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 ℂn − {0}. The authors show that there exists a line bundle L over X such that EL 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.

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Correspondence to Ning Gan or Xiangyu Zhou.

Additional information

This work was supported by the National Natural Science Foundation of China (Nos. 11671330, 11688101, 11431013).

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Gan, N., Zhou, X. The Structure of Vector Bundles on Non-primary Hopf Manifolds. Chin. Ann. Math. Ser. B 41, 929–938 (2020). https://doi.org/10.1007/s11401-020-0239-0

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Keywords

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

2000 MR Subject Classification

  • 32L05
  • 32L10
  • 32Q55