Isotopic embeddings of affine algebraic varieties into \(\mathbb{C}^ n\). (English) Zbl 0768.14005
Complex analysis, Proc. Symp., Madison/WI (USA) 1992, Contemp. Math. 137, 291-295 (1992).
[For the entire collection see Zbl 0755.00016.]
Let \(X\) be a closed algebraic subvariety of \(\mathbb{C}^ m\). Consider two proper algebraic embeddings \(f,g:X\to\mathbb{C}^ n\), where \(n\geq\max(1+2\dim X,\dim TX)\).
The main result says that \(f,g\) are isotopic via proper algebraic embeddings. If \(X\) is non-singular, the estimate \(n\geq 2\dim X+1\) cannot be improved. The author also proves that for every algebraic embedding \(f:\mathbb{C}\to\mathbb{C}^ n\) there exists a holomorphic automorphism \(\alpha\) of \(\mathbb{C}^ n\) such that \(\alpha\circ f\) is a linear embedding, and that for every \(n\geq 3\) it is not true for \(f\) being a proper holomorphic embedding.
In this way the author improves a result of Nori and disproves a conjecture by Bell and Narasimhan.
The paper is a continuation of the author’s previous work [Proc. Am. Math. Soc. 113, No. 2, 325-334 (1991; Zbl 0743.14011)].
Let \(X\) be a closed algebraic subvariety of \(\mathbb{C}^ m\). Consider two proper algebraic embeddings \(f,g:X\to\mathbb{C}^ n\), where \(n\geq\max(1+2\dim X,\dim TX)\).
The main result says that \(f,g\) are isotopic via proper algebraic embeddings. If \(X\) is non-singular, the estimate \(n\geq 2\dim X+1\) cannot be improved. The author also proves that for every algebraic embedding \(f:\mathbb{C}\to\mathbb{C}^ n\) there exists a holomorphic automorphism \(\alpha\) of \(\mathbb{C}^ n\) such that \(\alpha\circ f\) is a linear embedding, and that for every \(n\geq 3\) it is not true for \(f\) being a proper holomorphic embedding.
In this way the author improves a result of Nori and disproves a conjecture by Bell and Narasimhan.
The paper is a continuation of the author’s previous work [Proc. Am. Math. Soc. 113, No. 2, 325-334 (1991; Zbl 0743.14011)].
Reviewer: K.Dabrowski (Piastow)
MSC:
| 14E25 | Embeddings in algebraic geometry |
| 32H02 | Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables |
| 14F35 | Homotopy theory and fundamental groups in algebraic geometry |
| 32C25 | Analytic subsets and submanifolds |
| 14H37 | Automorphisms of curves |
