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A differential form of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d0321302.png" />, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d0321304.png" />-form, on a differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d0321305.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d0321306.png" /> times covariant tensor field on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d0321307.png" />. It may also be interpreted as a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d0321308.png" />-linear (over the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d0321309.png" /> of smooth real-valued functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213010.png" />) mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213012.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213013.png" />-module of smooth vector fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213014.png" />. Forms of degree one are also known as Pfaffian forms. An example of such a form is the differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213015.png" /> of a smooth function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213016.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213017.png" />, which is defined as follows: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213019.png" />, is the derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213020.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213021.png" /> in the direction of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213022.png" />. Riemannian metrics on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213023.png" /> serve as examples of symmetric differential forms of degree two. However, the term "differential form" is often used to denote skew-symmetric or exterior differential forms, which have the greatest number of applications.
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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213024.png" /> is a local system of coordinates in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213025.png" />, the forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213026.png" /> constitute a basis of the cotangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213028.png" />. For this reason (cf. [[Exterior algebra|Exterior algebra]]) any exterior <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213029.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213030.png" /> may be written in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213031.png" /> in the form
+
{{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/d/d032/d032130/d03213032.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
A differential form of degree  $  p $,
 +
a  $  p $-
 +
form, on a differentiable manifold  $  M $
 +
is a  $  p $
 +
times covariant tensor field on  $  M $.
 +
It may also be interpreted as a  $  p $-
 +
linear (over the algebra  $  F( M) $
 +
of smooth real-valued functions on  $  M $)
 +
mapping  $  {\mathcal X} ( M)  ^ {p} \rightarrow F( M) $,
 +
where  $  {\mathcal X} ( M) $
 +
is the  $  F( M) $-
 +
module of smooth vector fields on  $  M $.
 +
Forms of degree one are also known as Pfaffian forms. An example of such a form is the differential  $  df $
 +
of a smooth function  $  f $
 +
on  $  M $,
 +
which is defined as follows: $  ( df  ) ( X) $,
 +
$  X \in {\mathcal X} ( M) $,
 +
is the derivative  $  Xf $
 +
of  $  f $
 +
in the direction of the field  $  X $.  
 +
Riemannian metrics on a manifold  $  M $
 +
serve as examples of symmetric differential forms of degree two. However, the term "differential form" is often used to denote skew-symmetric or exterior differential forms, which have the greatest number of applications.
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213033.png" /> are functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213034.png" />. In particular,
+
If  $  ( x  ^ {1} \dots x  ^ {n} ) $
 +
is a local system of coordinates in a domain  $  U \subset  M $,
 +
the forms  $  dx  ^ {1} \dots dx  ^ {n} $
 +
constitute a basis of the cotangent space  $  T _ {x} ( M)  ^ {*} $,
 +
$  x \in U $.  
 +
For this reason (cf. [[Exterior algebra|Exterior algebra]]) any exterior  $  p $-
 +
form  $  \alpha $
 +
may be written in  $  U $
 +
in the form
  
<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/d/d032/d032130/d03213035.png" /></td> </tr></table>
+
$$ \tag{1 }
 +
\alpha  = \sum _ {i _ {1} \dots i _ {p} } a _ {i _ {1}  \dots i _ {p} }  dx ^ {i _ {1} } \wedge \dots \wedge dx ^ {i _ {p} } ,
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213036.png" /> be the space of all exterior <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213037.png" />-forms of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213038.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213039.png" />. The exterior multiplication <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213040.png" /> converts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213041.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213042.png" />) to an associative graded algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213043.png" /> which satisfies the condition of graded commutativity
+
where the $  a _ {i _ {1}  \dots i _ {p} } $
 +
are functions on  $  U $.  
 +
In particular,
  
<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/d/d032/d032130/d03213044.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$
 +
df  =
 +
\frac{\partial  f }{\partial  x  ^ {i} }
 +
  dx  ^ {i} .
 +
$$
  
A smooth mapping between manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213045.png" /> induces a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213046.png" /> between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213047.png" />-algebras.
+
Let  $  E  ^ {p} = E  ^ {p} ( M) $
 +
be the space of all exterior  $  p $-
 +
forms of class  $  C  ^  \infty  $,
 +
where  $  E  ^ {0} ( M) = F ( M) $.  
 +
The exterior multiplication  $  \alpha \wedge \beta $
 +
converts  $  E  ^ {*} ( M) = \sum _ {p = 0 }  ^ {n} E  ^ {p} ( M) $(
 +
where  $  n = \mathop{\rm dim}  M $)
 +
to an associative graded algebra over  $  F ( M) $
 +
which satisfies the condition of graded commutativity
  
The concept of the differential of a function is generalized as follows. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213048.png" /> there exists a unique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213049.png" />-linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213050.png" /> (exterior differentiation), which for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213051.png" /> coincides with the differential introduced above, with the following properties:
+
$$ \tag{2 }
 +
\alpha \wedge \beta  = ( - 1 ) ^ {pq} \beta \wedge \alpha ,\ \
 +
\alpha \in E  ^ {p} ,\  \beta \in E  ^ {q} .
 +
$$
  
<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/d/d032/d032130/d03213052.png" /></td> </tr></table>
+
A smooth mapping between manifolds  $  f : M \rightarrow N $
 +
induces a homomorphism  $  f ^ { * } : E  ^ {*} ( N) \rightarrow E  ^ {*} ( M) $
 +
between  $  \mathbf R $-
 +
algebras.
  
<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/d/d032/d032130/d03213053.png" /></td> </tr></table>
+
The concept of the differential of a function is generalized as follows. For any  $  p \geq  0 $
 +
there exists a unique  $  \mathbf R $-
 +
linear mapping  $  d : E  ^ {p} \rightarrow E  ^ {p+} 1 $(
 +
exterior differentiation), which for  $  p = 0 $
 +
coincides with the differential introduced above, with the following properties:
  
The exterior differential of a form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213054.png" /> written in in local coordinates (see (1)) is expressed by the formula
+
$$
 +
d ( \alpha \wedge \beta )  = d \alpha \wedge \beta + ( - 1 ) ^ {p}
 +
\alpha \wedge d \beta ,
 +
$$
  
<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/d/d032/d032130/d03213055.png" /></td> </tr></table>
+
$$
 +
\alpha  \in  E  ^ {p} ,\  \beta  \in  E  ^ {q} ,\  d ( d \alpha )  = 0 .
 +
$$
 +
 
 +
The exterior differential of a form  $  \alpha $
 +
written in in local coordinates (see (1)) is expressed by the formula
 +
 
 +
$$
 +
d \alpha  =  \sum _ {i _ {1} \dots i _ {p} } da _ {i _ {1}  \dots i _ {p} } \wedge dx ^ {i _ {1} } \wedge \dots \wedge dx ^ {i _ {p} } .
 +
$$
  
 
Its coordinate-free notation is
 
Its coordinate-free notation is
  
<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/d/d032/d032130/d03213056.png" /></td> </tr></table>
+
$$
 +
d \alpha ( X _ {1} \dots X _ {p+} 1 ) =
 +
$$
  
<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/d/d032/d032130/d03213057.png" /></td> </tr></table>
+
$$
 +
= \
 +
\sum _ {i = 1 } ^ { {p }  + 1 } (- 1)  ^ {i+} 1 X _ {i} \alpha ( X _ {1} \dots \widehat{X}  _ {i} \dots X _ {p+} 1 ) +
 +
$$
  
<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/d/d032/d032130/d03213058.png" /></td> </tr></table>
+
$$
 +
- \sum _ {i < j } (- 1)  ^ {i+} j \alpha ( [ X _ {i} , X _ {j} ]
 +
, X _ {1} \dots \widehat{X}  _ {i} \dots \widehat{X}  _ {j} \dots X _ {p+} 1 ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213059.png" />. The [[Lie derivative|Lie derivative]] operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213061.png" />, on differential forms is connected with the exterior differentiation operator by the relation
+
where $  X _ {1} \dots X _ {p+} 1 \in {\mathcal X} ( M) $.  
 +
The [[Lie derivative|Lie derivative]] operator $  L _ {X} $,  
 +
$  X \in {\mathcal X} ( M) $,  
 +
on differential forms is connected with the exterior differentiation operator by the relation
  
<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/d/d032/d032130/d03213062.png" /></td> </tr></table>
+
$$
 +
L _ {X}  = d \circ \iota _ {X} + \iota _ {X} \circ d ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213063.png" /> is the operator of interior multiplication by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213064.png" />:
+
where $  \iota _ {X} : E  ^ {p} \rightarrow E  ^ {p-} 1 $
 +
is the operator of interior multiplication by $  X $:
  
<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/d/d032/d032130/d03213065.png" /></td> </tr></table>
+
$$
 +
( \iota _ {X} \alpha ) ( X _ {1} \dots X _ {p-} 1 )  = \alpha ( X , X _ {1} \dots X _ {p-} 1 ) ,
 +
$$
  
