Challenging problems on affine $n$-space

Challenging problems on affine

n

-space

Kraft, Hanspeter

Séminaire Bourbaki : volume 1994/95, exposés 790-804, Astérisque, no. 237 (1996), Exposé no. 802, 23 p.

EuDML MR Zbl | 2 citations dans Numdam

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     author = {Kraft, Hanspeter},
     title = {Challenging problems on affine $n$-space},
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     series = {Ast\'erisque},
     note = {talk:802},
     pages = {295--317},
     year = {1996},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {237},
     mrnumber = {1423629},
     zbl = {0892.14003},
     language = {en},
     url = {https://www.numdam.org/item/SB_1994-1995__37__295_0/}
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Kraft, Hanspeter. Challenging problems on affine $n$-space, dans Séminaire Bourbaki : volume 1994/95, exposés 790-804, Astérisque, no. 237 (1996), Exposé no. 802, 23 p.. https://www.numdam.org/item/SB_1994-1995__37__295_0/

Bibliographie
Cité par

[AC95] N. A'Campo, M. Oka - Geometry of plane curves via Tschirnhausen resolution towers. J. Alg. Geometry (1995) (to appear). | Zbl

[AbM75] S. S. Abhyankar, T. T. Moh - Embeddings of the line in the plane. J. Reine Angew. Math. 276 (1975) 149-166. | Zbl | MR | EuDML

[As94] T. Asanuma - Non-linearizable algebraic group actions on $𝔸^{n}$ . J. Algebra 166 (1994) 72-79. | Zbl | DOI

[Ba84] H. Bass - A non-triangular action of $G_{a}$ on $𝔸^{3}$ . J. Pure Appl. Algebra 33 (1984) 1-5. | Zbl | MR | DOI

[BaH85] H. Bass, W. Haboush - Linearizing certain reductive group actions. Trans. Amer. Math. Soc. 292 (1985) 463-482. | Zbl | MR | DOI

[BaH87] H. Bass, W. Haboush - Some equivariant $K$ -theory of affine algebraic group actions. Comm. Algebra 15 (1987) 181-217. | Zbl | MR | DOI

[Bi66] A. Białynicki-Birula - Remarks on the action of an algebraic torus on $k^{n}$ , I. Bull. Acad. Polon. Sci. Sér. Sci. Math. 14 (1966) 177-181. | Zbl

[Bi67] A. Białynicki-Birula - Remarks on the action of an algebraic torus on $k^{n}$ , II. Bull. Acad. Polon. Sci. Sér. Sci. Math. 15 (1967) 123-125. | Zbl

[Bi76] A. Białynicki-Birula - Some properties of the decompositions of algebraic varieties determined by actions of a torus. Bull. Acad. Polon. Sci. Sér. Sci. Math. 24 (1976) 667-674. | Zbl

[CaL70] I. Canals, E. Lluis - Acerca de un resultado de Segre. Anales del Inst. de Metam., Univ. Nac. Aut. México 10 (1970) 1-15. | Zbl | MR

[Da89] W. Danielewski - On the cancellation problem and automorphism group of affine algebraic varieties. Preprint 1989. (Appendix by K. Fieseler)

[Di90] A. Dimca - Hypersurfaces in $ℂ^{n}$ diffeomorphic to $ℝ^{4 n - 2}$ . Max-Planck Institut Bonn, preprint 1990.

[Di92] A. Dimca - Singularities and topology of hypersurfaces. Universitext, Springer Verlag, Berlin-Heidelberg-New York, 1992. | Zbl | MR | DOI

[DMP89] K.-H. Dovermann, M. Masuda, T. Petrie - Fixed point free algebraic actions on varieties diffeomorphic to $ℝ^{n}$ . In: Topological methods in algebraic transformation groups (H. Kraft et al., eds.). Progress in Math. vol. 80, Birkhäuser Verlag, Basel - Boston, 1989, pp. 47-80. | Zbl | MR

[EaH73] P. Eakin, W. Heinzer - A cancellation problem for rings. In: Conference on Commutative Algebra (Brewer, J.W.; Rutter, E.A., eds.). Lecture Notes in Mathematics vol. 311, Springer Verlag, Berlin-Heidelberg-New York, 1973, pp. 61-77. | Zbl | MR | DOI

