Algebraic Geometry over the Complex Numbers

This is a relatively fast paced graduate level introduction to complex algebraic geometry, from the basics to the frontier of the subject. It covers sheaf theory, cohomology, some Hodge theory, as well as some of the more algebraic aspects of algebraic geometry. The author frequently refers the reader if the treatment of a certain topic is readily available elsewhere but goes into considerable detail on topics for which his treatment puts a twist or a more transparent viewpoint. His cases of exploration and are chosen very carefully and deliberately. The textbook achieves its purpose of taking new students of complex algebraic geometry through this a deep yet broad introduction to a vast subject, eventually bringing them to the forefront of the topic via a non-intimidating style.

Preface.- 1. Plane Curves.- 2. Manifolds and Varieties via Sheaves.- 3. More Sheaf Theory.- 4. Sheaf Cohomology.- 5. de Rham Cohomoloy of Manifolds.- 6. Riemann Surfaces.- 7. Simplicial Methods.- 8. The Hodge Theorem for Riemann Manifolds.- 9. Toward Hodge Theory for Complex Manifolds.- 10. Kahler Manifolds.- 11. A Little Algebraic Surface Theory.- 12. Hodge Structures and Homological Methods.- 13. Topology of Families.- 14. The Hard Lefschez Theorem.- 15. Coherent Sheaves.- 16. Computation of Coherent Sheaves.- 17. Computation of some Hodge numbers.- 18. Deformation Invariance of Hodge Numbers.- 19. Analogies and Conjectures.- References.- Index.