<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/d/d032/d032130/d03213066.png" /></td> </tr></table>
+
$$
 +
\alpha  \in  E  ^ {p} ( M) ,\  X _ {1} \dots X _ {p-} 1  \in  {\mathcal X} ( M) .
 +
$$
  
The complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213067.png" /> is a cochain complex (the de Rham complex). The cocycles of this complex are said to be closed forms, while the coboundaries are known as exact forms. According to the [[De Rham theorem|de Rham theorem]], the cohomology algebra
+
The complex $  ( E  ^ {*} ( M) , d ) $
 +
is a cochain complex (the de Rham complex). The cocycles of this complex are said to be closed forms, while the coboundaries are known as exact forms. According to the [[De Rham theorem|de Rham theorem]], the cohomology algebra
  
<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/d/d032/d032130/d03213068.png" /></td> </tr></table>
+
$$
 +
H  ^ {*} ( M)  = \sum _ {p = 0 } ^ { n }  H  ^ {p} ( M)
 +
$$
  
of the de Rham complex is isomorphic to the real cohomology algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213069.png" /> of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213070.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213071.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213072.png" /> (Poincaré's lemma).
+
of the de Rham complex is isomorphic to the real cohomology algebra $  H  ^ {*} ( M, \mathbf R ) $
 +
of the manifold $  M $.  
 +
In particular, $  H  ^ {p} ( \mathbf R  ^ {n} ) = 0 $
 +
if  $  p > 0 $(
 +
Poincaré's lemma).
  
The de Rham theorem is closely connected with another operation, that of integration of differential forms. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213073.png" /> be a bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213074.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213075.png" /> be a smooth mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213076.png" />, defined in a neighbourhood of the closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213077.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213078.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213079.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213080.png" /> is a smooth function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213081.png" />. The integral of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213082.png" /> over the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213083.png" /> is defined by the formula:
+
The de Rham theorem is closely connected with another operation, that of integration of differential forms. Let $  D $
 +
be a bounded domain in $  \mathbf R  ^ {p} $
 +
and let $  s $
 +
be a smooth mapping $  \mathbf R  ^ {p} \rightarrow M $,  
 +
defined in a neighbourhood of the closure $  \overline{D}\; $.  
 +
If $  \alpha \in E  ^ {p} ( M) $,  
 +
then $  s {}  ^  \star  \alpha = a  dx  ^ {1} \wedge \dots \wedge dx  ^ {p} $,  
 +
where $  a $
 +
is a smooth function in $  \overline{D}\; $.  
 +
The integral of the form $  \alpha $
 +
over the surface $  s $
 +
is defined by the formula:
  
<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/d/d032/d032130/d03213084.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { s } \alpha  = \int\limits _ { D } a ( x _ {1} \dots x _ {p} )  dx  ^ {1} \dots dx  ^ {p} .
 +
$$
  
If the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213085.png" /> is piecewise smooth, the formula
+
If the boundary of $  D $
 +
is piecewise smooth, the formula
  
<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/d/d032/d032130/d03213086.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
\int\limits _ { s }  d \alpha  = \int\limits _ {\partial  s } \alpha ,\ \
 +
\alpha \in E  ^ {p-} 1 ( M) ,
 +
$$
  
is valid; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213087.png" /> is defined as the sum of the integrals of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213088.png" /> over the smooth pieces of the boundary, provided with their natural parametrizations. The classical formulas of Newton–Leibniz, Green–Ostrogradski and Stokes (see also [[Stokes theorem|Stokes theorem]]) are all special cases of this formula. By virtue of formula (3) each closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213089.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213090.png" /> defines a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213091.png" />-dimensional singular cocycle whose value on the simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213092.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213093.png" />. This correspondence is a realization of the isomorphism given by de Rham's theorem.
+
is valid; here $  \int _ {\partial  s }  \alpha $
 +
is defined as the sum of the integrals of the form $  \alpha $
 +
over the smooth pieces of the boundary, provided with their natural parametrizations. The classical formulas of Newton–Leibniz, Green–Ostrogradski and Stokes (see also [[Stokes theorem|Stokes theorem]]) are all special cases of this formula. By virtue of formula (3) each closed $  p $-
 +
form $  \alpha $
 +
defines a $  p $-
 +
dimensional singular cocycle whose value on the simplex $  s $
 +
is $  \int _ {s} \alpha $.  
 +
This correspondence is a realization of the isomorphism given by de Rham's theorem.
  
 
Formula (3) was published in 1899 by H. Poincaré [[#References|[2]]], who regarded exterior forms as integrand expressions in integral invariants. At the same time E. Cartan [[#References|[3]]] gave an almost-modern definition of exterior forms and of the exterior differentiation operator (at first on Pfaffian forms), stressing the connection between his own construction and exterior algebra.
 
Formula (3) was published in 1899 by H. Poincaré [[#References|[2]]], who regarded exterior forms as integrand expressions in integral invariants. At the same time E. Cartan [[#References|[3]]] gave an almost-modern definition of exterior forms and of the exterior differentiation operator (at first on Pfaffian forms), stressing the connection between his own construction and exterior algebra.
  
As well as the exterior scalar forms defined above, one may also study exterior differential forms with values in a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213094.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213095.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213096.png" /> is an algebra, then a natural multiplication (an extension of the exterior multiplication) is defined on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213097.png" /> of forms with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213098.png" />. If the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d03213099.png" /> is also associative, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130100.png" /> is associative as well; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130101.png" /> is commutative, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130102.png" /> is graded-commutative (formula (2)); if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130103.png" /> is a Lie algebra, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130104.png" /> is a graded Lie algebra. The following, even more general, concept is also often considered. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130105.png" /> be a smooth vector bundle with base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130106.png" />. If for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130107.png" /> there is given a skew-symmetric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130108.png" />-linear function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130109.png" /> with values in the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130110.png" /> of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130111.png" />, a so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130114.png" />-valued <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130115.png" />-form is obtained. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130116.png" />-valued <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130117.png" />-form can also be interpreted as a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130118.png" />-linear (over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130119.png" />) mapping of the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130120.png" /> into the module of smooth sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130121.png" />. The space of such forms is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130122.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130123.png" /> is given by locally constant transition functions or, which amounts to the same thing, if a flat connection is specified on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130124.png" />, it is possible to define the de Rham complex and to generalize the de Rham theorem to this case.
+
As well as the exterior scalar forms defined above, one may also study exterior differential forms with values in a vector space $  V $
 +
over $  \mathbf R $.  
 +
If $  V $
 +
is an algebra, then a natural multiplication (an extension of the exterior multiplication) is defined on the space $  E ( M , V ) $
 +
of forms with values in $  V $.  
 +
If the algebra $  V $
 +
is also associative, $  E ( M , V ) $
 +
is associative as well; if $  V $
 +
is commutative, $  E ( M , V ) $
 +
is graded-commutative (formula (2)); if $  V $
 +
is a Lie algebra, then $  E ( M , V ) $
 +
is a graded Lie algebra. The following, even more general, concept is also often considered. Let $  F $
 +
be a smooth vector bundle with base $  M $.  
 +
If for each point $  x \in M $
 +
there is given a skew-symmetric $  p $-
 +
linear function on $  T _ {x} ( M) $
 +
with values in the fibre $  F _ {x} $
 +
of the bundle $  F $,  
 +
a so-called $  F $-
 +
valued $  p $-
 +
form is obtained. An $  F $-
 +
valued $  p $-
 +
form can also be interpreted as a $  p $-
 +
linear (over $  F ( M) $)  
 +
mapping of the module $  {\mathcal X} ( M)  ^ {p} $
 +
into the module of smooth sections of $  F $.  
 +
The space of such forms is denoted by $  E  ^ {p} ( F  ) $.  
 +
If $  F $
 +
is given by locally constant transition functions or, which amounts to the same thing, if a flat connection is specified on $  F $,  
 +
it is possible to define the de Rham complex and to generalize the de Rham theorem to this case.
  
Forms with values in the tangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130125.png" /> are also called vector differential forms; these forms may be identified with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130126.png" /> times covariant and one time contravariant tensor fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130127.png" /> which are skew-symmetric with respect to the covariant indices. Vector differential forms are used to describe the derivations of the algebra of exterior forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130128.png" /> [[#References|[4]]]. Vector forms (as well as their generalization — jet forms) are used in the theory of deformations of complex and other differential-geometric structures on manifolds.
+
Forms with values in the tangent bundle $  T ( M) $
 +
are also called vector differential forms; these forms may be identified with $  p $
 +
times covariant and one time contravariant tensor fields on $  M $
 +
which are skew-symmetric with respect to the covariant indices. Vector differential forms are used to describe the derivations of the algebra of exterior forms $  E ( M) $[[#References|[4]]]. Vector forms (as well as their generalization — jet forms) are used in the theory of deformations of complex and other differential-geometric structures on manifolds.
  
