Analytic extensions of algebraic isomorphisms. (English) Zbl 1331.14058
The paper under review deals with the following question:
Let \(\Phi: X_1 \rightarrow X_2\) be an isomorphism of closed affine algebraic sub-varieties of \(\mathbb{C}^n\). When can \(\Phi\) be extended to an algebraic/holomorphic automorphism of \(\mathbb{C}^n\)?
It was proved earlier by S. Kaliman [Proc. Am. Math. Soc. 113, No. 2, 325-334 (1991; Zbl 0743.14011)] and V. Srinivas [Math. Ann. 289, 125–132 (1991; Zbl 0725.14003)] that \(\Phi\) can be extended to an algebraic automorphism if \(n > \max(2\dim(X_1)+1, \dim(TX_1))\).
The bound on \(\dim(TX_1)\) is optimal. But we have the following open question:
{ Question 1}: Is the term \(2\dim(X_1)+1\) the best possible bound?
To this end the author proves the following
{ Theorem} Let \(\Phi: X_1 \rightarrow X_2\) be an isomorphism of closed affine algebraic sub-varieties of \(\mathbb{C}^n\) and \(n > \max(2\dim(X_1), \dim(TX_1))\). Then \(\Phi\) can be extended to a holomorphic automorphism of \(\mathbb{C}^n\).
The author also mentions the following un-resolved problem:
{ Question 2}: Can an isomorphism of two closed curves in \(\mathbb{C}^3\) be extend to an algebraic automorphism of \(\mathbb{C}^3\)?
The paper under review answers Question 2 in the affirmative with algebraic automorphism replaced by holomorphic automorphism. This is interesting in the sense that it demonstrates the absence of any analytic obstructions in answering Question 2.
The properties of semi-isomorphisms and pseudo-isomorphisms are used in the proofs of the above results.
Let \(\Phi: X_1 \rightarrow X_2\) be an isomorphism of closed affine algebraic sub-varieties of \(\mathbb{C}^n\). When can \(\Phi\) be extended to an algebraic/holomorphic automorphism of \(\mathbb{C}^n\)?
It was proved earlier by S. Kaliman [Proc. Am. Math. Soc. 113, No. 2, 325-334 (1991; Zbl 0743.14011)] and V. Srinivas [Math. Ann. 289, 125–132 (1991; Zbl 0725.14003)] that \(\Phi\) can be extended to an algebraic automorphism if \(n > \max(2\dim(X_1)+1, \dim(TX_1))\).
The bound on \(\dim(TX_1)\) is optimal. But we have the following open question:
{ Question 1}: Is the term \(2\dim(X_1)+1\) the best possible bound?
To this end the author proves the following
{ Theorem} Let \(\Phi: X_1 \rightarrow X_2\) be an isomorphism of closed affine algebraic sub-varieties of \(\mathbb{C}^n\) and \(n > \max(2\dim(X_1), \dim(TX_1))\). Then \(\Phi\) can be extended to a holomorphic automorphism of \(\mathbb{C}^n\).
The author also mentions the following un-resolved problem:
{ Question 2}: Can an isomorphism of two closed curves in \(\mathbb{C}^3\) be extend to an algebraic automorphism of \(\mathbb{C}^3\)?
The paper under review answers Question 2 in the affirmative with algebraic automorphism replaced by holomorphic automorphism. This is interesting in the sense that it demonstrates the absence of any analytic obstructions in answering Question 2.
The properties of semi-isomorphisms and pseudo-isomorphisms are used in the proofs of the above results.
Reviewer: Sagar Kolte (Mumbai)
MSC:
| 14R20 | Group actions on affine varieties |
| 32M17 | Automorphism groups of \(\mathbb{C}^n\) and affine manifolds |
References:
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| [8] | Kaliman, Shulim, Extensions of isomorphisms between affine algebraic subvarieties of \(k^n\) to automorphisms of \(k^n\), Proc. Amer. Math. Soc., 113, 2, 325-334 (1991) · Zbl 0743.14011 · doi:10.2307/2048516 |
| [9] | Kaliman, Shulim, Isotopic embeddings of affine algebraic varieties into \({\bf C}^n\). The Madison Symposium on Complex Analysis, Madison, WI, 1991, Contemp. Math. 137, 291-295 (1992), Amer. Math. Soc., Providence, RI · Zbl 0768.14005 · doi:10.1090/conm/137/1190990 |
| [10] | Srinivas, V., On the embedding dimension of an affine variety, Math. Ann., 289, 1, 125-132 (1991) · Zbl 0725.14003 · doi:10.1007/BF01446563 |
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