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''of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h0475402.png" /> on a complex manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h0475403.png" />''
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$#C+1 = 61 : ~/encyclopedia/old_files/data/H047/H.0407540 Holomorphic form
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A [[Differential form|differential form]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h0475404.png" /> of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h0475405.png" /> that satisfies the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h0475406.png" />, i.e. a form that can be written in the local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h0475407.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h0475408.png" /> as
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h0475409.png" /></td> </tr></table>
+
''of degree  $  p $
 +
on a complex manifold  $  M $''
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754010.png" /> are holomorphic functions (cf. [[Holomorphic function|Holomorphic function]]). The holomorphic forms of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754011.png" /> form a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754012.png" /> over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754013.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754014.png" /> is the space of holomorphic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754015.png" />.
+
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
  
On a compact [[Kähler manifold|Kähler manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754016.png" /> the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754017.png" /> coincides with the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754018.png" /> of harmonic forms of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754019.png" /> (cf. [[Harmonic form|Harmonic form]]), hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754020.png" /> is the first [[Betti number|Betti number]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754021.png" /> [[#References|[1]]]. Holomorphic forms on a [[Riemann surface|Riemann surface]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754022.png" /> are also known as differentials of the first kind; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754023.png" /> is compact, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754024.png" /> is equal to its genus (cf. [[Genus of a curve|Genus of a curve]]).
+
$$
 +
\alpha  = \sum _ {i _ {1} \dots i _ {p} }
 +
a _ {i _ {1}  \dots i _ {p} } \
 +
dz ^ {i _ {1} } \wedge \dots \wedge dz ^ {i _ {p} } ,
 +
$$
  
The spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754026.png" />, form a locally exact complex with respect to the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754027.png" />, known as the holomorphic de Rham complex. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754028.png" /> is a [[Stein manifold|Stein manifold]], then the cohomology spaces of this complex are isomorphic to the complex cohomology spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754029.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754030.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754031.png" /> [[#References|[2]]].
+
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 $.
  
Holomorphic forms with values in some analytic vector bundle (cf. [[Vector bundle, analytic|Vector bundle, analytic]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754032.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754033.png" /> are defined in the same manner (here, holomorphic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754034.png" />-forms are holomorphic sections of the bundle). The germs of holomorphic forms of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754035.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754036.png" /> form a locally free analytic sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754037.png" />. The Dolbeault complex of forms of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754039.png" />, with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754040.png" /> is a fine resolution of this sheaf, so that
+
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]]).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754041.png" /></td> </tr></table>
+
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 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754042.png" /> that is an analytic subspace of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754043.png" />. The sheaf of germs of holomorphic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754044.png" />-forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754045.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754046.png" /> is defined as
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754047.png" /></td> </tr></table>
+
$$
 +
\left . \Omega _ {G}  ^ {p} / K  ^ {p} \right | _ {X} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754048.png" /> is the sheaf of germs of holomorphic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754049.png" />-forms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754050.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754051.png" /> consists of the germs of forms of the type
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754052.png" /></td> </tr></table>
+
$$
 +
\sum _ {k = 1 } ^ { r }
 +
f _ {k} \alpha _ {k} +
 +
\sum _ {l = 1 } ^ { s }
 +
dg _ {l} \wedge \beta _ {l} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754053.png" /></td> </tr></table>
+
$$
 +
f _ {k} , g _ {l}  \in  I,\  \alpha _ {k}  \in  \Omega _ {G}  ^ {p} ,\  \beta _ {l}  \in  \Omega _ {G} ^ {p - 1 } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754054.png" /> is the sheaf of ideals which define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754055.png" />. The holomorphic de Rham complex of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754056.png" /> is also defined, but it is not locally exact. For this complex to be locally exact at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754057.png" /> starting from the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754058.png" />-th degree it is sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754059.png" /> has, in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754060.png" />, a holomorphic contraction onto a local analytic set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754061.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047540/h04754062.png" /> [[#References|[3]]].
+
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====
 
====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>
 
<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>

Latest revision as of 22:10, 5 June 2020


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