Several Complex Variables VII

1. Front Matter

   Pages i-viii

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2. Introduction

   • H. Grauert, Th. Peternell, R. Remmert

   Pages 1-5

3. Local Theory of Complex Spaces

   • R. Remmert

   Pages 7-96

4. Differential Calculus, Holomorphic Maps and Linear Structures on Complex Spaces

   • Th. Peternell, R. Remmert

   Pages 97-144

5. Cohomology

   • Th. Peternell

   Pages 145-182

6. Seminormal Complex Spaces

   • G. Dethloff, H. Grauert

   Pages 183-220

7. Pseudoconvexity, the Levi Problem and Vanishing Theorems

   • Th. Peternell

   Pages 221-257

8. Theory of q-Convexity and q-Concavity

   • H. Grauert

   Pages 259-284

9. Modifications

   • Th. Peternell

   Pages 285-317

10. Cycle Spaces

    • F. Campana, Th. Peternell

    Pages 319-349

11. Extension of Analytic Objects

    • H. Grauert, R. Remmert

    Pages 351-360

12. Back Matter

    Pages 361-372

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Of making many books there is no end; and much study is a weariness of the flesh. Eccl. 12.12. 1. In the beginning Riemann created the surfaces. The periods of integrals of abelian differentials on a compact surface of genus 9 immediately attach a g­ dimensional complex torus to X. If 9 ~ 2, the moduli space of X depends on 3g - 3 complex parameters. Thus problems in one complex variable lead, from the very beginning, to studies in several complex variables. Complex tori and moduli spaces are complex manifolds, i.e. Hausdorff spaces with local complex coordinates Z 1, ... , Zn; holomorphic functions are, locally, those functions which are holomorphic in these coordinates. th In the second half of the 19 century, classical algebraic geometry was born in Italy. The objects are sets of common zeros of polynomials. Such sets are of finite dimension, but may have singularities forming a closed subset of lower dimension; outside of the singular locus these zero sets are complex manifolds.

• Complex analysis
• Complex spaces
• Kohomologie
• Kohärente Garben
• Konvexität und Konkavität
• Modifikation
• Pseudoconvexity
• calculus
• coherent sheaves
• cohomology
• convexity and concavity
• komplexe Räume
• modifications

• Mathematisches Institut, Universität Göttingen, Göttingen, Germany

  H. Grauert

• Mathematisches Institut, Universität Bayreuth, Bayreuth, Germany

  Th. Peternell

• Mathematisches Institut, Universität Münster, Münster, Germany

  R. Remmert

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