(see also Chern-Weil theory, parameterized homotopy theory)
geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
A complex vector bundle over a complex manifold such that it admits transition functions that are holomorphic functions.
By the GAGA-principle holomorphic vector bundles and more generally analytic coherent sheaves over a projetive smooth complex variety coincide with complex algebraic vector bundles/coherent sheaves.
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 $\bar \partial$ acting on the sections of the holomorphic vector bundle is identified with the holomorphic component of the covariant derivative of the given connection.
The analogous statement is true for generalization of vector bundles to chain complexes of module sheaves with coherent cohomology.
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).
Over Riemann surfaces holomorphic vector bundles are a central part of the theory of the moduli space of flat connections. See at Narasimhan-Seshadri theorem.
A key observation here is (Atiyah-Bott 83, section 7), that a $U(n)$-principal connection induces a holomorphic structure on the associated complex vector bundle by taking the $(0,1)$-part of the connection 1-form as the Dolbeault operator. For review of the statement and its proof see (Evans, lecture 10).
The classical statement of theorem 1 is due to Alexander Grothendieck and
Over Riemann surfaces and in the context of the moduli space of flat connections:
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)
Jonathan Evans, Aspects of Yang-Mills theory, lecture notes, (lecture 10, lecture 11, lecture 12)
Generalization to coherent sheaves is due to
Further Generalization to chain complexes of holomorphic vector bundles is discussed in
in terms of Lie infinity-algebroid representations of the holomorphic tangent Lie algebroid.
Generalization to infinite-dimensional vector bundles is in