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Holomorphic form

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of degree on a complex manifold

A differential form of type that satisfies the condition , i.e. a form that can be written in the local coordinates on as

where are holomorphic functions (cf. Holomorphic function). The holomorphic forms of degree form a vector space over the field ; is the space of holomorphic functions on .

On a compact Kähler manifold the space coincides with the space of harmonic forms of type (cf. Harmonic form), hence is the first Betti number of [1]. Holomorphic forms on a Riemann surface are also known as differentials of the first kind; if is compact, is equal to its genus (cf. Genus of a curve).

The spaces , , form a locally exact complex with respect to the operator , known as the holomorphic de Rham complex. If is a Stein manifold, then the cohomology spaces of this complex are isomorphic to the complex cohomology spaces , and if [2].

Holomorphic forms with values in some analytic vector bundle (cf. Vector bundle, analytic) over are defined in the same manner (here, holomorphic -forms are holomorphic sections of the bundle). The germs of holomorphic forms of degree with values in form a locally free analytic sheaf . The Dolbeault complex of forms of type , , with values in is a fine resolution of this sheaf, so that

(the Dolbeault–Serre theorem [1], [4]).

The definition of holomorphic forms can be extended to complex-analytic spaces. It is sufficient to do this for local models, i.e. for the case of a space that is an analytic subspace of a domain . The sheaf of germs of holomorphic -forms in is defined as

where is the sheaf of germs of holomorphic -forms in , while consists of the germs of forms of the type

where is the sheaf of ideals which define . The holomorphic de Rham complex of is also defined, but it is not locally exact. For this complex to be locally exact at a point starting from the -th degree it is sufficient that has, in a neighbourhood of , a holomorphic contraction onto a local analytic set for which [3].

References

[1] S.S. Chern, "Complex manifolds without potential theory" , Springer (1979)
[2] R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965)
[3] H.J. Reiffen, "Das Lemma von Poincaré für holomorphe Differentialformen auf komplexen Räumen" Math. Z. , 101 (1967) pp. 269–284
[4] R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980)
How to Cite This Entry:
Holomorphic form. A.L. Onishchik (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Holomorphic_form&oldid=15455
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098