Seminar Algebra and Geometry: Immanuel Stampfli (Jacobs University, Bremen)

Holomorphically Equivalent Algebraic Embeddings

This is joint work with Peter Feller (University of Bern). Let X be a smooth affine algebraic variety. A natural question is, whether two algebraic embeddings f, g : X → C^m are algebraically equivalent, i.e. whether there exists an algebraic automorphism φ of C^m such that φ◦f=g. Kaliman, Nori and Srinivas gave an affirmative answer, provided that 2 dimX+ 2≤m. In this talk we discuss the following one-dimensional improvement under a relaxed equivalence condition.

Theorem. If f, g : X → C^m are algebraic embeddings and 2 dimX+ 1≤m, then there exists a holomorphic automorphism φ of C^m such that φ◦f=g.

In fact, the proof is based on an idea of Kaliman, with which he proved that two algebraic embeddings of C into C³ are holomorphically equivalent. In the course of this talk, we discuss this idea. Moreover, we provide examples of algebraic embeddings into C^m that are holomorphically non-equivalent.

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