 Solution

A complex manifold of complex dimension $n$ is a Hausdorff, second-countable topological manifold with charts to open subsets of ${\mathbb{C}}^n$ whose transition maps are biholomorphic. A rank-$r$ holomorphic vector bundle is a complex vector bundle with local trivializations $E{|}_U\cong U\times {\mathbb{C}}^r$ whose transition matrices are holomorphic maps to ${\mathrm{GL}}_r(\mathbb{C})$.