Manifolds and Modular Forms

1. Front Matter

   Pages i-xi

   PDF

2. Background

   • Friedrich Hirzebruch, Thomas Berger, Rainer Jung

   Pages 1-21

3. Elliptic genera

   • Friedrich Hirzebruch, Thomas Berger, Rainer Jung

   Pages 23-33

4. A universal addition theorem for genera

   • Friedrich Hirzebruch, Thomas Berger, Rainer Jung

   Pages 35-40

5. Multiplicativity in fibre bundles

   • Friedrich Hirzebruch, Thomas Berger, Rainer Jung

   Pages 41-56

6. The Atiyah-Singer index theorem

   • Friedrich Hirzebruch, Thomas Berger, Rainer Jung

   Pages 57-72

7. Twisted operators and genera

   • Friedrich Hirzebruch, Thomas Berger, Rainer Jung

   Pages 73-95

8. Riemann-Roch and elliptic genera in the complex case

   • Friedrich Hirzebruch, Thomas Berger, Rainer Jung

   Pages 97-112

9. A divisibility theorem for elliptic genera

   • Friedrich Hirzebruch, Thomas Berger, Rainer Jung

   Pages 113-120

10. Back Matter

    Pages 121-212

    PDF

During the winter term 1987/88 I gave a course at the University of Bonn under the title "Manifolds and Modular Forms". I wanted to develop the theory of "Elliptic Genera" and to learn it myself on this occasion. This theory due to Ochanine, Landweber, Stong and others was relatively new at the time. The word "genus" is meant in the sense of my book "Neue Topologische Methoden in der Algebraischen Geometrie" published in 1956: A genus is a homomorphism of the Thorn cobordism ring of oriented compact manifolds into the complex numbers. Fundamental examples are the signature and the A-genus. The A-genus equals the arithmetic genus of an algebraic manifold, provided the first Chern class of the manifold vanishes. According to Atiyah and Singer it is the index of the Dirac operator on a compact Riemannian manifold with spin structure. The elliptic genera depend on a parameter. For special values of the parameter one obtains the signature and the A-genus. Indeed, the universal elliptic genus can be regarded as a modular form with respect to the subgroup r (2) of the modular group; the two cusps 0 giving the signature and the A-genus. Witten and other physicists have given motivations for the elliptic genus by theoretical physics using the free loop space of a manifold.

• Signatur
• algebra
• manifold
• material

• Max-Planck-Institut für Mathematik, Bonn, Germany

  Friedrich Hirzebruch, Thomas Berger, Rainer Jung

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