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at holomorphic vector bundle I have started a section titled As complex vector bundles with holomorphically flat connections.
This deserves much more discussion (and maybe in a dedicated entry), but for the moment I have there the following paragraphs (with lots of room for further improvement):
+– {: .num_theorem #KoszulMalgrangeTheorem}
Holomorphic vector bundles over a complex manifold are equivalently complex vector bundles which are equipped with a holomorphic flat connection. 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.
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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).
+– {: .num_remark}
The equivalence in theorem \ref{KoszulMalgrangeTheorem} 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).
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Ah, and added a pointer to (Koszul-Malgrange 58)
Okay, I have given it its own entry, Koszul-Malgrange theorem
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