riemann surfaces, algebraic curves & moduli spaces [pdf] martin schlichenmaier
An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces (2nd Edition) (Theoretical and Mathematical Physics) by Martin Schlichenmaier
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An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces (2nd Edition) (Theoretical and Mathematical Physics) by
Martin Schlichenmaier .
This book gives an introduction to modern geometry. Starting from an elementary level, the author develops deep geometrical concepts that play an important role in contemporary theoretical physics, presenting various techniques and viewpoints along the way. This second edition contains two additional, more advanced geometric techniques: the modern language and modern view of Algebraic Geometry and Mirror Symmetry.
An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces (2nd Edition) (Theoretical and Mathematical Physics) by
Martin Schlichenmaier
cover the following topics.
1. Manifolds
1.1 Generalities
1.2 Complex Manifolds
1.3 The Classification Problem
Hints for Further Reading
2. Topology of Riemann Surfaces
2.1 Fundamental Group
2.2 Simplicial Homology
2.3 Universal Covering Space
Hints for Further Reading
3. Analytic Structure
3.1 Holomorphic and Meromorphic Functions
3.2 Divisors and the Theorem of Riemann–Roch
3.3 Meromorphic Functions on the Torus
Hints for Further Reading
4. Differentials and Integration
4.1 Tangent Space and Differentials
4.2 Differential Forms of Second Order
4.3 Integration
Hints for Further Reading
5. Tori and Jacobians
5.1 Higher Dimensional Tori
5.2 Jacobians
Hints for Further Reading
6. Projective Varieties
6.1 Generalities
6.2 Embedding of One-Dimensional Tori
6.3 Theta Functions
Hints for Further Reading
7. Moduli Spaces of Curves
7.1 The Definition
7.2 Methods of Construction
7.3 The Geometry of the Moduli Space and Its Compactification
Hints for Further Reading
8. Vector Bundles, Sheaves And Cohomology
8.1 Vector Bundles
8.2 Sheaves
8.3 Cohomology
Hints for Further Reading
9. The Theorem of Riemann–Roch for Line Bundles
9.1 Divisors and Line Bundles
9.2 An Application: The Krichever–Novikov Algebra
Hints for Further Reading
10. The Mumford Isomorphism on the Moduli Space
10.1 The Mumford Isomorphism
10.2 The Grothendieck–Riemann–Roch Theorem
Hints for Further Reading
11. Modern Algebraic Geometry
11.1 Varieties
11.2 The Spectrum of a Ring
11.3 Homomorphisms
11.4 Noncommutative Spaces
Hints for Further Reading
12. Schemes
12.1 Affine Schemes
12.2 General Schemes
12.3 The Structure Sheaf OR
12.4 Examples of Schemes
Hints for Further Reading
13. Hodge Decomposition and K¨ahler Manifold
13.1 Some Introductory Remarks on Mirror Symmetry
13.2 Compact Complex Manifolds and Hodge Decomposition
13.3 K¨ahler Manifolds
13.4 Hodge Numbers of the Projective Space
Hints for Further Reading
14. Calabi-Yau Manifolds and Mirror Symmetry
14.1 Calabi-Yau Manifolds
14.2 K3 Surfaces, Hypersurfaces and Complete Intersections
14.3 Geometric Mirror Symmetry
14.4 Example of a Calabi-Yau Three-fold and Its Mirror: Results of Givental
Hints for Further Reading
Appendix p-adic Numbers
Index
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