H. CARTAN and J.-P. SERRE have shown how fundamental theoremson holomorphically complete manifolds （STEIN manifolds） can be for-mulated in terms of sheaf theory. These theorems imply many facts offunction theory because the domains of holomorphy are holomorphicallycomplete. They can also be applied to algebraic geometry because thecomplement of a hyperplane section of an algebraic manifold is holo-morphically complete. J.-P. SERRE has obtained important results onalgebraic manifolds by these and other methods. Recently many of hisresults have been proved for algebraic varieties defined over a field ofarbitrary characteristic. K. KODAIRA and D. C. SPENCER have alsoapplied sheaf theory to algebraic geometry with great success. Theirmethods differ from those of SERRE in that they use techniques fromdifferential geometry （harmonic integrals etc.） but do not make any useof the theory of STEIN manifolds. M. F. ATIVAH and W. V. D. HODGE have dealt successfully with problems on integrals of the second kind onalgebraic manifolds with the help of sheaf theory.
目录
Introduction Chapter One. Preparatory material 1. Multiplicative sequences 2. Sheaves 3. Fibre bundles 4. Characteristic classes Chapter Two. The cobordism ring 5. PONTRJAGIN numbers 6. The ring 7. The cobordism ring 8. The index of a 4 k-dimensional manifold 9. The virtual index Chapter Three. The TODD genus 10. Definition of the TODD genus 11. The virtual generalised TODD genus 12. The T-characteristic of a G L (q， C)-bundle 13. Split manifolds and splitting methods 14. Multiplicative properties of the TODD genus Chapter Four. The RIEMANN-ROCH theorem for algebraic manifolds 15. Cohomology of compact complex manifolds 16. Further properties of the Xy-characteristic 17. The virtual Xy-characteristic 18. Some fundamental theorems of KODAIRA 19. The virtual Xy-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 Appendix One by R. L. E. SCHWARZENBERGER 22. Applications of the RIEMANN-ROCH theorem 23. The RIEMANN-ROCH theorem of GROTHENDIECK 24. The GROTHENDIECK ring of continuous vector bundles 25. The ATIYAH-SINGER index theorem 26. Integrality theorems for differentiable manifolds Appendix Two by A. BOREL A spectral sequence for complex analytic bundles Bibliography Index
