The first contribution of this EMS volume on complex algebraic geometry touches upon many of the central problems in this vast and very active area of current research. While it is much too short to provide complete coverage of this subject, it provides a succinct summary of the areas it covers, while providing in-depth coverage of certain very important fields.

The second part provides a brief and lucid introduction to the recent work on the interactions between the classical area of the geometry of complex algebraic curves and their Jacobian varieties,and partial differential equations of mathematical physics. The paper discusses the.work of Mumford, Novikov, Krichever, and Shiota,and would be an excellent companion to the older classics on the subject.

Introduction

Chapter 1. Classical Hodge Theory

1. Algebraic Varieties

2. Complex Manifolds

3. A Comparison Between Algebraic Varieties and Analytic Spaces

4. Complex Manifolds as C Manifolds

5. Connections on Holomorphic Vector Bundles

6. Hermitian Manifolds

7. Kahler Manifolds

8. Line Bundles and Divisors

9. The Kodaira Vanishing Theorem

10. Monodromy

Chapter 2. Periods of Integrals on Algebraic Varieties

1. Classifying Space

2. Complex Tori

3. The Period Mapping

4. Variation of Hodge Structures

~5. Torelli Theorems

6. Infinitesimal Variation of Hodge Structures

Chapter 3. Torelli Theorems

1. Algebraic Curves

2. The Cubic Threefold

3. K3 Surfaces and Elliptic Pencils

4. Hypersurfaces

5. Counterexamples to Torelli Theorems

Chapter 4. Mixed Hodge Structures

1. Definition of mixed Hodge structures

2. Mixed Hodge structure on the Cohomology of a Complete Variety with Normal Crossings

3. Cohomology of Smooth Varieties

4. The Invariant Subspace Theorem

5. Hodge Structure on the Cohomology of Smooth Hypersurfaces

6. Further Development of the Theory of Mixed Hodge Structures

Chapter 5. Degenerations of Algebraic Varieties

1. Degenerations of Manifolds

2. The Limit Hodge Structure

3. The Clemens-Schmid Exact Sequence

4. An Application of the Clemens-Schmid Exact Sequence to the Degeneration of Curves

5. An Application of the Clemens-Schmid Exact Sequence to

  Surface Degenerations. The Relationship Between the Numerical

  Invariants of the Fibers Xt and X0

6. The Epimorphicity of the Period Mapping for K3 Surfaces

Comments on the bibliography

References

Index