Analogues of differential forms are also constructed in simplicial theory. One such construction, whose idea is due to H. Whitney [[#References|[5]]], may be used to calculate the rational cohomology of a simplicial complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130129.png" />. A piecewise-linear form (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130131.png" />-form) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130132.png" /> is a compatible family of differential forms defined on the simplices of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130133.png" />, with polynomials with rational coefficients as coefficients when written in barycentric coordinates. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130134.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130135.png" /> form a graded commutative differential algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130136.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130137.png" />. The integration of forms determines an isomorphism of the cohomology algebra of this algebra onto the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130138.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130139.png" /> is the polyhedron corresponding to the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130140.png" />. The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130141.png" /> also completely defines the rational homotopy type (in particular, the ranks of homotopy groups) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130142.png" />. In a similar manner, the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130143.png" /> on a differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130144.png" /> defines the real homotopy type of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130145.png" /> [[#References|[9]]], [[#References|[11]]].
+
Analogues of differential forms are also constructed in simplicial theory. One such construction, whose idea is due to H. Whitney [[#References|[5]]], may be used to calculate the rational cohomology of a simplicial complex $  K $.  
 +
A piecewise-linear form (or $  PL $-
 +
form) on $  K $
 +
is a compatible family of differential forms defined on the simplices of the complex $  K $,  
 +
with polynomials with rational coefficients as coefficients when written in barycentric coordinates. The $  PL $-
 +
forms on $  K $
 +
form a graded commutative differential algebra $  E _ {PL}  ^ {*} ( K) $
 +
over $  \mathbf Q $.  
 +
The integration of forms determines an isomorphism of the cohomology algebra of this algebra onto the algebra $  H  ^ {*} ( | K | , \mathbf Q ) $,  
 +
where $  | K | $
 +
is the polyhedron corresponding to the complex $  K $.  
 +
The algebra $  E _ {PL}  ^ {*} ( K) $
 +
also completely defines the rational homotopy type (in particular, the ranks of homotopy groups) of $  | K | $.  
 +
In a similar manner, the algebra $  E  ^ {*} ( M) $
 +
on a differentiable manifold $  M $
 +
defines the real homotopy type of $  M $[[#References|[9]]], [[#References|[11]]].
  
The calculus of exterior forms on a complex analytic manifold has a number of special features [[#References|[6]]]. In this situation it is usual to consider the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130146.png" /> of complex-valued forms, or the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130147.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130148.png" /> is a holomorphic vector bundle on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130149.png" />. The following decomposition is valid:
+
The calculus of exterior forms on a complex analytic manifold has a number of special features [[#References|[6]]]. In this situation it is usual to consider the space $  E  ^ {p} ( M , \mathbf C ) $
 +
of complex-valued forms, or the spaces $  E  ^ {p} ( F  ) $
 +
where $  F $
 +
is a holomorphic vector bundle on $  M $.  
 +
The following decomposition is valid:
  
<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/d/d032/d032130/d032130150.png" /></td> </tr></table>
+
$$
 +
E  ^ {p} ( M , \mathbf C )  = \sum _ {r+ s = p } E ^ {r , s } ( M) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130151.png" /> is the space of forms of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130153.png" />, i.e. of forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130154.png" /> which are locally representable as
+
where $  E ^ {r , s } ( M) $
 +
is the space of forms of type $  ( r , s ) $,  
 +
i.e. of forms $  \alpha $
 +
which are locally representable 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/d/d032/d032130/d032130155.png" /></td> </tr></table>
+
$$
 +
\sum a _ {i _ {1}  \dots i _ {r} , j _ {1} \dots j _ {s} } \
 +
dz ^ {i _ {1} } \wedge \dots \wedge dz ^ {i _ {r} } \wedge d {\overline{z}\; } {} ^
 +
{j _ {1} } \wedge \dots \wedge d {\overline{z}\; } {} ^ {j _ {s} } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130156.png" /> is a local analytic coordinate system on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130157.png" />. Similarly,
+
where $  ( z  ^ {1} \dots z  ^ {n} ) $
 +
is a local analytic coordinate system on $  M $.  
 +
Similarly,
  
<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/d/d032/d032130/d032130158.png" /></td> </tr></table>
+
$$
 +
E  ^ {p} ( F  )  = \sum _ {r + s = p } E ^ {r , s } ( F  ) .
 +
$$
  
Further, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130159.png" />, where
+
Further, $  d = d ^  \prime  + d  ^ {\prime\prime} $,  
 +
where
  
<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/d/d032/d032130/d032130160.png" /></td> </tr></table>
+
$$
 +
d  ^  \prime  : E ^ {r , s } ( M)  \rightarrow  E ^ {r + 1 , s } ( M) ,\ \
 +
d  ^ {\prime\prime} : E ^ {r , s } ( M)  \rightarrow  E ^ {r , s + 1 }
 +
( M) .
 +
$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130161.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130162.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130163.png" /> define cochain complexes. The best known is the complex of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130164.png" /> (the Dolbeault complex), the cohomology of which is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130165.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130166.png" />-cocycles of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130167.png" /> are holomorphic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130169.png" />-forms (cf. [[Holomorphic form|Holomorphic form]]). The following Grothendieck lemma is valid for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130170.png" />: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130171.png" /> is a form of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130172.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130173.png" /> in a neighbourhood of zero of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130174.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130175.png" />, then a smaller neighbourhood of zero contains a form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130176.png" /> of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130177.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130178.png" />. The Dolbeault complex may also be defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130179.png" />-valued forms where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130180.png" /> is a holomorphic vector bundle. This leads to the cohomology spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130181.png" />. The Grothendieck lemma implies the following isomorphism:
+
Here $  {d }  ^  \prime  2 = {d }  ^ {\prime\prime} 2 = 0 $,  
 +
so that d ^  \prime  $
 +
and d ^ {\prime\prime} $
 +
define cochain complexes. The best known is the complex of the operator d ^ {\prime\prime} $(
 +
the Dolbeault complex), the cohomology of which is denoted by $  H ^ {r , s } ( M) $.  
 +
d ^ {\prime\prime} $-
 +
cocycles of type $  ( p , 0 ) $
 +
are holomorphic $  p $-
 +
forms (cf. [[Holomorphic form|Holomorphic form]]). The following Grothendieck lemma is valid for d ^ {\prime\prime} $:  
 +
If $  \alpha $
 +
is a form of type $  ( r , s ) $
 +
with  $  s > 0 $
 +
in a neighbourhood of zero of the space $  \mathbf C  ^ {n} $
 +
and  $  d ^ {\prime\prime} \alpha = 0 $,  
 +
then a smaller neighbourhood of zero contains a form $  \beta $
 +
of type $  ( r , s - 1 ) $
 +
such that $  \alpha = d ^ {\prime\prime} \beta $.  
 +
The Dolbeault complex may also be defined for $  F $-
 +
valued forms where $  F $
 +
is a holomorphic vector bundle. This leads to the cohomology spaces $  H ^ {r , s } ( F  ) $.  
 +
The Grothendieck lemma implies the following isomorphism:
  
<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/d/d032/d032130/d032130182.png" /></td> </tr></table>
+
$$
 +
H ^ {r , s } ( F  )  \cong  H  ^ {s} ( M , \Omega  ^ {r} ( F  ) ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130183.png" /> is the sheaf of germs of holomorphic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130184.png" />-valued <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130185.png" />-forms (Dolbeault's theorem). In particular,
+
where $  \Omega  ^ {r} ( F  ) $
 +
is the sheaf of germs of holomorphic $  F $-
 +
valued $  r $-
 +
forms (Dolbeault's theorem). In particular,
  
<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/d/d032/d032130/d032130186.png" /></td> </tr></table>
+
$$
 +
H ^ {r , s } ( M)  \cong  H  ^ {s} ( M , \Omega  ^ {r} ( M) ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130187.png" /> is the sheaf of germs of holomorphic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130188.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130189.png" />. There exists a spectral sequence with first term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130190.png" /> and converging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130191.png" />. The Euler characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130192.png" /> of a compact complex manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130193.png" /> is expressed in terms of the Dolbeault cohomology spaces by the formula
+
where $  \Omega  ^ {r} ( M) $
 +
is the sheaf of germs of holomorphic $  r $-
 +
forms on $  M $.  
 +
There exists a spectral sequence with first term $  \sum _ {r , s }  H ^ {r , s } ( M) $
 +
and converging to $  H  ^ {*} ( M , \mathbf C ) $.  
 +
The Euler characteristic $  \chi ( M) $
 +
of a compact complex manifold $  M $
 +
is expressed in terms of the Dolbeault cohomology spaces by the formula
  
<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/d/d032/d032130/d032130194.png" /></td> </tr></table>
+
$$
 +
\chi ( M)  = \sum _ {r , s } (- 1)  ^ {r+} s  \mathop{\rm dim}  H ^
 +
{r , s } ( M) .
 +
$$
  
 
Differential forms are an important component of the apparatus of differential geometry [[#References|[7]]], . They are also systematically employed in topology, in the theory of differential equations, in mechanics, in the theory of complex manifolds, and in the theory of functions of several complex variables. Currents are a generalization of differential forms, similar to generalized functions. The algebraic analogue of the theory of differential forms (cf. [[Derivations, module of|Derivations, module of]]) makes it possible to define differential forms on algebraic varieties and analytic spaces (cf. [[Differential calculus (on analytic spaces)|Differential calculus (on analytic spaces)]]). See also [[De Rham cohomology|de Rham cohomology]]; [[Differential on a Riemann surface|Differential on a Riemann surface]]; [[Harmonic form|Harmonic form]]; [[Holomorphic form|Holomorphic form]]; [[Laplace operator|Laplace operator]].
 
