nLab
Koszul-Malgrange theorem

Contents

Context

Complex geometry

Differential cohomology

Contents

Statement

Theorem

Holomorphic vector bundles over a (compact) complex manifold XX are equivalently complex vector bundles EE which are equipped with a holomorphically flat connection \nabla, hence with a covariant derivative

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

such that ¯ E¯ E=0\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)D_{coh}^b(X) of 𝒪 X\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 XX a complex manifold, write

CE(T holX)(Ω 0,(X),¯) 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 holX)Rep(T_{hol} X) whose

(Block 05, def. 2.3.2 in theorem 4.1.3).

Theorem

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

HoRep(T holX)D coh b(X) 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 𝒪 X\mathcal{O}_X-module sheaves with coherent cohomology.

(Block 05, theorem 4.1.3).

References

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:

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

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)

Last revised on January 4, 2019 at 17:05:23. See the history of this page for a list of all contributions to it.