On holomorphic vector bundles on Hopf manifolds with trivial pullback on the universal covering
The main objective of the paper is to study vector bundles \(E\) of rank \(r\) over a Hopf manifold \(X\) of dimension \(n\geq 2\) whose pullbacks \(\pi^*(E)\) over the universal covering \(\mathbb{C}^ n-\{0\}\) of the Hopf manifold are trivial. Because Hopf manifolds are never algebraic it is not self-evident that these bundles are filtrable. A principal filtrability property for these bundles is proven: there is a line bundle \(L\) such that \(L\otimes E\) has a nowhere vanishing holomorphic section. Applying this property we can prove a structure theorem for all Hopf manifolds save exceptional surfaces, which expresses the triviality of \(\pi^*(E)\) by an intrinsic property of \(E\), the existence of a filtration by subvector bundles: \(0\subset E^ 1\subset E^ 2\subset\cdots\subset E^ r=E\) with \(E^ i\) of rank \(i\). The bundles under study can be described by a matrix valued holomorphic function \(A\) on \(\mathbb{C}^ n\): \(A\in GL(r,\Gamma(\mathbb{C}^ n,{\mathcal O})\), the automorphy factor of \(E\). The isomorphism classes of the bundles \(E\) with holomorphically trivial \(\pi^*(E)\) correspond univocally to the orbits of a certain action of \(GL(r,\Gamma(\mathbb{C}^ n,{\mathcal O})\) on itself. This action is given by \(\mapsto B(f(z))\cdot A\cdot B^{-1}\), \(B\in GL(r,\Gamma(\mathbb{C}^ n,{\mathcal O})\), where \(f\) is any (fixed) contraction which induces the Hopf manifold \(X\). Therefore the classification problem is reduced to the problem of normal forms for matrix-valued functions on \(\mathbb{C}^ n\). In the special case of flat bundles this problem is solved by the classical Jordan normal forms for constant matrices. The second part of the paper is concerned with finding normal forms for the automorphy factors in general. Among various applications some invariants for the bundles studied are introduced and a decomposability criterium is deduced from the normal form. In an appendix elliptic curves are considered as one dimensional Hopf manifolds and holomorphic vector bundles defined over them are partly described by an automorphy factor over \(\mathbb{C}^*\). As a by-product a fragment of Atiyah's classification is derived.
scientific article; zbMATH DE number 86214
scientific article