A direct proof that toric rank 2 bundles on projective space split
DOI:
https://doi.org/10.7146/math.scand.a-121452Abstract
The point of this paper is to give a short, direct proof that rank 2 toric vector bundles on n-dimensional projective space split once n is at least 3. This result is originally due to Bertin and Elencwajg, and there is also related work by Kaneyama, Klyachko, and Ilten-Süss. The idea is that, after possibly twisting the vector bundle, there is a section which is a complete intersection.
References
Barth, W., Transplanting cohomology classes in complex-projective space, Amer. J. Math. 92 (1970), 951–967. https://doi.org/10.2307/2373404
Bertin, J. and Elencwajg, G., Symétries des fibrés vectoriels sur Pn et nombre d'Euler, Duke Math. J. 49 (1982), no. 4, 807–831.
Hartshorne, R., Varieties of small codimension in projective space, Bull. Amer. Math. Soc. 80 (1974), 1017–1032. https://doi.org/10.1090/S0002-9904-1974-13612-8
Horrocks, G. and Mumford, D., A rank 2 vector bundle on P4 with 15,000 symmetries, Topology 12 (1973), 63–81. https://doi.org/10.1016/0040-9383(73)90022-0
Ilten, N. and Süss, H., Equivariant vector bundles on T-varieties, Transform. Groups 20 (2015), no. 4, 1043–1073. https://doi.org/10.1007/s00031-015-9312-2
Kaneyama, T., Torus-equivariant vector bundles on projective spaces, Nagoya Math. J. 111 (1988), 25–40. https://doi.org/10.1017/S0027763000000982
Klyachko, A. A., Equivariant bundles over toric varieties, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 5, 1001–1039. https://doi.org/10.1070/IM1990v035n02ABEH000707
Bertin, J. and Elencwajg, G., Symétries des fibrés vectoriels sur Pn et nombre d'Euler, Duke Math. J. 49 (1982), no. 4, 807–831.
Hartshorne, R., Varieties of small codimension in projective space, Bull. Amer. Math. Soc. 80 (1974), 1017–1032. https://doi.org/10.1090/S0002-9904-1974-13612-8
Horrocks, G. and Mumford, D., A rank 2 vector bundle on P4 with 15,000 symmetries, Topology 12 (1973), 63–81. https://doi.org/10.1016/0040-9383(73)90022-0
Ilten, N. and Süss, H., Equivariant vector bundles on T-varieties, Transform. Groups 20 (2015), no. 4, 1043–1073. https://doi.org/10.1007/s00031-015-9312-2
Kaneyama, T., Torus-equivariant vector bundles on projective spaces, Nagoya Math. J. 111 (1988), 25–40. https://doi.org/10.1017/S0027763000000982
Klyachko, A. A., Equivariant bundles over toric varieties, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 5, 1001–1039. https://doi.org/10.1070/IM1990v035n02ABEH000707
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Published
2020-09-03
How to Cite
Stapleton, D. (2020). A direct proof that toric rank 2 bundles on projective space split. MATHEMATICA SCANDINAVICA, 126(3), 493–496. https://doi.org/10.7146/math.scand.a-121452
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