geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
Holomorphic vector bundles over a (compact) complex manifold $X$ are equivalently complex vector bundles $E$ which are equipped with a holomorphically flat connection $\nabla$, hence with a covariant derivative
such that $\bar \partial_E \circ \bar \partial_E = 0$, hence with a compatible Dolbeault operator
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.
As an equivalence of categories and with vector bundles generalized to coherent sheaves the statement appears in (Pali 06). This is generalized to an equivalence of homotopy categories of categories of chain complexes of coherent sheaves, hence to the derived category $D_{coh}^b(X)$ of $\mathcal{O}_X$-modules with coherent cohomology, in (Block 05, theorem 4.1.3).
The equivalence in theorem 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).
We state the derived version in more detail:
For $X$ a complex manifold, write
for its Dolbeault complex regarded as a differential graded algebra (the Chevalley-Eilenberg algebra of its holomorphic tangent Lie algebroid). Define a dg-category $Rep(T_{hol} X)$ whose
objects are finite type ∞-Lie algebroid representations of $T_{hol} X$, hence cochain complexes of complex vector bundles $E^{\bullet}$ over $X$ equipped with a $\mathbb{C}$-linear map
that extend to a graded derivation on $\Gamma(E)^\bullet \otimes_{C^\infty(X,\mathbb{C})} CE(T_{hol}X)$;
hom-objects are the evident chain complexes (…).
(Block 05, def. 2.3.2 in theorem 4.1.3).
For $X$ a compact complex manifold, there is an equivalence of categories
between the homotopy category of the dg-category of complexes of holomorphic flat connections, def. , and the bounded derived category of $\mathcal{O}_X$-module sheaves with coherent cohomology.
The classical statement of theorem is due (according to Pali 06) 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
Last revised on January 4, 2019 at 17:05:23. See the history of this page for a list of all contributions to it.