[Fa95] M. Fankhauser - Fixed points for reductive group actions on acyclic varieties. Ann. Inst. Fourier (1995) to appear. (Thesis Basel 1994) | Numdam | MR | EuDML

[Fi94] K.-H. Fieseler - On complex affine surfaces with $ℂ^{+}$ -action. Comment. Math. Helv. 69 (1994) 5-27. | Zbl | MR | EuDML | DOI

[Fu79] T. Fujita - On Zariski problem. Proc. Japan Acad. Ser. A. Math. Sci. 55 (1979) 106-110. | Zbl | MR | DOI

[Fu81] T. Fujita - Cancellation problem for complete varieties. Invent. Math. 64 (1981) 119-121. | Zbl | MR | EuDML | DOI

[Fu82] T. Fujita - On the topology of non-complete surfaces. J. Fac. Sci. Univ. Tokyo 29 (1982) 503-566. | Zbl | MR

[GiD75] M. H. Gizatullin, V. I. Danilov - Automorphisms of affine surfaces, I. Math. USSR-Izv. 9 (1975) 493-534. | Zbl | MR | DOI

[Gu88] J. Gubeladze - Anderson's conjecture and the maximal monoid class over which projective modules are free. Math. USSR-Sb. 63 (1988) 165-180. | Zbl | MR | DOI

[GuM87] R. V. Gurjar, M. Miyanishi - Affine surfaces with $\bar{κ} \leq 1$ . In: "Algebraic Geometry and Commutative Algebra" in Honor of Masayoshi Nagata., 1987, pp. 99-124. | Zbl

[GuM95] R. V. Gurjar, M. Miyanishi - On contractible curves in $ℂ^{2}$ . Preprint 1995.

[HeK94] P. Heinzner, F. Kutzschebauch - An equivariant version of Grauert's Oka principle. Invent. Math. 119 (1994) 317-346. | Zbl | MR | EuDML | DOI

[Ho72] M. Hochster - Nonuniqueness of coefficient rings in a polynomial ring. Proc. Amer. Math. Soc. 34 (1972) 81-82. | Zbl | MR | DOI

[HsS86] W. Hsiang, E. Straume - Actions of compact connected Lie groups on acyclic manifolds with low dimensional orbit space. J. Reine Angew. Math. 369 (1986) 21-39. | Zbl | MR | EuDML

[Ii82] S. Iitaka - Algebraic Geometry : An introduction to birational geometry of algebraic varieties. Graduate Texts in Mathematics, vol. 76, Springer Verlag, Berlin-Heidelberg-New York, 1982. | Zbl | MR | DOI

[IiF77] S. Iitaka, T. Fujita - Cancellation theorem for algebraic varieties. J. Fac. Sci. Univ. Tokyo 24 (1977) 123-127. | Zbl | MR

[Je87] Z. Jelonek - The extension of regular and rational embeddings. Math. Ann. 113 (1987) 113-120. | Zbl | MR | EuDML | DOI

[Je91] Z. Jelonek - Identity sets for polynomial automorphisms. J. Pure Appl. Algebra 76 (1991) 333-339. | Zbl | MR | DOI

[Je92] Z. Jelonek - The Jacobian Conjecture and the extension of a polynomial embeddings. Math. Annalen 294 (1992) 289-293. | Zbl | MR | EuDML | DOI

[Je93] Z. Jelonek - Irreducible identity sets for polynomial automorphisms. Math. Zeitschrift 212 (1993) 601-617. | Zbl | MR | EuDML | DOI

[Je94] Z. Jelonek - Affine smooth varieties with finite group of automorphisms. Math. Zeitschrift 216 (1994) 575-591. | Zbl | MR | EuDML | DOI

[Ju42] H. W. E. Jung - Über ganze birationale Transformationen der Ebene. J. Reine Angew. Math. 184 (1942) 161-174. | MR | JFM | EuDML

[Ka91] S. Kaliman - Extensions of isomorphisms between affine algebraic subvarieties of $k^{n}$ to automorphisms of $k^{n}$ . Proc. Amer. Math. Soc. 113 (1991) 325-334. | Zbl | MR

