Ebook: 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.

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Part 1. Introduction through Examples -- Plane Curves -- Part 2. Sheaves and Geometry -- Manifolds and Varieties via Sheaves -- More Sheaf Theory -- Sheaf Cohomology -- De Rham Cohomology of Manifolds -- Riemann Surfaces -- Simplicial Methods -- Part 3. Hodge Theory -- The Hodge Theorem for Riemannian Manifolds -- Toward Hodge Theory for Complex Manifolds -- Kähler Manifolds -- A Little Algebraic Surface Theory -- Hodge Structures and Homological Methods -- Topology of Families -- The Hard Lefschetz Theorem -- Part 4. Coherent Cohomology -- Coherent Sheaves -- Cohomology of Coherent Sheaves -- Computation of Some Hodge Numbers -- Deformations and Hodge Theory -- Part 5. Analogies and Conjectures -- Analogies and Conjectures