Differential forms are an important component of the apparatus of differential geometry [[#References|[7]]], . They are also systematically employed in topology, in the theory of differential equations, in mechanics, in the theory of complex manifolds, and in the theory of functions of several complex variables. Currents are a generalization of differential forms, similar to generalized functions. The algebraic analogue of the theory of differential forms (cf. [[Derivations, module of|Derivations, module of]]) makes it possible to define differential forms on algebraic varieties and analytic spaces (cf. [[Differential calculus (on analytic spaces)|Differential calculus (on analytic spaces)]]). See also [[De Rham cohomology|de Rham cohomology]]; [[Differential on a Riemann surface|Differential on a Riemann surface]]; [[Harmonic form|Harmonic form]]; [[Holomorphic form|Holomorphic form]]; [[Laplace operator|Laplace operator]].
Line 99: Line 329:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> G. de Rham, "Differentiable manifolds" , Springer (1984) (Translated from French) {{MR|}} {{ZBL|0534.58003}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> H. Poincaré, "Les méthodes nouvelles de la mécanique céleste" , '''3''' , Gauthier-Villars (1899) pp. Chapt. 26 {{MR|0926908}} {{MR|0926907}} {{MR|0926906}} {{MR|0087814}} {{MR|0087813}} {{MR|0087812}} {{ZBL|30.0834.08}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E. Cartan, "Sur certaines expressions différentielles et le problème de Pfaff" , ''Oeuvres complètes'' , '''1, Pt. 2''' , Gauthier-Villars pp. 303–396 {{MR|1508969}} {{ZBL|30.0313.04}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A. Frölicher, A. Nijenhuis, "Theory of vector-valued differential forms. I. Derivations in the graded ring of differential forms" ''Proc. Koninkl. Ned. Akad. Wet. Ser. A'' , '''59''' : 3 (1956) pp. 338–359 {{MR|0082554}} {{ZBL|0079.37502}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> H. Whitney, "Geometric integration theory" , Princeton Univ. Press (1957) {{MR|0087148}} {{ZBL|0083.28204}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980) {{MR|0608414}} {{ZBL|0435.32004}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964) {{MR|0193578}} {{ZBL|0129.13102}} </TD></TR><TR><TD valign="top">[8a]</TD> <TD valign="top"> H. Cartan, "Calcul différentiel" , Hermann (1967) {{MR|0223194}} {{ZBL|0156.36102}} </TD></TR><TR><TD valign="top">[8b]</TD> <TD valign="top"> H. Cartan, "Formes différentielles" , Hermann (1967) {{MR|0231303}} {{ZBL|0184.12701}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> P. Deligne, P. Griffiths, J. Morgan, D. Sullivan, "The real homotopy of Kaehler manifolds" ''Invent. Math.'' , '''29''' (1975) pp. 245–274  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> R. Bott, L.W. Tu, "Differential forms in algebraic topology" , Springer (1982) {{MR|0658304}} {{ZBL|0496.55001}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> D. Sullivan, "Infinitesimal computations in topology" ''Publ. Math. IHES'' , '''47''' (1977) pp. 269–331 {{MR|0646078}} {{ZBL|0374.57002}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> G. de Rham, "Differentiable manifolds" , Springer (1984) (Translated from French) {{MR|}} {{ZBL|0534.58003}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> H. Poincaré, "Les méthodes nouvelles de la mécanique céleste" , '''3''' , Gauthier-Villars (1899) pp. Chapt. 26 {{MR|0926908}} {{MR|0926907}} {{MR|0926906}} {{MR|0087814}} {{MR|0087813}} {{MR|0087812}} {{ZBL|30.0834.08}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> E. Cartan, "Sur certaines expressions différentielles et le problème de Pfaff" , ''Oeuvres complètes'' , '''1, Pt. 2''' , Gauthier-Villars pp. 303–396 {{MR|1508969}} {{ZBL|30.0313.04}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A. Frölicher, A. Nijenhuis, "Theory of vector-valued differential forms. I. Derivations in the graded ring of differential forms" ''Proc. Koninkl. Ned. Akad. Wet. Ser. A'' , '''59''' : 3 (1956) pp. 338–359 {{MR|0082554}} {{ZBL|0079.37502}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> H. Whitney, "Geometric integration theory" , Princeton Univ. Press (1957) {{MR|0087148}} {{ZBL|0083.28204}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980) {{MR|0608414}} {{ZBL|0435.32004}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964) {{MR|0193578}} {{ZBL|0129.13102}} </TD></TR><TR><TD valign="top">[8a]</TD> <TD valign="top"> H. Cartan, "Calcul différentiel" , Hermann (1967) {{MR|0223194}} {{ZBL|0156.36102}} </TD></TR><TR><TD valign="top">[8b]</TD> <TD valign="top"> H. Cartan, "Formes différentielles" , Hermann (1967) {{MR|0231303}} {{ZBL|0184.12701}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> P. Deligne, P. Griffiths, J. Morgan, D. Sullivan, "The real homotopy of Kaehler manifolds" ''Invent. Math.'' , '''29''' (1975) pp. 245–274  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> R. Bott, L.W. Tu, "Differential forms in algebraic topology" , Springer (1982) {{MR|0658304}} {{ZBL|0496.55001}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> D. Sullivan, "Infinitesimal computations in topology" ''Publ. Math. IHES'' , '''47''' (1977) pp. 269–331 {{MR|0646078}} {{ZBL|0374.57002}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
In the Western literature, by a differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130195.png" />-form on a differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130196.png" /> one always means a smooth section of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130197.png" />-th exterior power of the tangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130198.png" />, i.e. a smooth section of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130199.png" />. (These are called exterior differential forms in the main article.)
+
In the Western literature, by a differential $  p $-
 +
form on a differentiable manifold $  M $
 +
one always means a smooth section of the $  p $-
 +
th exterior power of the tangent bundle $  T M $,  
 +
i.e. a smooth section of $  \Lambda  ^ {p} ( T M ) $.  
 +
(These are called exterior differential forms in the main article.)
  
The value of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130200.png" /> on a differential form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130201.png" /> is also called the pullback of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130202.png" /> under the infinitesimal transformation generated by the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130203.png" />.
+
The value of the operator $  L _ {X} $
 +
on a differential form $  x $
 +
is also called the pullback of $  x $
 +
under the infinitesimal transformation generated by the vector field $  X $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130204.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130205.png" />-manifold of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130206.png" />. A current on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130207.png" /> is a linear functional defined on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130208.png" /> of smooth (anti-symmetric) differential forms of compact support on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130209.png" />. The current <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130210.png" /> is said to be homogeneous of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130211.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130212.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130213.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130214.png" />.
+
Let $  X $
 +
be a $  C  ^  \infty  $-
 +
manifold of dimension $  n $.  
 +
A current on $  X $
 +
is a linear functional defined on the space $  \Omega _ {c} ( X) $
 +
of smooth (anti-symmetric) differential forms of compact support on $  X $.  
 +
The current $  T $
 +
is said to be homogeneous of dimension $  p $
 +
if $  T ( \phi ) = 0 $
 +
for all $  \phi \in \Omega _ {c}  ^ {q} ( X) $
 +
with $  q \neq p $.
  
The degree of a current <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130215.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130216.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130217.png" /> is its dimension (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130218.png" /> is homogeneous).
+
The degree of a current $  T $
 +
is $  n - p $,  
 +
if $  p $
 +
is its dimension (if $  T $
 +
is homogeneous).
  