[Ka93] S. Kaliman - Smooth contractible hypersurfaces in $ℂ^{n}$ and exotic algebraic structures on $ℂ^{3}$ . Math. Zeitschrift 214 (1993) 499-510. | Zbl | MR | EuDML | DOI

[Ka95] S. Kaliman - Exotic structures on $ℂ^{n}$ and $ℂ^{*}$ -actions on $ℂ^{3}$ . In: Proceedings on "Complex Analysis and Geometry". Lecture Notes in Pure and Appl. Math., Marcel Dekker Inc., 1995 (to appear) | Zbl | MR

[KaM95a] S. Kaliman, L. Makar-Limanov - Affine algebraic manifolds without dominant morphisms from euclidean space. Preprint 1995. | MR | Zbl

[KaM95b] S. Kaliman, L. Makar-Limanov - On Russell's contractible threefolds. Preprint 1995.

[Kam75] T. Kambayashi - On the absence of nontrivial separable forms of the affine plane. J. Algebra 35 (1975) 449-456. | Zbl | MR | DOI

[Kam79] T. Kambayashi - Automorphism group of a polynomial ring and algebraic group action on an affine space. J. Algebra 60 (1979) 439-451. | Zbl | MR | DOI

[Kam80] T. Kambayashi - On Fujita's strong cancellation theorem for the affine plane. J. Fac. Sci. Univ. Tokyo, Sec. IA 27 (1980) 535-548. | Zbl | MR

[KaR82] T. Kambayashi, P. Russell - On linearizing algebraic torus actions. J. Pure Appl. Algebra 23 (1982) 243-250. | Zbl | MR | DOI

[Kn91] F. Knop - Nichtlinearisierbare Operationen halbeinfacher Gruppen auf affinen Räumen. Invent. Math. 105 (1991) 217-220. | Zbl | MR | EuDML | DOI

[KoR86] M. Koras, P. Russell - $G_{m}$ -actions on $𝔸^{3}$ . In: Canad. Math. Soc. Confer. Proc.. vol. 6, 1986, pp. 269-276. | Zbl | MR

[KoR89a] M. Koras, P. Russell - Codimension $2$ torus actions on affine $n$ -space. In: Canad. Math. Soc. Confer. Proc.. Proceedings of the Conference on "Group Actions and Invariant Theory", Montreal 1988 vol. 10, 1989, pp. 103-110. | Zbl | MR

[KoR89b] M. Koras, P. Russell - On linearizing "good" $ℂ^{*}$ -actions on $ℂ^{3}$ . In: Canad. Math. Soc. Confer. Proc.. Proceedings of the Conference on "Group Actions and Invariant Theory" , Montreal 1988 vol. 10, 1989, pp. 93-102. | Zbl | MR

[KoR95] M. Koras, P. Russell - Contractible algebraic threefolds and $ℂ^{*}$ -actions on $ℂ^{3}$ . Preprint 1995. | Zbl

[Kr89a] H. Kraft - Algebraic automorphisms of affine space. In: Topological methods in algebraic transformation groups (H. Kraft et al., eds.). Progress in Math. vol. 80, Birkhäuser Verlag, Basel-Boston, 1989, pp. 81-105. | Zbl | MR

[Kr89b] H. Kraft - $G$ -vector bundles and the linearization problem. In: Proceedings of the Conference on "Group Actions and Invariant Theory" , Montreal 1988. Can. Math. Soc. Conf. Proc. vol. 10, 1989, pp. 111-123. | Zbl | MR

[Kr90] H. Kraft - $ℂ^{*}$ -actions on affine space. In: Operator Algebras, Unitary Representations, Envelopping Algebras, and Invariant Theory (A. Connes, M. Duflo, A. Joseph, R. Rentschler, eds.). Progress in Mathematics vol. 92, Birkhäuser Verlag, Basel-Boston, 1990, pp. 561-579. | Zbl | MR

[KrP85] H. Kraft, V. L. Popov - Semisimple group actions on the three dimensional affine space are linear. Comment. Math. Helv. 60 (1985) 466-479. | Zbl | MR | EuDML | DOI

[KrS89] H. Kraft, G. W. Schwarz - Linearizing reductive group actions on affine space with one-dimensional quotient. In: Proceedings of the Conference on "Group Actions and Invariant Theory," Montreal 1988. Contemp. Math. vol. 10, 1989 | Zbl | MR

