A direct proof that toric rank $2$ bundles on projective space split

  • David Stapleton


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.


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 $\mathbf P^n$ 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 $\mathbf P^4$ 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
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