View source for Holomorphic form

<!--
h0475402.png
$#A+1 = 61 n = 0
$#C+1 = 61 : ~/encyclopedia/old_files/data/H047/H.0407540 Holomorphic form
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

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

A [[Differential form|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|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|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|Harmonic form]]), hence  $  2  \mathop{\rm dim}  \Omega  ^ {1} ( M) $
is the first [[Betti number|Betti number]] of  $  M $[[#References|[1]]]. Holomorphic forms on a [[Riemann surface|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|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|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 $[[#References|[2]]].

Holomorphic forms with values in some analytic vector bundle (cf. [[Vector bundle, analytic|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 [[#References|[1]]], [[#References|[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 $[[#References|[3]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.S. Chern,   "Complex manifolds without potential theory" , Springer  (1979)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.C. Gunning,   H. Rossi,   "Analytic functions of several complex variables" , Prentice-Hall  (1965)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H.J. Reiffen,   "Das Lemma von Poincaré für holomorphe Differentialformen auf komplexen Räumen"  ''Math. Z.'' , '''101'''  (1967)  pp. 269–284</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.O. Wells jr.,   "Differential analysis on complex manifolds" , Springer  (1980)</TD></TR></table>