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More about usAn 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.

1.1 Generalities

1.2 Complex Manifolds

1.3 The Classification Problem

Hints for Further Reading

2.1 Fundamental Group

2.2 Simplicial Homology

2.3 Universal Covering Space

Hints for Further Reading

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

4.1 Tangent Space and Differentials

4.2 Differential Forms of Second Order

4.3 Integration

Hints for Further Reading

5.1 Higher Dimensional Tori

5.2 Jacobians

Hints for Further Reading

6.1 Generalities

6.2 Embedding of One-Dimensional Tori

6.3 Theta Functions

Hints for Further Reading

7.1 The Definition

7.2 Methods of Construction

7.3 The Geometry of the Moduli Space and Its Compactification

Hints for Further Reading

8.1 Vector Bundles

8.2 Sheaves

8.3 Cohomology

Hints for Further Reading

9.1 Divisors and Line Bundles

9.2 An Application: The Krichever–Novikov Algebra

Hints for Further Reading

10.1 The Mumford Isomorphism

10.2 The Grothendieck–Riemann–Roch Theorem

Hints for Further Reading

11.1 Varieties

11.2 The Spectrum of a Ring

11.3 Homomorphisms

11.4 Noncommutative Spaces

Hints for Further Reading

12.1 Affine Schemes

12.2 General Schemes

12.3 The Structure Sheaf OR

12.4 Examples of Schemes

Hints for Further Reading

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

14.1 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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