On criteria of algebraicity of analytic sets and algebraic embeddings of complex manifolds

The talk is devoted to classical criteria for the algebraicity of analytic subsets in complex Euclidean spaces and to conditions ensuring algebraic embeddability of complex manifolds. Special emphasis is placed on the local criterion of Sadullaev expressed in terms of the Siciak—Zahariuta extremal function from pluripotential theory. As an application of this result, we obtain a new criterion for the algebraic embeddability of parabolic Stein manifolds.