Griffiths and Harris, Principles of Algebraic Geometry, Wiley Also recommended: Cornalba et al, Lectures on Riemann Surfaces, World Scientific Griffiths, Lectures on Algebraic Curves, AMS

1. Holomorphic functions in one and several variables. Definition of a complex manifold. Canonical splitting of the complexified tangent and cotangent space. T R ⊂ T ⊗ C = T ′ ⊕ T ′′ , where T R is the real tangent space at a point, T ′′ = ∂/∂z i annihilates holomorphic functions and T ′ = ∂/∂z i annihilates antiholomorphic functions. On C, Df (w) = ∂f ∂z w + ∂f ∂z w. Quasiconformal maps. Stone-Weierstrass theorem: a continuous function can be approximated by a polynomial in z and z.