Define a chain element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130219.png" /> (of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130220.png" />) as a smooth mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130221.png" /> of the standard cube of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130222.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130223.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130224.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130225.png" />-form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130226.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130227.png" /> is defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130228.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130229.png" /> (cf. the main article above). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130230.png" /> is smooth and orientation-preserving, the integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130231.png" /> only depends on the image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130232.png" /> (and on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130233.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130234.png" />). More generally, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130236.png" />-chain on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130237.png" /> is a formal linear combination <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130238.png" /> of chain elements. The corresponding integral is defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130239.png" />. Each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130240.png" />-chain thus defines a current on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130241.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130242.png" /> and degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130243.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130244.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130245.png" />-form, it defines a homogeneous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130246.png" /> current by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130247.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130248.png" /> be a contravariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130249.png" />-vector with local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130250.png" />. Let the local coordinates of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130251.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130252.png" /> be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130253.png" />; then
+
Define a chain element in $  X $(
 +
of dimension $  p $)  
 +
as a smooth mapping $  s : \Delta _ {p} \rightarrow X $
 +
of the standard cube of dimension $  p $
 +
into $  X $.  
 +
If $  \omega $
 +
is a $  p $-
 +
form on $  X $,  
 +
then $  \int _ {s} \omega $
 +
is defined as $  \int _ {\Delta _ {p}  } f $
 +
if  $  s {}  ^  \star  ( \omega ) = d x _ {1} \wedge \dots \wedge d x _ {p} $(
 +
cf. the main article above). If $  s $
 +
is smooth and orientation-preserving, the integral $  \int _ {s} \omega $
 +
only depends on the image $  s ( \Delta _ {p} ) \subset  X $(
 +
and on $  \omega $
 +
over $  s ( \Delta _ {p} ) $).  
 +
More generally, a $  p $-
 +
chain on a manifold $  X $
 +
is a formal linear combination $  c = \sum a _ {i} s _ {i} $
 +
of chain elements. The corresponding integral is defined as $  \int _ {c} \omega = \sum a _ {i} \int _ {s _ {i}  } \omega $.  
 +
Each $  p $-
 +
chain thus defines a current on $  X $
 +
of dimension $  p $
 +
and degree $  n - p $.  
 +
If $  \alpha $
 +
is a $  q $-
 +
form, it defines a homogeneous $  n - q $
 +
current by the formula $  T _  \alpha  ( \omega ) = \int _ {V} \alpha \wedge \omega $.  
 +
Let $  b $
 +
be a contravariant $  p $-
 +
vector with local coordinates $  b ^ {i _ {1} \dots i _ {p} } $.  
 +
Let the local coordinates of the $  p $-
 +
form $  \omega $
 +
be $  \omega _ {i _ {1}  \dots i _ {p} } $;  
 +
then
  
<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/d/d032/d032130/d032130254.png" /></td> </tr></table>
+
$$
 +
T _ {b} ( \omega )  = \sum _ {i _ {1} < \dots < i _ {p} } \omega _ {i _ {1}  \dots i _ {p} } b ^ {i _ {1} \dots i _ {p} }
 +
$$
  
defines a current of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130255.png" />. Thus, currents generalize both forms (i.e. covariant vectors) and contravariant vectors. They are also a global generalization of the idea of distributions (as a generalization of functions). The role of a space of test functions is played by the smooth forms of compact support. The name "current" comes from the fact that in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130256.png" /> the currents of dimension 1 can be interpreted as electric currents.
+
defines a current of dimension $  p $.  
 +
Thus, currents generalize both forms (i.e. covariant vectors) and contravariant vectors. They are also a global generalization of the idea of distributions (as a generalization of functions). The role of a space of test functions is played by the smooth forms of compact support. The name "current" comes from the fact that in $  \mathbf R  ^ {3} $
 +
the currents of dimension 1 can be interpreted as electric currents.
  
In the theory of several complex variables one defines the currents of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130257.png" /> over a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130258.png" /> as the (complex-valued) linear functionals on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130259.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130260.png" /> (complex-valued) forms with compact support. In this area currents and their applications are a very active field of research at the moment [[#References|[a2]]].
+
In the theory of several complex variables one defines the currents of type $  ( p , q ) $
 +
over a domain $  D $
 +
as the (complex-valued) linear functionals on the space $  \Omega _ {c} ^ {n - p , n - q } ( D) $
 +
of $  ( n - p , n - q ) $(
 +
complex-valued) forms with compact support. In this area currents and their applications are a very active field of research at the moment [[#References|[a2]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E.M. Chirka, "Complex analytic sets" , Kluwer (1989) (Translated from Russian) {{MR|1111477}} {{ZBL|0683.32002}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P. Lelong, L. Gruman, "Entire functions of several complex variables" , Springer (1986) {{MR|0837659}} {{ZBL|0583.32001}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E.M. Chirka, "Complex analytic sets" , Kluwer (1989) (Translated from Russian) {{MR|1111477}} {{ZBL|0683.32002}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P. Lelong, L. Gruman, "Entire functions of several complex variables" , Springer (1986) {{MR|0837659}} {{ZBL|0583.32001}} </TD></TR></table>
  
A differential form on an algebraic variety is the analogue of the concept of a differential form on a differentiable manifold. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130261.png" /> be an irreducible algebraic variety of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130262.png" /> over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130263.png" /> (cf. [[Irreducible variety|Irreducible variety]]) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130264.png" /> be its field of rational functions. A differential form of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130266.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130267.png" /> is an element of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130268.png" />-space
+
A differential form on an algebraic variety is the analogue of the concept of a differential form on a differentiable manifold. Let $  X $
 +
be an irreducible algebraic variety of dimension d $
 +
over an algebraically closed field $  k $(
 +
cf. [[Irreducible variety|Irreducible variety]]) and let $  K $
 +
be its field of rational functions. A differential form of degree $  r $
 +
on $  X $
 +
is an element of the $  K $-
 +
space
  
<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/d/d032/d032130/d032130269.png" /></td> </tr></table>
+
$$
 +
\Omega  ^ {r} ( X)  = \wedge ^ { r }  \Omega _ {K / k }  ^ {1} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130270.png" /> is the module of derivations (cf. [[Derivations, module of|Derivations, module of]]) of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130271.png" /> over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130272.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130273.png" /> is a separable basis of transcendency of the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130274.png" />, any differential form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130275.png" /> can be written as
+
where $  \Omega _ {K/k} $
 +
is the module of derivations (cf. [[Derivations, module of|Derivations, module of]]) of the field $  K $
 +
over the field $  k $.  
 +
If $  x _ {1} \dots x _ {d} $
 +
is a separable basis of transcendency of the extension $  K/k $,  
 +
any differential form $  \omega \in \Omega  ^ {r} ( X) $
 +
can be written 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/d/d032/d032130/d032130276.png" /></td> </tr></table>
+
$$
 +
\omega  = \sum a _ {i _ {1}  \dots i _ {r} }  dx _ {i} \wedge \dots
 +
\wedge dx _ {i _ {r}  } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130277.png" />. A differential form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130278.png" /> is called regular on an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130279.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130280.png" /> belongs to the submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130281.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130282.png" />, regarded as a module over the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130283.png" /> of regular functions on the subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130284.png" />. A differential form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130285.png" /> is called regular if any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130286.png" /> has a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130287.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130288.png" /> is regular on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130289.png" />. The regular differential forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130290.png" /> form a module over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130291.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130292.png" />. Its elements are identified with the sections of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130293.png" /> over the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130294.png" />. In a neighbourhood of each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130295.png" /> a regular differential form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130296.png" /> is written as
+
where $  a _ {i _ {1}  \dots i _ {r} } \in K $.  
 +
A differential form $  \omega $
 +
is called regular on an open set $  U \subset  X $
 +
if $  \omega $
 +
belongs to the submodule $  \Omega _ {k [ U] / k }  ^ {r} $
 +
of $  \Omega  ^ {r} ( X) $,  
 +
regarded as a module over the ring $  k [ U] $
 +
of regular functions on the subset $  U $.  
 +
A differential form $  \omega $
 +
is called regular if any point $  x \in X $
 +
has a neighbourhood $  U $
 +
such that $  \omega $
 +
is regular on $  U $.  
 +
The regular differential forms on $  X $
 +
form a module over $  k [ X] $,  
 +
denoted by $  \Omega  ^ {r} [ X] $.  
 +
Its elements are identified with the sections of the sheaf $  \Omega _ {X/k}  ^ {r} $
 +
over the variety $  X $.  
 +
In a neighbourhood of each point $  x \in X $
 +
a regular differential form $  \omega \subset  \Omega  ^ {r} [ X] $
 +
is written 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/d/d032/d032130/d032130297.png" /></td> </tr></table>
+
$$
 +
\omega  = \sum \alpha _ {i _ {1}  \dots i _ {r} }  df _ {i _ {1}  }
 +
\wedge \dots \wedge df _ {i _ {r}  } ,
 +
$$
  
