Holomorphic vector bundles over a complex manifold are equivalently complex vector bundles which are equipped with a holomorphically flat connection (hence a connection whose holomorphic component vanishes). Under this identification the Dolbeault operator acting on the sections of the holomorphic vector bundle is identified with the holomorphic component of the covariant derivative of the given connection.
For complex vector bundles over complex varieties this statement is due to Alexander Grothendieck and (Koszul-Malgrange 58), recalled for instance as (Pali 06, theorem 1). It may be understood as a special case of the Newlander-Nirenberg theorem, see (Delzant-Py 10, section 6), which also generalises the proof to infinite-dimensional vector bundles. Over Riemann surfaces, see below, the statement was highlighted in (Atiyah-Bott 83) in the context of the Narasimhan–Seshadri theorem.
The generalization from vector bundles to coherent sheaves is due to (Pali 06). In the genrality of (∞,1)-categories of chain complexes (dg-categories) of holomorphic vector bundles the statement is discussed in (Block 05).
The equivalence in theorem 1 serves to relate a fair bit of differential geometry/differential cohomology with constructions in algebraic geometry. For instance intermediate Jacobians arise in differential geometry and quantum field theory as moduli spaces of flat connections equipped with symplectic structure and Kähler polarization, all of which in terms of algebraic geometry directly comes down moduli spaces of abelian sheaf cohomology with coefficients in the structure sheaf (and/or some variants of that, under the exponential exact sequence).
A key observation here is (Atiyah-Bott 83, section 7), that a -principal connection induces a holomorphic structure on the associated complex vector bundle by taking the -part of the connection 1-form as the Dolbeault operator. For review of the statement and its proof see (Evans, lecture 10).
Michael Atiyah, Raoul Bott, The Yang-Mills equations over Riemann surfaces, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences Vol. 308, No. 1505 (Mar. 17, 1983), pp. 523-615 (jstor, lighning summary)
Generalization to coherent sheaves is due to
Further Generalization to chain complexes of holomorphic vector bundles is discussed in
Generalization to infinite-dimensional vector bundles is in