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On equivariant Serre problem for principal bundles. (English) Zbl 1394.14026

Summary: Let \(E_G\) be a \(\Gamma\)-equivariant algebraic principal \(G\)-bundle over a normal complex affine variety \(X\) equipped with an action of \(\Gamma\), where \(G\) and \(\Gamma\) are complex linear algebraic groups. Suppose \(X\) is contractible as a topological \(\Gamma\)-space with a dense orbit, and \(x_0 \in X\) is a \(\Gamma\)-fixed point. We show that if \(\Gamma\) is reductive, then \(E_G\) admits a \(\Gamma\)-equivariant isomorphism with the product principal \(G\)-bundle \(X \times_\rho E_G(x_0)\), where \(\rho : \Gamma \rightarrow G\) is a homomorphism between algebraic groups. As a consequence, any torus equivariant principal \(G\)-bundle over an affine toric variety is equivariantly trivial. This leads to a classification of torus equivariant principal \(G\)-bundles over any complex toric variety, generalizing the main result of the author et al. [ibid. 27, No. 14, Article ID 1650115, 16 p. (2016; Zbl 1360.32014)].

MSC:

14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
32L05 Holomorphic bundles and generalizations
14M25 Toric varieties, Newton polyhedra, Okounkov bodies

Citations:

Zbl 1360.32014
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References:

[1] Atiyah, M. F.; Macdonald, I. G., Introduction to Commutative Algebra, (1969), Addison-Wesley Publishing, Reading · Zbl 0175.03601
[2] Biswas, I.; Dey, A.; Poddar, M., A classification of equivariant principal bundles over nonsingular toric varieties, Internat. J. Math., 27, 14, 16, (2016) · Zbl 1360.32014
[3] Dieck, T. T., Transformation Groups, 8, 312, (1987), Walter de Gruyter, Berlin
[4] Eisenbud, D., Commutative algebra with a view toward algebraic geometry, 150, (1995), Springer-Verlag, New York · Zbl 0819.13001
[5] Fulton, W., Introduction to Toric Varieties, 131, (1993), Princeton University Press, Princeton
[6] J. Gubeladze, The Anderson conjecture and a maximal class of monoids over which projective modules are free, Mat. Sb. (N.S.)135(177)(2) (1988) 169-185 (in Russian); Math. USSR-Sb.63(1) (1989) 165-180. · Zbl 0654.13013
[7] Heinzner, P.; Kutzschebauch, F., An equivariant version of grauert’s Oka principle, Invent. Math., 119, 317-346, (1995) · Zbl 0837.32004
[8] Kaneyama, T., On equivariant vector bundles on an almost homogeneous variety, Nagoya Math. J., 57, 65-86, (1975) · Zbl 0283.14008
[9] A. A. Klyachko, Equivariant bundles over toric varieties, Izv. Akad. Nauk SSSR Ser. Mat.53 (1989) 1001-1039, 1135 (in Russian); Math. USSR-Izv.35(2) (1990) 337-375.
[10] Kutzschebauch, F.; Lárusson, F.; Schwarz, G. W., An Oka principle for equivariant isomorphisms, J. Reine Angew. Math., 706, 193-214, (2015) · Zbl 1331.32004
[11] Kutzschebauch, F.; Lárusson, F.; Schwarz, G. W., Homotopy principles for equivariant isomorphisms, Trans. Amer. Math. Soc., 369, 10, 7251-7300, (2017) · Zbl 1394.32016
[12] Kutzschebauch, F.; Lárusson, F.; Schwarz, G. W., Sufficient conditions for holomorphic linearisation, Transform. Groups, 22, 2, 475-485, (2017) · Zbl 1379.32019
[13] Kutzschebauch, F.; Lárusson, F.; Schwarz, G. W., An equivariant parametric Oka principle for bundles of homogeneous spaces, Math. Ann., 370, 1-2, 819-839, (2018) · Zbl 1395.32015
[14] Lashof, R. K., Equivariant bundles, Illinois J. Math., 26, 2, 257-271, (1982) · Zbl 0458.55009
[15] Masuda, M., Transformation Group Theory, Equivariant algebraic vector bundles over affine toric varieties, 76-84, (1997), Korea Advanced Institute of Science and Technology, Taejon
[16] Masuda, M.; Moser-Jauslin, L.; Petrie, T., The equivariant Serre problem for abelian groups, Topology, 35, 329-334, (1996) · Zbl 0884.14007
[17] Masuda, M.; Petrie, T., Stably trivial equivariant algebraic vector bundles, J. Amer. Math. Soc., 8, 3, 687-714, (1995) · Zbl 0862.14009
[18] Palais, R. S., On the existence of slices for actions of non-compact Lie groups, Ann. Math., 73, 295-323, (1961) · Zbl 0103.01802
[19] Quillen, D., Projective modules over polynomial rings, Invent. Math., 36, 1, 167-171, (1976) · Zbl 0337.13011
[20] Raghunathan, M. S., C. P. Ramanujam-A Tribute, 8, Principal bundles on affine space, 187-206, (1978), Springer, Berlin · Zbl 0417.14037
[21] Schwarz, G. W., Exotic algebraic group actions, C. R. Acad. Sci. Paris, 309, 2, 89-94, (1989) · Zbl 0688.14040
[22] Serre, J.-P., Faisceaux algébriques cohérents, Ann. Math., 61, 197-278, (1955) · Zbl 0067.16201
[23] A. A. Suslin, Projective modules over polynomial rings are free, Dokl. Akad. Nauk SSSR229(5) (1976) 1063-1066 (in Russian), Soviet Math.17(4) (1976) 1160-1164.
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