where the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130298.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130299.png" /> are regular at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130300.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130301.png" /> is a [[Complete algebraic variety|complete algebraic variety]], the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130302.png" /> are finite-dimensional, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130303.png" /> is non-singular, the dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130304.png" /> is known as the geometric genus of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130305.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130306.png" /> is a complete variety over the field of complex numbers, the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130307.png" /> is identical with the space of holomorphic differential forms of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130308.png" /> on the corresponding [[Analytic space|analytic space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130309.png" />.
+
where the functions $  \alpha _ {i _ {1}  \dots i _ {r} } $,  
 +
$  f _ {i _ {1}  } \dots f _ {i _ {r}  } $
 +
are regular at the point $  x $.  
 +
If $  X $
 +
is a [[Complete algebraic variety|complete algebraic variety]], the spaces $  \Omega  ^ {r} [ X] $
 +
are finite-dimensional, and if $  X $
 +
is non-singular, the dimension $  p _ {g} ( X) = \mathop{\rm dim} _ {k}  \Omega  ^ {d} [ X] $
 +
is known as the geometric genus of the variety $  X $.  
 +
If $  X $
 +
is a complete variety over the field of complex numbers, the space $  \Omega  ^ {r} [ X] $
 +
is identical with the space of holomorphic differential forms of degree $  r $
 +
on the corresponding [[Analytic space|analytic space]] $  X ^ {\textrm{ an } } $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130310.png" /> be a normal variety and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130311.png" />; for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130312.png" /> of codimension one the differential form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130313.png" /> may be written as
+
Let $  X $
 +
be a normal variety and let $  \omega \in \Omega  ^ {d} [ X] $;  
 +
for any point $  x \in X  ^ {(} 1) $
 +
of codimension one the differential form $  \omega $
 +
may be written 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/d/d032/d032130/d032130314.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
\omega  = a  dt \wedge dt _ {1} \wedge \dots \wedge dt _ {d- 1 }  ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130315.png" /> belongs to the field of fractions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130316.png" /> of the local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130317.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130318.png" /> is the generator of its maximal ideal, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130319.png" /> is a separable basis of transcendency over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130320.png" /> of the residue field of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130321.png" />. The value of the valuation at the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130322.png" />, as defined by the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130323.png" />, does not depend on the choice of the representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130324.png" /> in the form (*) and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130325.png" />. The divisor
+
where $  a $
 +
belongs to the field of fractions $  K _ {x} $
 +
of the local ring $  {\mathcal O} _ {X,x }  $,  
 +
$  t $
 +
is the generator of its maximal ideal, and $  t _ {1} \dots t _ {d-} 1 $
 +
is a separable basis of transcendency over $  k $
 +
of the residue field of the ring $  {\mathcal O} _ {X,x }  $.  
 +
The value of the valuation at the element $  a $,  
 +
as defined by the ring $  {\mathcal O} _ {X,x }  $,  
 +
does not depend on the choice of the representation of $  \omega $
 +
in the form (*) and is denoted by $  \nu _ {x} ( \omega ) $.  
 +
The divisor
  
<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/d/d032/d032130/d032130326.png" /></td> </tr></table>
+
$$
 +
= \sum _ {x \in X  ^ {(} 1) } \nu _ {x} ( \omega ) \{ \overline{x}\; \}
 +
$$
  
is defined and is known as divisor of the differential form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130327.png" />. A differential form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130328.png" /> is regular if and only if its divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130329.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130330.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130331.png" />. The divisors of any two differential forms are equivalent and, moreover, the divisors of all differential forms on a given algebraic variety form a divisor class with respect to linear equivalence. This class is known as the [[canonical class]] of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130332.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130333.png" />. For a non-singular variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130334.png" /> the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130335.png" /> is identical with the first Chern class of the invertible sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130336.png" />; in particular,
+
is defined and is known as divisor of the differential form $  \omega $.  
 +
A differential form $  \omega $
 +
is regular if and only if its divisor $  D \geq  0 $,  
 +
i.e. $  \nu _ {x} ( \omega ) \geq  0 $
 +
for all $  x \in X  ^ {(} 1) $.  
 +
The divisors of any two differential forms are equivalent and, moreover, the divisors of all differential forms on a given algebraic variety form a divisor class with respect to linear equivalence. This class is known as the [[canonical class]] of the variety $  X $
 +
and is denoted by $  K _ {X} $.  
 +
For a non-singular variety $  X $
 +
the class $  K _ {X} $
 +
is identical with the first Chern class of the invertible sheaf $  \Omega _ {X/k }  ^ {d} $;  
 +
in particular,
  
<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/d/d032/d032130/d032130337.png" /></td> </tr></table>
+
$$
 +
\Omega _ {X / k }  ^ {d}  \simeq  {\mathcal O} _ {X} ( D )
 +
$$
  
for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130338.png" />.
+
for any $  D \in K _ {X} $.
  
Any dominant rational mapping between algebraic varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130339.png" /> induces a canonical homomorphism
+
Any dominant rational mapping between algebraic varieties $  f : X ^ { \prime } \rightarrow X $
 +
induces a canonical homomorphism
  
<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/d/d032/d032130/d032130340.png" /></td> </tr></table>
+
$$
 +
f ^ { * } : \Omega  ^ {r} ( X) \rightarrow \Omega  ^ {r} ( X ^ { \prime } ) .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130341.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130342.png" /> are non-singular and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130343.png" /> is complete, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130344.png" /> maps regular differential forms to regular ones. In particular, if two non-singular complete varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130345.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130346.png" /> are birationally isomorphic, the vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130347.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130348.png" /> are isomorphic over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130349.png" />.
+
If $  X $
 +
and $  X ^ { \prime } $
 +
are non-singular and $  X $
 +
is complete, $  f ^ { * } $
 +
maps regular differential forms to regular ones. In particular, if two non-singular complete varieties $  X $
 +
and $  X ^ { \prime } $
 +
are birationally isomorphic, the vector spaces $  \Omega  ^ {r} [ X] $
 +
and $  \Omega  ^ {r} [ X ^ { \prime } ] $
 +
are isomorphic over the field $  k $.
  
For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130350.png" /> the elements of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130351.png" />-th symmetric power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130352.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130353.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130354.png" /> are known as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130357.png" />-tuple differential forms of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130358.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130359.png" />. Each such differential form may be considered as a rational section of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130360.png" />. The regular sections
+
For any $  i > 1 $
 +
the elements of the $  i $-
 +
th symmetric power $  S  ^ {i} ( \Omega  ^ {r} ( X) ) $
 +
of the $  K $-
 +
space $  \Omega  ^ {r} ( X) $
 +
are known as $  i $-
 +
tuple differential forms of degree $  r $
 +
on $  X $.  
 +
Each such differential form may be considered as a rational section of the sheaf $  S  ^ {i} ( \Omega _ {X/k }  ^ {r} ) $.  
 +
The regular sections
  
<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/d/d032/d032130/d032130361.png" /></td> </tr></table>
+
$$
 +
\omega  \in  \Gamma ( X , S  ^ {i} ( \Omega _ {X/k }  ^ {r} ) )
 +
$$
  
are known as regular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130364.png" />-tuple differential forms of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130365.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130366.png" />. For a non-singular complete variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130367.png" /> the dimension
+
are known as regular $  i $-
 +
tuple differential forms of degree $  r $
 +
on $  X $.  
 +
For a non-singular complete variety $  X $
 +
the dimension
  
<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/d/d032/d032130/d032130368.png" /></td> </tr></table>
+
$$
 +
P _ {i} ( X)  =   \mathop{\rm dim} _ {k}  \Gamma ( X , S  ^ {i} ( \Omega _ {X/k }  ^ {d} ))
 +
$$
  
is known as the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130370.png" /> genus of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130371.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130372.png" /> genera of birationally isomorphic varieties are identical.
+
is known as the $  i $
 +
genus of the variety $  X $.  
 +
The $  i $
 +
genera of birationally isomorphic varieties are identical.
  
