Filtrable bundles of rank 2 on Hopf surfaces
A Hopf surface is a complex structure on \(S^ 1 \times S^ 3\). The classical Hopf surfaces are obtained as quotients of \(\mathbb{C}^ 2 \backslash \{0\}\) under \(z \mapsto \mu z\) for \(\mu\) a nonzero complex number of absolute value less than one. The general Hopf surface may also be realised as a quotient of \(\mathbb{C}^ 2 \backslash \{0\}\) and there is a complete classification. A holomorphic rank-2 vector bundle \(E\) on a complex surface \(X\) is called filtrable if there is a line bundle \(L\) on \(X\) and a nonzero homomorphism \(L \to E\). This does not mean that \(L\) is a subbundle of \(E\) but it is so outside a finite number of points. If \(c_ 2 (E) = 0\), these points are eliminated and \(E\) may be written as an extension of line-bundles. Being non-Kähler, Hopf surfaces are a good place to begin a study of holomorphic vector bundles on nonalgebraic surfaces. In this article the author does just this. The existence of a holomorphic structure on a smooth complex vector bundle \(E\) on a Hopf surface is assured if \(c_ 2 (E) \geq 0\) (more generally, see \textit{C. Bănică} and \textit{J. LePotier} [J. Reine Angew. Math. 378, 1-31 (1987; Zbl 0624.32017)]) but the classification of such structures is more involved. In this article the author classifies the holomorphic structures on a topologically trivial rank-2 bundle on a Hopf surface assuming the bundle to be filtrable (and, hence, an extension of line bundles). It is a clearly written and informative article, pointing the way for further work.
scientific article; zbMATH DE number 549731
scientific article