The Structure of Vector Bundles on Non-primary Hopf Manifolds
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Chinese Annals of Mathematics, Series B c The Editorial Office of CAM and
Springer-Verlag Berlin Heidelberg 2020
The Structure of Vector Bundles on Non-primary Hopf Manifolds∗ Ning GAN1
Xiangyu ZHOU2
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 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 2000 MR Subject Classification 32L05, 32L10, 32Q55
1 Introduction In recent years holomorphic vector bundles on non-algebraic surfaces have received increasing attentions (see [1–3]). Hopf surfaces constitute a simple but important class of compact non-K¨ ahlian surfaces. The geometry of Hopf manifolds has been studied by many authors, see [4–8] for instance. For the results of holomorphic vector bundles in Hopf manifolds, the reader is referred to [9–14]. In this paper we investigate the structure of holomorphic vector bundles with trivial pullback on non-primary Hopf manifolds with non-Abelian fundamental group. It generalizes the results of [9] on the primary Hopf manifolds and [10] on the non-primary Hopf manifolds with Abelian fundamental group. The universal covering of an n-dimensional Hopf manifold X, n ≥ 2, is biholomorphic to the punctured n-space W = Cn − {0}. X can be written as a quotient space W/G with a group G generated by some biholomorphic transformations of W which act on W properly discontinuous and free. When n ≥ 2, W is simply connected and G is the fundamental group of X. When G is the infinite cyclic group Z, the Hopf manifold is called primary, otherwise it is called nonprimary. By a contraction f : (Cn , 0) → (Cn , 0), n ≥ 2, we mean an automorphism of Cn fixing 0 with the property that the eigenvalues µ1 , · · · , µn of f ′ (0), the differential of f at 0, are inside the unit circle. We always assume, without loss of generality, that 0 ≤ |µ1 | ≤ |µ2 | ≤ · · · ≤ |µn | < 1. A contraction of the form f : (z1 , · · · , zn ) → (µ1 z1 , · · · , µn zn ) is called diagonal. Similar to Kodaira’s argument for Hopf surface in [8], the fundamental group G of X has the following properties: G contains a contraction f and the infinite cyclic subgroup Z generated by f has Manuscript received February 19, 2019. of Sciences, Jimei University, Xiamen 361021, Fujian, China. E-mail: [email protected] 2 Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China. E-mail: [email protected] ∗ This work was supported by the National Natural Science Foundation of China (Nos. 11671330, 11688101, 11431013). 1 School
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