Holomorphically Equivalent Algebraic Embeddings

TL;DR

This paper proves that under certain dimension conditions, two algebraic embeddings of a smooth variety into complex space are equivalent via a holomorphic change of coordinates, extending previous algebraic results.

Contribution

It extends the algebraic equivalence of embeddings to a holomorphic setting for varieties with specific dimension constraints.

Findings

Two algebraic embeddings are holomorphically equivalent if 2*dim(X)+1 ≤ m.

The proof extends Kaliman's technique using generic linear projections.

Improves upon previous algebraic embedding equivalence results.

Abstract

We prove that two algebraic embeddings of a smooth variety $X$ in $\mathbb{C}^m$ are the same up to a holomorphic coordinate change, provided that $2 \dim X + 1$ is smaller than or equal to $m$. This improves an algebraic result of Nori and Srinivas. For the proof we extend a technique of Kaliman using generic linear projections of $\mathbb{C}^m$.