指标定理（一） (11-2)

2.3 Statement of the theorem and idea of the proof............... 14

3 Examples: de Rham Complex and Dolbeault Complex 15

3.1 de Rham complex................................ 15

3.2 Dolbeault complex............................... 17

4 Hirzebruch Signature Theorem 19

4.1 Multiplicative sequence............................. 19

4.2 Hirzebruch signature theorem......................... 19

5 Riemann-Roch Theorem 22

5.1 Divisors on Riemann surfaces......................... 22

5.2 Divisors and line bundles............................23

5.3 Hirzebruch-Riemann-Roch theorem...................... 24

6 Further Developments 25

2

1 Introduction and Preliminaries

1.1 Introduction

The main question in index theory is to provide index formulas for classes of Fredholm operators. Index theory has become a subject on its own only after M.F.Atiyah and I. Singer published their index theorems in a sequence of papers. Among them, Hirzebruch's signature theorem occupies a special place.Hirzebruch's theorem was generalized by A. Grothendieck, who introduced many of the ideas that proved to be fundamental for the proof of the index theorems. All these theorems turned out to be consequences of the Atiyah-Singer index theorems. ¹

In this review we give a brief introduction to some index theorems, Readers who are familiar with materials in this section can skip to the next section.

1.2 Fibre bundles and vector bundles

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