Deformations of \(\mathbb A^1\)-fibrations. (English) Zbl 1326.14146
Cheltsov, Ivan (ed.) et al., Automorphisms in birational and affine geometry. Papers based on the presentations at the conference, Levico Terme, Italy, October 29 – November 3, 2012. Cham: Springer (ISBN 978-3-319-05680-7/hbk; 978-3-319-05681-4/ebook). Springer Proceedings in Mathematics & Statistics 79, 327-361 (2014).
Summary: Let \(B\) be an integral domain which is finitely generated over a subdomain \(R\) and let \(D\) be an \(R\)-derivation on \(B\) such that the induced derivation \(D_{\mathfrak m}\) on \(B\otimes_RR/\mathfrak m\) is locally nilpotent for every maximal ideal \(\mathfrak m\). We ask if \(D\) is locally nilpotent. Theorem 2.1 asserts that this is the case if \(B\) and \(R\) are affine domains. We next generalize the case of \(G_a\)-action treated in Theorem 2.1 to the case of \(\mathbb A^1\)-fibrations and consider the log deformations of affine surfaces with \(\mathbb A^1\)-fibrations. The case of \(\mathbb A^1\)-fibrations of affine type behaves nicely under log deformations, while the case of \(\mathbb A^1\)-fibrations of complete type is more involved (see [A. Dubouloz and T. Kishimoto, Bull. Soc. Math. Fr. 143, No. 2, 383–401 (2015; Zbl 1327.14196)]. As a corollary, we prove the generic triviality of \(\mathbb A^2\)-fibration over a curve and generalize this result to the case of affine pseudo-planes of \(\mathrm{ML}_0\)-type under a suitable monodromy condition.
For the entire collection see [Zbl 1291.14005].
For the entire collection see [Zbl 1291.14005].
Citations:
Zbl 1327.14196References:
| [1] | G. Bredon, \(Topology and Geometry\). Graduate Texts in Mathematics, vol. 139 (Springer, New York, 1993) |
| [2] | C.H. Clemens, P.A. Griffiths, The intermediate Jacobian of the cubic threefold. Ann. Math. 95(2), 281-356 (1972) · Zbl 0214.48302 · doi:10.2307/1970801 |
| [3] | A. Dubouloz, T. Kishimoto, Log-uniruled affine varieties without cylinder-like open subsets. arXiv: 1212.0521 (2012) · Zbl 1327.14196 |
| [4] | T. Fujita, On the topology of noncomplete algebraic surfaces. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29(3), 503-566 (1982) · Zbl 0513.14018 |
| [5] | H. Flenner, M. Zaidenberg, |
| [6] | H. Flenner, Sh. Kaliman, M. Zaidenberg, Deformation equivalence of affine ruled surfaces. arXiv:1305.5366v1, 23 May 2013 |
| [7] | A. Grothendieck, Techniques de construction et théorèmes d’existence en géométrie algébrique IV: Les schémas de Hilbert, Séminaire Bourbaki, 13e année, 1960/61, no. 221, 1-28 |
| [8] | R.V. Gurjar, K. Masuda, M. Miyanishi, |
| [9] | R.V. Gurjar, K. Masuda, M. Miyanishi, Surjective derivations in small dimensions. J. Ramanujan Math. Soc. 28A(Special Issue), 221-246 (2013) · Zbl 1327.13094 |
| [10] | R.V. Gurjar, K. Masuda, M. Miyanishi, P. Russell, Affine lines on affine surfaces and the Makar-Limanov invariant. Can. J. Math. 60(1), 109-139 (2008) · Zbl 1137.14049 · doi:10.4153/CJM-2008-005-8 |
| [11] | R. Hartshorne, \(Algebraic Geometry\) (Springer, New York, 1977) · Zbl 0367.14001 · doi:10.1007/978-1-4757-3849-0 |
| [12] | S. Iitaka, \(Deformations of Compact Complex Surfaces\). Global Analysis (Papers in Honor of K. Kodaira) (Univ. Tokyo Press, Tokyo, 1969), pp. 267-272 |
| [13] | T. Kambayashi, On the absence of nontrivial separable forms of the affine plane. J. Algebra 35, 449-456 (1975) · Zbl 0309.14029 · doi:10.1016/0021-8693(75)90058-7 |
| [14] | T. Kambayashi, M. Miyanishi, On flat fibrations by the affine line. Illinois J. Math. 22, 662-671 (1978) · Zbl 0406.14012 |
| [15] | T. Kambayashi, D. Wright, Flat families of affine lines are affine line bundles. Illinois J. Math. 29, 672-681 (1985) · Zbl 0599.14010 |
| [16] | Sh. Kaliman, M. Zaidenberg, Families of affine planes: the existence of a cylinder. Michigan Math. J. 49(2), 353-367 (2001) · Zbl 1071.14522 · doi:10.1307/mmj/1008719778 |
| [17] | Y. Kawamata, On deformations of compactifiable complex manifolds. Math. Ann. 235(3), 247-265 (1978) · Zbl 0363.32015 · doi:10.1007/BF01420124 |
| [18] | Y. Kawamata, On the classification of non-complete algebraic surfaces, in \(Algebraic Geometry\) (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978). Lecture Notes in Mathematics, vol. 732 (Springer, Berlin, 1979), pp. 215-232 |
| [19] | K. Kodaira, On the stability of compact submanifolds of complex manifolds. Am. J. Math. 85, 79-94 (1963) · Zbl 0173.33101 · doi:10.2307/2373187 |
| [20] | K. Kodaira, \(Complex Manifolds and Deformation of Complex Structures\). Classics in Mathematics (Springer, Berlin, 1986) · Zbl 0581.32012 |
| [21] | J. Kollar, \(Rational Curves on Algebraic Varieties\). Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 32 (Springer, Berlin, 1996) |
| [22] | J. Kollar, S. Mori, \(Birational Geometry of Algebraic Varieties\). Cambridge Tracts in Mathematics, vol. 134 (Cambridge University Press, Cambridge, 1998) |
| [23] | M. Miyanishi, An algebro-topological characterization of the affine space of dimension three. Am. J. Math. 106, 1469-1486 (1984) · Zbl 0595.14025 · doi:10.2307/2374401 |
| [24] | M. Miyanishi, Affine pseudo-coverings of algebraic surfaces. J. Algebra 294, 156-176 (2005) · Zbl 1086.14051 · doi:10.1016/j.jalgebra.2005.01.042 |
| [25] | J.A. Morrow, Minimal normal compactifications of · Zbl 0277.32019 |
| [26] | W.D. Neumann, On the topology of curves in complex surfaces, in \(Topological Methods in Algebraic Transformation Groups\). Progress in Mathematics, vol. 80 (Birkhauser, Boston, 1989), pp. 117-133 |
| [27] | C.P. Ramanujam, A topological characterization of the affine plane as an algebraic variety. Ann. Math. 94, 69-88 (1971) · Zbl 0218.14021 · doi:10.2307/1970735 |
| [28] | P. Russell, Embeddings problems in affine algebraic geometry, in \(Polynomial Automorphisms and Related Topics\) (Publishing House for Science and Technology, Hanoi, 2007), pp. 113-135 |
| [29] | A. Sathaye, Polynomial ring in two variables over a DVR: a criterion. Invent. Math. 74(1), 159-168 (1983) · Zbl 0538.13006 · doi:10.1007/BF01388536 |
| [30] | R.O. Wells, \(Differential Analysis on Complex Manifolds\). Graduate Texts in Mathematics, vol. 65 (Springer, New York, 2008) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