[KrS92] H. Kraft, G. W. Schwarz - Reductive group actions with one-dimensional quotient. Publ. Math. IHES 76 (1992) 1-97. | Numdam | Zbl | MR | EuDML | DOI

[KrS95] H. Kraft, G. W. Schwarz - Finite automorphisms of affine $n$ -space. In: Proceedings of the Curaçao Conference on "Automorphisms of Affine Space". to appear, 1995 | Zbl | MR | DOI

[Li77] A. Libgober - A geometrical procedure for killing the middle dimensional homology groups of algebraic hypersurfaces. Proc. Amer. Math. Soc. 63 (1977) 198-202. | Zbl | MR | DOI

[Lu73] D. Luna - Slices étales. Bull. Soc. Math. France, Mémoire 33 (1973) 81-105. | Numdam | Zbl | MR | EuDML | DOI

[Ma94] L. Makar-Limanov - On the hypersurface $x + x^{2} y + z^{2} + t^{3} = 0$ in $ℂ^{4}$ . Preprint 1994. | MR

[MMP91] M. Masuda, L. Moser-Jauslin, T. Petrie - Equivariant algebraic vector bundles over representations of reductive groups: Applications. Proc. Natl. Acad. Sci. USA 88 (1991) 9065-9066. | Zbl | MR | DOI

[MMP95] M. Masuda, L. Moser-Jauslin, T. Petrie - The equivariant Serre Problem for abelian groups. J. Amer. Math. Soc (1995). | MR | Zbl

[MaP91] M. Masuda, T. Petrie - Equivariant algebraic vector bundles over representations of reductive groups: Theory. Proc. Natl. Acad. Sci. USA 88 (1991) 9061-9064. | Zbl | MR

[Me95] K. Mederer - Moduli of equivariant $G$ -vector bundles. Thesis Brandeis 1995.

[Mi87] M. Miyanishi - Algebraic characterization of the affine 3-space. Proc. Alg. Geometry Seminar, Singapore, World Scientific, 1987, pp. 53-67. | MR

[MiS80] M. Miyanishi, T. Sugie - Affine surfaces containing cylinderlike open sets. J. Math. Kyoto Univ. 20 (1980) 11-42. | Zbl | MR | DOI

[MiT87] M. Miyanishi, S. Tsunoda - Open algebraic surfaces with Kodaira dimension - $\infty$ . Proc. Sympos. Pure Math. 46 (1987) 435-450. | Zbl | DOI

[Na67] M. Nagata - A theorem on valuation rings and its applications. Nagoya Math. J. 29 (1967) 85-91. | Zbl | MR | DOI

[Ne89a] W. D. Neumann - On the topology of curves in complex surfaces. In: Topological methods in algebraic transformation groups (H. Kraft et al, eds.). Progress in Math. vol. 80, Birkhäuser Verlag, Basel-Boston, 1989, pp. 117-133. | Zbl | MR

[Ne89b] W. D. Neumann - Complex algebraic plane curves via their links at infinity. Invent. math. 98 (1989) 445-489. | Zbl | MR | EuDML | DOI

[Pa84] D. I. Panyushev - Semisimple automorphism groups of four-dimensional affine space. Math. USSR-Izv. 23 (1984) 171-183. | Zbl | DOI

[PeR86] T. Petrie, J. D. Randall - Finite-order algebraic automorphisms of affine varieties. Comment. Math. Helv. 61 (1986) 203-221. | Zbl | MR | EuDML | DOI

[Ra71] C. P. Ramanujam - A topological characterization of the affine plane as an algebraic variety. Ann. of Math. 94 (1971) 69-88. | Zbl | MR | DOI

[Re68] R. Rentschler - Opérations du groupe additif sur le plan affine. C. R. Acad. Sci. Paris 267 A (1968) 384-387. | Zbl | MR

[Rud82] L. Rudolph - Embedding of the line in the plane. J. Reine Angew. Math. 337 (1982) 113-118. | Zbl | MR | EuDML

[Ru81] P. Russell - On affine-ruled rational surfaces. Math. Ann. 255 (1981) 287-302. | Zbl | MR | EuDML | DOI

[Ru92] P. Russell - On a class of $ℂ^{3}$ -like threefolds. Preliminary report 1992.

