Holomorphic form

of degree $ p $ on a complex manifold $ M $

A differential form $ \alpha $ of type $ ( p, 0) $ that satisfies the condition $ d ^ {\prime\prime} \alpha = 0 $, i.e. a form that can be written in the local coordinates $ z _ {1} \dots z _ {n} $ on $ M $ as

$$ \alpha = \sum _ {i _ {1} \dots i _ {p} } a _ {i _ {1} \dots i _ {p} } \ dz ^ {i _ {1} } \wedge \dots \wedge dz ^ {i _ {p} } , $$

where $ a _ {i _ {1} \dots i _ {p} } $ are holomorphic functions (cf. Holomorphic function). The holomorphic forms of degree $ p $ form a vector space $ \Omega ^ {p} ( M) $ over the field $ \mathbf C $; $ \Omega ^ {0} ( M) $ is the space of holomorphic functions on $ M $.

On a compact Kähler manifold $ M $ the space $ \Omega ^ {p} ( M) $ coincides with the space $ H ^ {p,0} ( M) $ of harmonic forms of type $ ( p, 0) $( cf. Harmonic form), hence $ 2 \mathop{\rm dim} \Omega ^ {1} ( M) $ is the first Betti number of $ M $[1]. Holomorphic forms on a Riemann surface $ M $ are also known as differentials of the first kind; if $ M $ is compact, $ \mathop{\rm dim} \Omega ^ {1} ( M) $ is equal to its genus (cf. Genus of a curve).

The spaces $ \Omega ^ {p} ( M) $, $ p = 0 \dots \mathop{\rm dim} _ {\mathbf C } M $, form a locally exact complex with respect to the operator $ d $, known as the holomorphic de Rham complex. If $ M $ is a Stein manifold, then the cohomology spaces of this complex are isomorphic to the complex cohomology spaces $ H ^ {p} ( M, \mathbf C ) $, and $ H ^ {p} ( M, \mathbf C ) = 0 $ if $ p > \mathop{\rm dim} _ {\mathbf C } M $[2].

Holomorphic forms with values in some analytic vector bundle (cf. Vector bundle, analytic) $ E $ over $ M $ are defined in the same manner (here, holomorphic $ 0 $- forms are holomorphic sections of the bundle). The germs of holomorphic forms of degree $ p $ with values in $ E $ form a locally free analytic sheaf $ \Omega _ {E} ^ {p} $. The Dolbeault complex of forms of type $ ( p, q) $, $ q = 0 \dots \mathop{\rm dim} _ {\mathbf C } M $, with values in $ E $ is a fine resolution of this sheaf, so that

$$ H ^ {p, q } ( M, E) \cong \ H ^ {q} ( M, \Omega _ {E} ^ {p} ) $$

(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 $ X $ that is an analytic subspace of a domain $ G \subset \mathbf C ^ {n} $. The sheaf of germs of holomorphic $ p $- forms $ \Omega _ {X} ^ {p} $ in $ X $ is defined as

$$ \left . \Omega _ {G} ^ {p} / K ^ {p} \right | _ {X} , $$

where $ \Omega _ {G} ^ {p} $ is the sheaf of germs of holomorphic $ p $- forms in $ G $, while $ K ^ {p} $ consists of the germs of forms of the type

$$ \sum _ {k = 1 } ^ { r } f _ {k} \alpha _ {k} + \sum _ {l = 1 } ^ { s } dg _ {l} \wedge \beta _ {l} , $$

$$ f _ {k} , g _ {l} \in I,\ \alpha _ {k} \in \Omega _ {G} ^ {p} ,\ \beta _ {l} \in \Omega _ {G} ^ {p - 1 } , $$

where $ I $ is the sheaf of ideals which define $ X $. The holomorphic de Rham complex of $ X $ is also defined, but it is not locally exact. For this complex to be locally exact at a point $ x \in X $ starting from the $ k $- th degree it is sufficient that $ X $ has, in a neighbourhood of $ x $, a holomorphic contraction onto a local analytic set $ Y \subset X $ for which $ { \mathop{\rm em} \mathop{\rm dim} } _ {x} Y = k $[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)