Holomorphic form

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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)
How to Cite This Entry:
Holomorphic form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Holomorphic_form&oldid=47244
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article