 
====References====
 
====References====
Line 172: Line 578:
  
 
====Comments====
 
====Comments====
In terms of the description of (Cartier) divisors by local functions (cf. [[Divisor|Divisor]]), the divisor associated to a differential form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130373.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130374.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130375.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130376.png" /> smooth, can be described as follows. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130377.png" /> there is an open affine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130378.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130379.png" /> can be represented in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130380.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130381.png" />. Now cover <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130382.png" /> by open affine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130383.png" />. Let the representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130384.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130385.png" /> be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130386.png" />. Then on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130387.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130388.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130389.png" /> times the Jacobian of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130390.png" /> with respect to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130391.png" />. Thus the local functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130392.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130393.png" /> define a divisor on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130394.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130395.png" />. One has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130396.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130397.png" /> and thus, because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130398.png" /> is one-dimensional over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130399.png" />, all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130400.png" /> define the same divisor class, the canonical class (canonical divisor class) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032130/d032130401.png" />.
+
In terms of the description of (Cartier) divisors by local functions (cf. [[Divisor|Divisor]]), the divisor associated to a differential form $  \omega $
 +
of degree $  n $,  
 +
$  n = \mathop{\rm dim}  X $,  
 +
$  X $
 +
smooth, can be described as follows. For each $  x \in X $
 +
there is an open affine $  U $
 +
such that $  \omega $
 +
can be represented in $  U $
 +
as  $  \omega = d u _ {1} \wedge \dots \wedge d u _ {n} $.  
 +
Now cover $  X $
 +
by open affine $  U _ {i} $.  
 +
Let the representation of $  \omega $
 +
in $  U _ {i} $
 +
be  $  \omega = g  ^ {(} i)  d u _ {1}  ^ {(} i) \wedge \dots \wedge d u _ {n}  ^ {(} i) $.  
 +
Then on $  U _ {i} \cap U _ {j} $,  
 +
$  g  ^ {(} j) $
 +
is equal to $  g  ^ {(} i) $
 +
times the Jacobian of the $  u _ {1}  ^ {(} i) \dots u _ {n}  ^ {(} i) $
 +
with respect to the $  u _ {1}  ^ {(} j) \dots u _ {n}  ^ {(} j) $.  
 +
Thus the local functions $  g  ^ {(} i) $
 +
on $  U _ {i} $
 +
define a divisor on $  X $,  
 +
denoted by $  ( \omega ) $.  
 +
One has $  ( f \omega ) = ( f  ) + ( \omega ) $
 +
for all $  f \in k ( X) $
 +
and thus, because $  \Omega  ^ {n} ( X) $
 +
is one-dimensional over $  k ( X) $,  
 +
all 0 \neq \omega \in \Omega  ^ {n} ( X) $
 +
define the same divisor class, the canonical class (canonical divisor class) of $  X $.

Revision as of 18:34, 5 June 2020


A differential form of degree $ p $, a $ p $- form, on a differentiable manifold $ M $ is a $ p $ times covariant tensor field on $ M $. It may also be interpreted as a $ p $- linear (over the algebra $ F( M) $ of smooth real-valued functions on $ M $) mapping $ {\mathcal X} ( M) ^ {p} \rightarrow F( M) $, where $ {\mathcal X} ( M) $ is the $ F( M) $- module of smooth vector fields on $ M $. Forms of degree one are also known as Pfaffian forms. An example of such a form is the differential $ df $ of a smooth function $ f $ on $ M $, which is defined as follows: $ ( df ) ( X) $, $ X \in {\mathcal X} ( M) $, is the derivative $ Xf $ of $ f $ in the direction of the field $ X $. Riemannian metrics on a manifold $ M $ serve as examples of symmetric differential forms of degree two. However, the term "differential form" is often used to denote skew-symmetric or exterior differential forms, which have the greatest number of applications.

If $ ( x ^ {1} \dots x ^ {n} ) $ is a local system of coordinates in a domain $ U \subset M $, the forms $ dx ^ {1} \dots dx ^ {n} $ constitute a basis of the cotangent space $ T _ {x} ( M) ^ {*} $, $ x \in U $. For this reason (cf. Exterior algebra) any exterior $ p $- form $ \alpha $ may be written in $ U $ in the form

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

where the $ a _ {i _ {1} \dots i _ {p} } $ are functions on $ U $. In particular,

$$ df = \frac{\partial f }{\partial x ^ {i} } dx ^ {i} . $$

Let $ E ^ {p} = E ^ {p} ( M) $ be the space of all exterior $ p $- forms of class $ C ^ \infty $, where $ E ^ {0} ( M) = F ( M) $. The exterior multiplication $ \alpha \wedge \beta $ converts $ E ^ {*} ( M) = \sum _ {p = 0 } ^ {n} E ^ {p} ( M) $( where $ n = \mathop{\rm dim} M $) to an associative graded algebra over $ F ( M) $ which satisfies the condition of graded commutativity

$$ \tag{2 } \alpha \wedge \beta = ( - 1 ) ^ {pq} \beta \wedge \alpha ,\ \ \alpha \in E ^ {p} ,\ \beta \in E ^ {q} . $$

A smooth mapping between manifolds $ f : M \rightarrow N $ induces a homomorphism $ f ^ { * } : E ^ {*} ( N) \rightarrow E ^ {*} ( M) $ between $ \mathbf R $- algebras.

The concept of the differential of a function is generalized as follows. For any $ p \geq 0 $ there exists a unique $ \mathbf R $- linear mapping $ d : E ^ {p} \rightarrow E ^ {p+} 1 $( exterior differentiation), which for $ p = 0 $ coincides with the differential introduced above, with the following properties:

$$ d ( \alpha \wedge \beta ) = d \alpha \wedge \beta + ( - 1 ) ^ {p} \alpha \wedge d \beta , $$

$$ \alpha \in E ^ {p} ,\ \beta \in E ^ {q} ,\ d ( d \alpha ) = 0 . $$

The exterior differential of a form $ \alpha $ written in in local coordinates (see (1)) is expressed by the formula

$$ d \alpha = \sum _ {i _ {1} \dots i _ {p} } da _ {i _ {1} \dots i _ {p} } \wedge dx ^ {i _ {1} } \wedge \dots \wedge dx ^ {i _ {p} } . $$

Its coordinate-free notation is

$$ d \alpha ( X _ {1} \dots X _ {p+} 1 ) = $$

$$ = \ \sum _ {i = 1 } ^ { {p } + 1 } (- 1) ^ {i+} 1 X _ {i} \alpha ( X _ {1} \dots \widehat{X} _ {i} \dots X _ {p+} 1 ) + $$

$$ - \sum _ {i < j } (- 1) ^ {i+} j \alpha ( [ X _ {i} , X _ {j} ] , X _ {1} \dots \widehat{X} _ {i} \dots \widehat{X} _ {j} \dots X _ {p+} 1 ) , $$

where $ X _ {1} \dots X _ {p+} 1 \in {\mathcal X} ( M) $. The Lie derivative operator $ L _ {X} $, $ X \in {\mathcal X} ( M) $, on differential forms is connected with the exterior differentiation operator by the relation

$$ L _ {X} = d \circ \iota _ {X} + \iota _ {X} \circ d , $$

where $ \iota _ {X} : E ^ {p} \rightarrow E ^ {p-} 1 $ is the operator of interior multiplication by $ X $:

$$ ( \iota _ {X} \alpha ) ( X _ {1} \dots X _ {p-} 1 ) = \alpha ( X , X _ {1} \dots X _ {p-} 1 ) , $$

$$ \alpha \in E ^ {p} ( M) ,\ X _ {1} \dots X _ {p-} 1 \in {\mathcal X} ( M) . $$

The complex $ ( E ^ {*} ( M) , d ) $ is a cochain complex (the de Rham complex). The cocycles of this complex are said to be closed forms, while the coboundaries are known as exact forms. According to the de Rham theorem, the cohomology algebra

$$ H ^ {*} ( M) = \sum _ {p = 0 } ^ { n } H ^ {p} ( M) $$

of the de Rham complex is isomorphic to the real cohomology algebra $ H ^ {*} ( M, \mathbf R ) $ of the manifold $ M $. In particular, $ H ^ {p} ( \mathbf R ^ {n} ) = 0 $ if $ p > 0 $( Poincaré's lemma).

The de Rham theorem is closely connected with another operation, that of integration of differential forms. Let $ D $ be a bounded domain in $ \mathbf R ^ {p} $ and let $ s $ be a smooth mapping $ \mathbf R ^ {p} \rightarrow M $, defined in a neighbourhood of the closure $ \overline{D}\; $. If $ \alpha \in E ^ {p} ( M) $, then $ s {} ^ \star \alpha = a dx ^ {1} \wedge \dots \wedge dx ^ {p} $, where $ a $ is a smooth function in $ \overline{D}\; $. The integral of the form $ \alpha $ over the surface $ s $ is defined by the formula:

$$ \int\limits _ { s } \alpha = \int\limits _ { D } a ( x _ {1} \dots x _ {p} ) dx ^ {1} \dots dx ^ {p} . $$

If the boundary of $ D $ is piecewise smooth, the formula

$$ \tag{3 } \int\limits _ { s } d \alpha = \int\limits _ {\partial s } \alpha ,\ \ \alpha \in E ^ {p-} 1 ( M) , $$

is valid; here $ \int _ {\partial s } \alpha $ is defined as the sum of the integrals of the form $ \alpha $ over the smooth pieces of the boundary, provided with their natural parametrizations. The classical formulas of Newton–Leibniz, Green–Ostrogradski and Stokes (see also Stokes theorem) are all special cases of this formula. By virtue of formula (3) each closed $ p $- form $ \alpha $ defines a $ p $- dimensional singular cocycle whose value on the simplex $ s $ is $ \int _ {s} \alpha $. This correspondence is a realization of the isomorphism given by de Rham's theorem.

