Topological Methods in Algebraic Geometry

Biography of Friedrich Hirzebruch Friedrich Hirzebruch was born on October 17, 1927 in Hamm, Germany. He studied mathematics at the University of Munster and the ETH Zurich, under Heinrich Behnke and Heinz Hopf. Shortly after the award of his doctoral degree in 1950, he obtained an assistantship in Erlangen and then a membership at the Institute for Advanced Study, Princeton, followed by an assistant professorship at Princeton University. In 1956 he returned to Germany to a chair at the University of Bonn, which he held until his retirement in 1993. Since 1980 he has been the Director of the Max Planck Institute for Mathematics in Bonn. Hirzebruch's work has been fundamental in combining topology, algebraic and differential geometry and number theory. It has had a deep and far-reaching influence on the work of many others, who have expanded and generalized his ideas. His most famous result is the theorem of Riemann-Roch-Hirzebruch.

• One. Preparatory material.- § 1. Multiplicative sequences.- §2. Sheaves.- §3. Fibre bundles.- § 4. Characteristic classes.- Two. The cobordism ring.- § 5. Pontrjagin numbers.- § 6. The ring $$\tilde \Omega \otimes \mathcal{Q}$$?Q.- § 7. The cobordism ring ?.- § 8. The index of a 4 k-dimensional manifold.- § 9. The virtual index.- Three. The Todd genus.- § 10. Definition of the Todd genus.- § 11. The virtual generalised Todd genus.- § 12. The T-characteristic of a GL(q, C)-bundle.- § 13. Split manifolds and splitting methods.- § 14. Multiplicative properties of the Todd genus.- Four. The Riemann-Roch theorem for algebraic manifolds.- § 15. Cohomology of compact complex manifolds.- § 16. Further properties of the ?y-characteristic.- § 17. The virtual ?y-characteristic.- § 18. Some fundamental theorems of Kodaira.- § 19. The virtual ?y-characteristic for algebraic manifolds.- § 20. The Riemann-Roch theorem for algebraic manifolds and complex analytic line bundles.- §21. The Riemann-Roch theorem for algebraic manifolds and complex analytic vector bundles.- § 26. Integrality theorems for differentiate manifolds.- A spectral sequence for complex analytic bundles.