# What´s essential to learn about complex spaces and several complex variables for an algebraic geometer?

Hi, I don´t know if this question is suitable for this site. The field of several complex variables is too broad, so I would like to know what´s essential to learn about complex spaces and several complex variables for an algebraic geometer? Any references?

• Griffiths-Harris: "Principles of algebraic geometry", Voisin: "Hodge theory and complex algebraic geometry", Huybrechts: "Complex geometry". The material covered in these books is more than enough in order to get started. – Francesco Polizzi Jun 10 '13 at 13:45
• Just this Francesco? :p – diverietti Jun 10 '13 at 21:20
• I said "to get started" :-) – Francesco Polizzi Jun 11 '13 at 8:36
• Instead of Huybrechts: "Complex geometry". it think chern's complex manifolds without potential theory is where he copies from most of the stuff! – Koushik Jul 12 '14 at 7:58

The basic yoga of positivity in complex geometry is that ampleness of a line bundle $L$ is equivalent to the positivity of the curvature form of a smooth hermitian metric on $L$. This allows us to treat global algebraic questions involving ampleness and cohomology by looking at pointwise estimates of positive differential forms on our manifold. Once there, all of the machinery of Riemannian and complex geometry is available and hard global questions get converted into extremely computationally messy problems of linear algebra. For certain things, like cohomology of adjoint bundles $K_X \otimes L$, these methods work very well, for others they work less well or not at all, but it's always good to have another tool with which to attack problems.