holomorphic vector bundle in nLab

Contents

Contents

Idea

A holomorphic vector bundle is a complex vector bundle over a complex manifold such that it admits transition functions that are holomorphic functions.

Properties

As complex algebraic vector bundles

By the GAGA-principle, holomorphic vector bundles and more generally analytic coherent sheaves over a projective smooth complex variety coincide with complex algebraic vector bundles/coherent sheaves.

As complex vector bundles with holomorphically flat connection

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).

Over Riemann surfaces

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)U(n)-principal connection induces a holomorphic structure on the associated complex vector bundle by taking the (0,1)(0,1)-part of the connection 1-form as the Dolbeault operator. For review of the statement and its proof see (Evans, lecture 10).

• Kodaira vanishing theorem

• Chern connection

• Kähler polarization

• stable vector bundle

• moduli space of bundles

• Deligne line bundle

• Higgs bundle

• algebraic line bundle

• holomorphic line 2-bundle, holomorphic line n-bundle

• complex analytic stack

• pseudoholomorphic vector bundle

• Oka principle

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References

General

Textbook accounts:

• Franc Forstnerič, Section 1.5 in: Stein manifolds and holomorphic mappings – The homotopy principle in complex analysis, Springer 2011 (doi:10.1007/978-3-642-22250-4)

  (in the context of the Oka principle)

See also

• Wikipedia, Holomorphic vector bundle

Relation to complex vector bundles with flat holomorphic connection

The classical statement of theorem is due 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)