The first contribution of this EMS volume on compilex 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 classicat 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 elassics on the subject.
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目录
I. Complex Algebraic Varieties：Periods of Integrals and Hodge Structures 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. Kähler 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 II. Algebraic Curves and Their Jacobians Introduction §1. Applications 1.1. Theory of Burnchall-Chaundy-Krichever 1.2. Deformation of Commuting Differential Operators 1.3. Kadomtsev-Petviashvili Equations 1.4. Finite Dimensional Solutions of the KP Hierarchy 1.5. Solutions of the Toda Lattice 1.6. Solution of Algebraic Equations Using Theta-Constants §2. Special Divisors 2.1. Varieties of Special Divisors and Linear Systems 2.2. The Brill-Noether Matrix. The Brill-Noether Numbers 2.3. Existence of Special Divisors 2.4. Connectedness 2.5. Special Curves. The General Case 2.6. Singularities 2.7. Infinitesimal Theory of Special Linear Systems 2.8. GaussMappings 2.9. Sharper Bounds on Dimensions 2.10.Tangent Cones §3. Prymians 3.1. Unbranched Double Covers 3.2. Prymians and Prym Varieties 3.3. Polarization Divisor 3.4. Singularities of the Polarization Divisor 3.5. Differences Between Prymians and Jacobians 3.6. The Prym Map §4. Characterizing Jacobians 4.1. The Variety of Jacobians 4.2. The Andreotti-Meyer Subvariety 4.3. Kummer Varieties 4.4. Reducedness of θ∩(θ+p) and Trisecants 4.5. The Characterization of Novikov-Krichever 4.6. Schottky Relations References Index
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