The Structure of Vector Bundles on Non-primary Hopf Manifolds

Abstract

Let X be a Hopf manifold with non-Abelian fundamental group and E be a holomorphic vector bundle over X, with trivial pull-back to ℂ ⁿ − {0}. The authors show that there exists a line bundle L over X such that E ⊗ L has a nowhere vanishing section. It is proved that in case dim(X) ≥ 3, π*(E) is trivial if and only if E is filtrable by vector bundles. With the structure theorem, the authors get the cohomology dimension of holomorphic bundle E over X with trivial pull-back and the vanishing of Chern class of E.

Keywords

Hopf manifolds / Holomorphic vector bundles / Exact sequence / Cohomology / Filtration / Chern class

References

[1] Brînzănescu V. Holomorphic Vector Bundles Over Compact Complex Surfaces, 1996, Berlin: Springer-Verlag

[2] Teleman A. The pseudo-effective cone of a non-Kahlerian surface and applications. Math. Ann., 2006, 335(4): 965-989

[3] Demailly J, Peternell T, Schneider M. Compact complex manifolds with numerically effective tangent bundles. J. Algebraic Geom., 1994, 3(2): 295-345

[4] Ise M. On the geometry of Hopf manifolds. Osaka Math. J., 1960, 12: 387-402

[5] Kodaira K. On the structure of compact complex analytic surfaces II. American Journal of Mathematics, 1966, 88(3): 682-721

[6] Barth W, Hulek K, Peters C, Van de Ven A. Compact Complex Surfaces, 2004 2nd ed. Berlin: Springer-Verlag

[7] Wehler J. Versal deformation of Hopf surfaces. J. Reine. angew. Math., 1981, 328: 22-32

[8] Levenberg N, Yamaguchi H. Pseudoconvex domains in the Hopf surface. J. Math. Soc. Japan, 2015, 67(1): 231-273

[9] Mall D. On holomorphic vector bundle on Hopf Manifolds with pullback on the universal covering. Math. Ann., 1992, 294: 719-740

[10] Liu W M, Zhou X Y. Holomorphic vector bundles on Hopf manifolds. Acta Mathematica Sinica, 2004, 20(4): 605-612

[11] Zhou X Y. A remark on the Douady sequence for non-primary Hopf manifolds. Asian. J. Math., 2004, 8(1): 131-136

[12] Zhou X Y. Some results about holomorphic vector bundles over general Hopf manifolds. Sci. China, Ser. A., 2009, 52(12): 2863-2866

[13] Gan N, Zhou X Y. The cohomology of line bundles on non-primary Hopf manifolds. Acta Math. Sin. (Engl. Ser.), 2007, 23(2): 289-296

[14] Gan N, Zhou X Y. The cohomology of vector bundles on general non-primary Hopf manifolds, Recent Progress on Some Problems in Several Complex Variables and Partial Differential Equations. Contemp. Math., 400, 2006, Providence, RI: Amer. Math. Soc. 107-115

[15] Grauert H, Remmert R. Coherent Analytic Sheaves, 1984, Berlin, Heidelberg, New York: Springer-Verlag

[16] Haefliger A. Deformation of transversely holomorphic flows on spheres and deformation of Hopf Manifolds. Compositio Math., 1985, 55: 241-251