nLab Koszul-Malgrange theorem

Statement

Theorem

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

$$\bar \partial_E \colon \Gamma(E)\longrightarrow \Gamma(E) \otimes_{C^\infty(X, \mathbb{C})}\Omega^{0,1}(X)$$

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 a discussion on this use of the terminology “holomorphically flat”, see this MO question.)

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).

Remark

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:

Definition

For $X$ a complex manifold, write

$$CE(T_{hol} X) \coloneqq (\Omega^{0,\bullet}(X), \bar \partial)$$

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

  $$\bar \partial_E \;\colon\; \Gamma(E)^\bullet \longrightarrow \Gamma(E)^\bullet \otimes_{C^\infty(X,\mathbb{C})} CE(T_{hol}X)$$

  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).

Theorem

For $X$ a compact complex manifold, there is an equivalence of categories

$$Ho Rep(T_{hol}X) \simeq D^b_{coh}(X)$$

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.

(Block 05, theorem 4.1.3).

Related theorems

• Narasimhan-Seshadri theorem

• Donaldson-Uhlenbeck-Yau theorem

• Kobayashi-Hitchin correspondence

References

The classical statement of theorem is due (according to Pali 06) to Alexander Grothendieck and

• Jean-Louis Koszul, Bernard Malgrange, Sur certaine structures fibrées complexes, arch. mat, vol IX, 1958

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

• N. Pali, Faisceaux $\bar \partial$-cohrents sur les variété complexes ( $\bar \partial$-Coherent sheaves on complex manifolds) Math. Ann. 336 (2006), no. 3, 571–615 (arXiv:math/0305422)

Further Generalization to chain complexes of holomorphic vector bundles is discussed in

• Jonathan Block, Duality and equivalence of module categories in noncommutative geometry I (arXiv:0509284)

in terms of Lie infinity-algebroid representations of the holomorphic tangent Lie algebroid.

Generalization to infinite-dimensional vector bundles is in

• Thomas Delzant, Pierre Py, Kähler groups, real hyperbolic spaces and the Cremona group, Compositio Math. 148, no. 1 (2012), 153–184 (arXiv:1012.1585)