Globally Generated Vector Bundles with c₁ = 1 over Grassmannians

Abstract

In this paper, we classify globally generated vector bundles with the first Chern class equal to 1 over Grassmannian G(d,n) (1 ≤ d ≤ n − d) over an algebraically closed field in characteristic zero, from the perspective of uniform vector bundles. In particular, we show that they are homogeneous.

Keywords

Uniform vector bundle / Harder–Narasimhan filtration / Grassmannian

References

[1] BallicoE. Uniform vector bundles of rank (n + 1) on ℙⁿ. Tsukuba J. Math., 1983, 7(2): 215-226

[2] DrézetJ-M. Exemples de fibrés uniformes non homogènes sur ℙ_n. C. R. Acad. Sci. Paris Sér. A-B, 1980, 291(2): A125-A128

[3] DuR, FangX, GaoY. Vector bundles on rational homogeneous spaces. Ann. Mat. Pura Appl. (4), 2021, 200(6): 2797-2827

[4] DuR, FangX, GaoY. Vector bundles on flag varieties. Math. Nachr., 2023, 296(2): 630-649

[5] Du R., Wang Y., Zhang D., On uniform and nonhomogeneous vector bundles over Grass-mannians. 2024, arXiv:2404.02593

[6] ElencwajgG. Les fibrés uniformes de rang 3 sur ℙ₂(C) sont homogènes. Math. Ann., 1978, 231(3): 217-227

[7] ElencwajgG, HirschowitzA, SchneiderM. Les fibrés uniformes de rang au plus n sur ℙ_n(ℂ) sont ceux qu’on croit. Vector Bundles and Differential Equations (Proc. Conf., Nice, 1979), 1980, Boston, MA, Birkhäuser: 37-63 7

[8] ElencwajgG. Des fibrés uniformes non homogènes. Math. Ann., 1979, 239(2): 185-192

[9] ElliaP. Sur les fibrés uniformes de rang (n + 1) sur ℙⁿ. Mém. Soc. Math. France (N.S.), 1982, 7: 60 pp.

[10] GuyotM. Caractérisation par l’uniformité des fibrés universels sur la Grassmanienne. Math. Ann., 1985, 270(1): 47-62

[11] OkonekC, SchneiderM, SpindlerHVector Bundles on Complex Projective Spaces, 2011, Basel, Birkhäuser

[12] SatoE. Uniform vector bundles on a projective space. J. Math. Soc. Japan, 1976, 28(1): 123-132

[13] SchwarzenbergerRLE. Vector bundles on the projective plane. Proc. London Math. Soc. (3), 1961, 11: 623-640

[14] Van de VenA. On uniform vector bundles. Math. Ann., 1971, 195: 245-248

RIGHTS & PERMISSIONS

Peking University