gives two applications: the multiplicative property of the xy-genus of a complex manifold (needed for the original proof of the Riemann-Roch theorem for vector bundles); a formula for the 5-cohomology of the Calabi-Eckmann manifolds (products of odd dimensional spheres with certain complex structures).