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The traditional Yang - Mills type massive gauge theories are interpreted in the geometrical frame of holomorphic vector bundles on a complex manifold of complex 2 -dimension. It is seen in this formalism that, although connection is flat the component (1, 1) of the curvature of this connection appears always owing to flat connections generated by holomorphic structure and it is possible to write a Lagrangian including massive and massless gauge fields. However, the mass matrix of a massive gauge field on such a bundle isn’t nilpotent and this field is generated by a noncommutative flat connection on the same bundle, then the structure group of this bundle is non - Abelian complex Lie group. However, if the gauge field is massless, then this is generated by commutative flat connection, then the structure group of the bundle is Abelian complex Lie group, so that this group may be candidate for the long - range gauge fields. In this context, the mass matrix is generated by real harmonic matrices. The Abelian groups have vanishing first and second characteristic classes on the complex 2 - dimension and non - Abelian ones have non-vanishing first and second characteristic classes and the first classes determines topologically the mass of the massive gauge field.