Holomorphic vector bundles on non-algebraic tori of dimension 2 (Q930580)

The authors prove the following result: Theorem 0.1. Let \(X\) be a non-algebraic complex torus of dimension 2. Let \(E\) be a topological vector bundle of rank \(r\) over \(X\). Then \(E\) has a holomorphic structure if and only if \(c_{1}(E)\in \text{NS}(X)\) and \(\Delta(E)\geq0\). The case \(r=2\) was proved by \textit{M. Toma} [Math. Z. 232, No. 3, 511--525 (1999; Zbl 0945.32007)]. A similar result is proved in this paper for non-algebraic \(K3\)-surfaces.