# Affine varieties as Stein surfaces

I have a somewhat general and vague question in mind. Is there anything in literature related to Affine varieties as examples of Stein manifolds? I know that there is a topological approach to Stein manifolds mainly using Eliashberg's characterisation. I want to know if there are other approaches using a bit or more of Algebraic Geometry (perhaps Lefschetz hyperplane theorem?).

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Affine varieties over complex numbers are always Stein spaces. –  Mohan Ramachandran May 5 at 0:38
Yes, there is a very effective approach using coherent analytic sheaves. The book "Theory of Stein Spaces" explains things rather well, including various ways of characterizing Stein spaces (cohomological, function-theoretic, etc.). The criteria apply to $\mathbf{C}^n$, and since the Stein property is inherited under closed immersions, you get that the analytifications of all affine algebraic $\mathbf{C}$-schemes are Stein. This has nothing to do with the Lefschetz hyperplane theorem, by the way. –  user29283 May 5 at 0:48
@ Mohan Ramachandran: I can see that I have written: "Affine varieties as examples of Stein manifolds", it is good idea to read the question carefully before making any comment. –  7779052 May 5 at 1:52
@kaavek. Varieties can be singular. –  Mohan Ramachandran May 5 at 2:20
@kaavek: There are many tools that are useful in studying Stein spaces (allowing for singularities is convenient for proving general theorems about even Stein manifolds, to say nothing of the fact that varieties can be singular). The irrelevance of the Lefschetz hyperplane theorem for addressing your question above isn't incompatible with the usefulness of that theorem for other purposes in the study of Stein manifolds. (Coherent analytic sheaves are very useful in complex-analytic geometry beyond Stein spaces, by the way.) –  user29283 May 5 at 2:34
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