Analytic extensions of algebraic isomorphisms

Kaliman, S.

doi:10.1090/proc/12684

Analytic extensions of algebraic isomorphisms
HTML articles powered by AMS MathViewer

by S. Kaliman PDF

Proc. Amer. Math. Soc. 143 (2015), 4571-4581 Request permission

Abstract:

Let $\Psi : X_1 \to X_2$ be an isomorphism of closed affine algebraic subvarieties of $\mathbb {C}^n$ such that $n > \max (2\dim X_1, \dim TX_1)$. We prove that $\Psi$ can be extended to a holomorphic automorphism of $\mathbb {C}^n$. Furthermore, when $\Psi$ is an isomorphism of curves, such an extension exists for every $n\geq 3$ even when $\dim TX_1=n$.

References

I. Arzhantsev, H. Flenner, S. Kaliman, F. Kutzschebauch, and M. Zaidenberg, Flexible varieties and automorphism groups, Duke Math. J. 162 (2013), no. 4, 767–823. MR 3039680, DOI 10.1215/00127094-2080132
Harm Derksen, Frank Kutzschebauch, and Jörg Winkelmann, Subvarieties of $\textbf {C}^n$ with non-extendable automorphisms, J. Reine Angew. Math. 508 (1999), 213–235. MR 1676877, DOI 10.1515/crll.1999.028
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
H. Flenner, S. Kaliman, M. Zaidenberg, A Gromov-Winkelmann type theorem for flexible varieites, arXiv:1305.6417, MPI preprint 2013-30, (to appear in J. of Eur. Math. Soc.).
Franc Forstnerič, Holomorphic flexibility properties of complex manifolds, Amer. J. Math. 128 (2006), no. 1, 239–270. MR 2197073
Franc Forstneric, Josip Globevnik, and Jean-Pierre Rosay, Nonstraightenable complex lines in $\mathbf C^2$, Ark. Mat. 34 (1996), no. 1, 97–101. MR 1396625, DOI 10.1007/BF02559509
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
Shulim Kaliman, Extensions of isomorphisms between affine algebraic subvarieties of $k^n$ to automorphisms of $k^n$, Proc. Amer. Math. Soc. 113 (1991), no. 2, 325–334. MR 1076575, DOI 10.1090/S0002-9939-1991-1076575-3
Shulim Kaliman, Isotopic embeddings of affine algebraic varieties into $\textbf {C}^n$, The Madison Symposium on Complex Analysis (Madison, WI, 1991) Contemp. Math., vol. 137, Amer. Math. Soc., Providence, RI, 1992, pp. 291–295. MR 1190990, DOI 10.1090/conm/137/1190990
V. Srinivas, On the embedding dimension of an affine variety, Math. Ann. 289 (1991), no. 1, 125–132. MR 1087241, DOI 10.1007/BF01446563