[Ru95] P. Russell - Open problems on open algebraic varieties (Montreal 1994 Problems). Prépublication de l'Institut Fourier, Laboratoire Mathématiques, n° 311, Grenoble 1995, 23 p.)

[Sa76] A. Sathaye - On linear planes. Proc. Amer. Math. Soc. 56 (1976) 1-7. | Zbl | MR | DOI

[Sc89] G. W. Schwarz - Exotic algebraic group actions. C. R. Acad. Sci. Paris 309 (1989) 89-94. | Zbl | MR

[Seg57] B. Segre - Corrispondenze di Möbius e transformazioni cremoniane intere. Atti Acad. Sci. Torino 91 (1957) 1-17. | Zbl | MR

[Se61] J.-P. Serre - Sur les modules projectifs. In Séminaire Dubreil-Pisot 1960/61, n°2. | Numdam | Zbl | EuDML

[Se80] J.-P. Serre - Trees. Springer Verlag, Berlin-Heidelberg-New York, 1980. | Zbl | MR

[Sh66] I. R. Shafarevich - On some infinite dimensional groups. Rend. Mat. e Appl. (5) 25 (1966) 208-212. | Zbl | MR

[Sr91] V. Srinivas - On the embedding dimension of an affine variety. Math. Annalen 289 (1991) 125-132. | Zbl | MR | EuDML | DOI

[Su89] T. Sugie - Algebraic characterization of the affine plane and the affine 3-space. In: Topological methods in algebraic transformation groups (H. Kraft et al., eds.). Progress in Math. vol. 80, Birkhäuser Verlag, Basel - Boston, 1989, pp. 177-190. | Zbl | MR

[Suz74] M. Suzuki - Propriétés topologiques des polynômes de deux variables complexes, et automorphismes algébriques de l'espace $ℂ^{2}$ . J. Math. Soc. Japan 26 (1974) 241-257. | Zbl | MR | DOI

[tDP89] T. Tom Dieck, T. Petrie - Homology planes: An announcement and survey. In: Topological methods in algebraic transformation groups (H. Kraft et al., eds.). Progress in Math. vol. 80, Birkhäuser Verlag, Basel - Boston, 1989, pp. 27-48. | Zbl | MR

[tDP90] T. Tom Dieck, T. Petrie - Contractible affine surfaces of Kodaira dimension one. Japan J. Math. 16 (1990) 147-169. | Zbl | MR | DOI

[Ku53] W. Van Der Kulk - On polynomial rings in two variables. Nieuw Arch. Wisk. 1 (1953) 33-41. | Zbl | MR

[Wi90] J. Winkelmann - On free holomorphic $ℂ$ -actions on $ℂ^{n}$ and homogeneous Stein manifolds. Math. Ann. 286 (1990) 593-612. | Zbl | MR | EuDML | DOI

[Wr79] D. Wright - Abelian subgroups of ${Aut}_{k} (k [X, Y])$ and applications to actions on the affine plane. Ill. J. Math. 23 (1979) 579-634. | Zbl | MR

[Za91] M. G. Zaidenberg - Ramanujam surfaces and exotic algebraic structures on $ℂ^{n}, n \geq 3$ . Soviet Math. Dokl. 42 (1991) 636-640. | Zbl

[Za93] M. G. Zaidenberg - An analytic cancellation theorem and exotic algebraic structures on $ℂ^{n}, n \geq 3$ . In: Astérisque 217 (1993), 251-282., 1993 (Max-Planck-Institut für Mathematik, Bonn, 1991. Preprint MPI/91-26, 33p.) | Zbl

[Za95] M. G. Zaidenberg - On exotic algebraic structures on affine spaces. In: Proceedings of a Conference "Geometric Complex Analysis". Tokyo, March 19-29, World Scientific, 1995 (Prépublication de l'Institut Fourier, Laboratoire de Mathématiques, n° 309, Grenoble 1995, 32 p.) | Zbl | MR

[ZaL83] M. G. Zaidenberg, V. Ya. Lin - An irreducible simply connected algebraic curve in $ℂ^{2}$ is a quasihomogeneous curve. Soviet Math. Dokl. 28 (1983). | Zbl