Sheaf theory has had profound effects on several mathematical subjects, especially, topology, differential and algebraic geometry. It gives a tool for dealing with problems by piecing together solutions of local problems in a coherent manner to get global solutions. For example, several geometric structures of a manifold, scheme on the space and general cohomology theory can be expressed in terms of sheaves. The sheaf cohomology measures the lack of exactness of the global solutions. In this book we introduce the foundations of sheaf theory to define the cohomologies of topological, differential and complex manifolds. In Chapter 1 we introduce Riemann surfaces and motivation of sheaf cohomology. In Chapter 2 we introduce direct limits, and in Chapter 3 presheaf, sheaf and examples. In Chapter 4 we introduce ringed spaces, geometric spaces and module over ringed spaces. In Chapter 5 we deal with �ech cohomology and Grothendieck cohomology. In Chapter 6 applications to Riemann surfaces. In Chapter 7 complex manifolds, in Chapter 8 the de Rham theory, and in Chapter 9 Hodge decomposition theorems on compact oriented manifolds and K�hler manifolds. In Appendix we introduce characteristic classes, low dimensional manifolds, and categories. The final manuscript and galley proof are read by my students Ahram Lim and Semin Yoo. I am sincerely grateful to them.

Chapter 1 Riemann Surface 1.1 Meromorphic Function / 3 1.2 Motivation / 7 1.3 Obstruction to the Construction of Function / 7 Chapter 2 Direct limit 2.1 Direct Limit of Set / 13 2.2 Direct Limit of Abelian Group / 16 Chapter 3 Sheaf 3.1 Sheaf / 21 3.2 Godement Sheaf and Cartan Sheaf / 27 3.3 Induced Sheaf / 33 Chapter 4 Geometric Space 4.1 Ringed Space / 41 4.2 Prime Spectrum of Ring / 51 4.3 Geometric Space / 58 4.4 Module over Ringed Space / 64 4.5 Locally Free Module / 73 Chapter 5 Sheaf Cohomology 5.1 Čech Cohomology / 82 5.2 Grothendieck Cohomology / 88 Chapter 6 Sheaf on Riemann Surface 6.1 Riemann-Roch Theorem / 101 6.2 Divisor Class Group / 102 6.3 Jacobian / 103 6.4 Differential / 106 6.5 Laurent Theorem / 109 6.6 Weierstrass Theorem / 110 Chapter 7 Complex Manifold 7.1 Compact Complex Manifold / 116 7.2 Stein Manifold / 119 Chapter 8 The de Rham Theorem 8.1 Exterior Product of Vector Space / 127 8.2 Differential Form / 129 8.3 The de Rham Theorem for Differential Manifold / 131 8.4 The de Rham Theorem for Stein Manifold / 138 8.5 Lefschetz Theorem / 140 Chapter 9 Hodge Theorem 9.1 Laplace Operator / 145 9.2 Hodge Decomposition Theorem on Compact Oriented Manifold / 148 9.3 Kähler Manifold / 156 9.4 Hodge Decomposition Theorem on Compact Kähler Manifold / 164 Appendix A: Characteristic Class 1. Vector Bundle / 171 2. Characteristic Class for Line Bundle / 173 3. Chern Class / 174 4. Gysin Homomorphism / 179 5. Euler Class / 184 6. Pontrjagin Class / 188 Appendix B: Low-Dimensional Manifold 1. One-Manifold / 191 2. Two-Manifold / 191 3. Three-Manifold / 193 4. Four-Manifold / 195 Appendix C: Category 1. Category / 200 2. Additive Category / 202 3. Abelian Category / 205 REFERENCES / 207 INDEX / 209

 

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