Andrei Moroianu is a Researcher at CNRS and a Professor of Mathematics at Ecole Polytechnique.

• Introduction; Part I. Basics on Differential Geometry: 1. Smooth manifolds; 2. Tensor fields on smooth manifolds; 3. The exterior derivative; 4. Principal and vector bundles; 5. Connections; 6. Riemannian manifolds; Part II. Complex and Hermitian Geometry: 7. Complex structures and holomorphic maps; 8. Holomorphic forms and vector fields; 9. Complex and holomorphic vector bundles; 10. Hermitian bundles; 11. Hermitian and Kähler metrics; 12. The curvature tensor of Kähler manifolds; 13. Examples of Kähler metrics; 14. Natural operators on Riemannian and Kähler manifolds; 15. Hodge and Dolbeault theory; Part III. Topics on Compact Kähler Manifolds: 16. Chern classes; 17. The Ricci form of Kähler manifolds; 18. The Calabi–Yau theorem; 19. Kähler–Einstein metrics; 20. Weitzenböck techniques; 21. The Hirzebruch–Riemann–Roch formula; 22. Further vanishing results; 23. Ricci–flat Kähler metrics; 24. Explicit examples of Calabi–Yau manifolds; Bibliography; Index.

"A concise and well-written modern introduction to the subject." Tatyana E. Foth, Mathematical Reviews