Algebraic geometry over the complex numbers

The book covers basic complex algebraic geometry. Here is the basic outline
  1. Plane curves
  2. Manifolds and varieties via sheaves
  3. More sheaf theory
  4. Sheaf cohomology
  5. De Rham cohomology of manifolds
  6. Riemann surfaces
  7. Simplicial methods
  8. The Hodge theorem for Riemannian 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 Lefschetz theorem
  15. Coherent sheaves
  16. Cohomology of coherent sheaves
  17. Computations of some Hodge numbers
  18. Deformations and Hodge theory
  19. Analogies and conjectures
Further details can be found at official website for the book at Springer.

Errata

My thanks to Lizhen Qin for the first comment, and Sandor Kovacs for the rest.
  1. p. 164. l.10 from the bottom: The definition of the approximate heat kernel \(K_0\) is wrong as written. There should be an extra cutoff factor: \[K_0(x,y,t) = (4 \pi t)^{-n/2} e^{-\delta(x,y)^2/4t}\delta(x,y)^2\] so that it dies off away from the diagonal.
  2. p.180, l.1: the 2nd and 4th _i should be _j
  3. p.180, l.4: at the end of the line "(X)" should be "_X"
  4. p.184, l.13: the third \(\partial\) should be \(\bar\partial\) and the third \(\bar\partial\) should be \(\partial\)
  5. p.185, displayed line in 10.2.6: second "q" should be "p", first "_Y" should be "_X"
  6. p.204: in the definition of a pure Hodge structure there is no \(H_{\mathbb C}\), just an H. Then in two displayed equations on this page it appears.
  7. p.204, l.18: \(H^{pq}\) should be \(H^{p'q}\)
  8. pp.203-205: There is some inconsistency about the symbol H. It should stand for the complex vector space and \(H_{ \mathbb{Z}}\) for the lattice.
  9. p.205, l.5: in the definition of a direct sum of Hodge structures. There shouldn't be a sum on the right side.
  10. p.208, l.12: \(\mathcal{H}\) should be regular \(H\).
  11. p.208, l.13-14: "I think here the superscript of the curly H''s should be p+q instead of q, because the Gr^p's are supported at the p^th term, so you get a shift in cohomology."
  12. p.208, end of proof of 12.2.4: something like "by Lemma 12.1.1" should be added.
  13. p.214,l.1: ^p should be ^i
  14. p 215, 12.4.6: it should read "\(< \infty\)"
  15. p 221, l4, should be "Carlson"