Formula (3) was published in 1899 by H. Poincaré [2], who regarded exterior forms as integrand expressions in integral invariants. At the same time E. Cartan [3] gave an almost-modern definition of exterior forms and of the exterior differentiation operator (at first on Pfaffian forms), stressing the connection between his own construction and exterior algebra.

As well as the exterior scalar forms defined above, one may also study exterior differential forms with values in a vector space $ V $ over $ \mathbf R $. If $ V $ is an algebra, then a natural multiplication (an extension of the exterior multiplication) is defined on the space $ E ( M , V ) $ of forms with values in $ V $. If the algebra $ V $ is also associative, $ E ( M , V ) $ is associative as well; if $ V $ is commutative, $ E ( M , V ) $ is graded-commutative (formula (2)); if $ V $ is a Lie algebra, then $ E ( M , V ) $ is a graded Lie algebra. The following, even more general, concept is also often considered. Let $ F $ be a smooth vector bundle with base $ M $. If for each point $ x \in M $ there is given a skew-symmetric $ p $- linear function on $ T _ {x} ( M) $ with values in the fibre $ F _ {x} $ of the bundle $ F $, a so-called $ F $- valued $ p $- form is obtained. An $ F $- valued $ p $- form can also be interpreted as a $ p $- linear (over $ F ( M) $) mapping of the module $ {\mathcal X} ( M) ^ {p} $ into the module of smooth sections of $ F $. The space of such forms is denoted by $ E ^ {p} ( F ) $. If $ F $ is given by locally constant transition functions or, which amounts to the same thing, if a flat connection is specified on $ F $, it is possible to define the de Rham complex and to generalize the de Rham theorem to this case.

Forms with values in the tangent bundle $ T ( M) $ are also called vector differential forms; these forms may be identified with $ p $ times covariant and one time contravariant tensor fields on $ M $ which are skew-symmetric with respect to the covariant indices. Vector differential forms are used to describe the derivations of the algebra of exterior forms $ E ( M) $[4]. Vector forms (as well as their generalization — jet forms) are used in the theory of deformations of complex and other differential-geometric structures on manifolds.

Analogues of differential forms are also constructed in simplicial theory. One such construction, whose idea is due to H. Whitney [5], may be used to calculate the rational cohomology of a simplicial complex $ K $. A piecewise-linear form (or $ PL $- form) on $ K $ is a compatible family of differential forms defined on the simplices of the complex $ K $, with polynomials with rational coefficients as coefficients when written in barycentric coordinates. The $ PL $- forms on $ K $ form a graded commutative differential algebra $ E _ {PL} ^ {*} ( K) $ over $ \mathbf Q $. The integration of forms determines an isomorphism of the cohomology algebra of this algebra onto the algebra $ H ^ {*} ( | K | , \mathbf Q ) $, where $ | K | $ is the polyhedron corresponding to the complex $ K $. The algebra $ E _ {PL} ^ {*} ( K) $ also completely defines the rational homotopy type (in particular, the ranks of homotopy groups) of $ | K | $. In a similar manner, the algebra $ E ^ {*} ( M) $ on a differentiable manifold $ M $ defines the real homotopy type of $ M $[9], [11].

The calculus of exterior forms on a complex analytic manifold has a number of special features [6]. In this situation it is usual to consider the space $ E ^ {p} ( M , \mathbf C ) $ of complex-valued forms, or the spaces $ E ^ {p} ( F ) $ where $ F $ is a holomorphic vector bundle on $ M $. The following decomposition is valid:

$$ E ^ {p} ( M , \mathbf C ) = \sum _ {r+ s = p } E ^ {r , s } ( M) , $$

where $ E ^ {r , s } ( M) $ is the space of forms of type $ ( r , s ) $, i.e. of forms $ \alpha $ which are locally representable as

$$ \sum a _ {i _ {1} \dots i _ {r} , j _ {1} \dots j _ {s} } \ dz ^ {i _ {1} } \wedge \dots \wedge dz ^ {i _ {r} } \wedge d {\overline{z}\; } {} ^ {j _ {1} } \wedge \dots \wedge d {\overline{z}\; } {} ^ {j _ {s} } , $$

where $ ( z ^ {1} \dots z ^ {n} ) $ is a local analytic coordinate system on $ M $. Similarly,

$$ E ^ {p} ( F ) = \sum _ {r + s = p } E ^ {r , s } ( F ) . $$

Further, $ d = d ^ \prime + d ^ {\prime\prime} $, where

$$ d ^ \prime : E ^ {r , s } ( M) \rightarrow E ^ {r + 1 , s } ( M) ,\ \ d ^ {\prime\prime} : E ^ {r , s } ( M) \rightarrow E ^ {r , s + 1 } ( M) . $$

Here $ {d } ^ \prime 2 = {d } ^ {\prime\prime} 2 = 0 $, so that $ d ^ \prime $ and $ d ^ {\prime\prime} $ define cochain complexes. The best known is the complex of the operator $ d ^ {\prime\prime} $( the Dolbeault complex), the cohomology of which is denoted by $ H ^ {r , s } ( M) $. $ d ^ {\prime\prime} $- cocycles of type $ ( p , 0 ) $ are holomorphic $ p $- forms (cf. Holomorphic form). The following Grothendieck lemma is valid for $ d ^ {\prime\prime} $: If $ \alpha $ is a form of type $ ( r , s ) $ with $ s > 0 $ in a neighbourhood of zero of the space $ \mathbf C ^ {n} $ and $ d ^ {\prime\prime} \alpha = 0 $, then a smaller neighbourhood of zero contains a form $ \beta $ of type $ ( r , s - 1 ) $ such that $ \alpha = d ^ {\prime\prime} \beta $. The Dolbeault complex may also be defined for $ F $- valued forms where $ F $ is a holomorphic vector bundle. This leads to the cohomology spaces $ H ^ {r , s } ( F ) $. The Grothendieck lemma implies the following isomorphism:

$$ H ^ {r , s } ( F ) \cong H ^ {s} ( M , \Omega ^ {r} ( F ) ) , $$

where $ \Omega ^ {r} ( F ) $ is the sheaf of germs of holomorphic $ F $- valued $ r $- forms (Dolbeault's theorem). In particular,

$$ H ^ {r , s } ( M) \cong H ^ {s} ( M , \Omega ^ {r} ( M) ) , $$

where $ \Omega ^ {r} ( M) $ is the sheaf of germs of holomorphic $ r $- forms on $ M $. There exists a spectral sequence with first term $ \sum _ {r , s } H ^ {r , s } ( M) $ and converging to $ H ^ {*} ( M , \mathbf C ) $. The Euler characteristic $ \chi ( M) $ of a compact complex manifold $ M $ is expressed in terms of the Dolbeault cohomology spaces by the formula

$$ \chi ( M) = \sum _ {r , s } (- 1) ^ {r+} s \mathop{\rm dim} H ^ {r , s } ( M) . $$

Differential forms are an important component of the apparatus of differential geometry [7], . They are also systematically employed in topology, in the theory of differential equations, in mechanics, in the theory of complex manifolds, and in the theory of functions of several complex variables. Currents are a generalization of differential forms, similar to generalized functions. The algebraic analogue of the theory of differential forms (cf. Derivations, module of) makes it possible to define differential forms on algebraic varieties and analytic spaces (cf. Differential calculus (on analytic spaces)). See also de Rham cohomology; Differential on a Riemann surface; Harmonic form; Holomorphic form; Laplace operator.

References

[1] G. de Rham, "Differentiable manifolds" , Springer (1984) (Translated from French) Zbl 0534.58003
[2] H. Poincaré, "Les méthodes nouvelles de la mécanique céleste" , 3 , Gauthier-Villars (1899) pp. Chapt. 26 MR0926908 MR0926907 MR0926906 MR0087814 MR0087813 MR0087812 Zbl 30.0834.08
[3] E. Cartan, "Sur certaines expressions différentielles et le problème de Pfaff" , Oeuvres complètes , 1, Pt. 2 , Gauthier-Villars pp. 303–396 MR1508969 Zbl 30.0313.04
[4] A. Frölicher, A. Nijenhuis, "Theory of vector-valued differential forms. I. Derivations in the graded ring of differential forms" Proc. Koninkl. Ned. Akad. Wet. Ser. A , 59 : 3 (1956) pp. 338–359 MR0082554 Zbl 0079.37502
[5] H. Whitney, "Geometric integration theory" , Princeton Univ. Press (1957) MR0087148 Zbl 0083.28204
[6] R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980) MR0608414 Zbl 0435.32004
[7] S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964) MR0193578 Zbl 0129.13102
[8a] H. Cartan, "Calcul différentiel" , Hermann (1967) MR0223194 Zbl 0156.36102
[8b] H. Cartan, "Formes différentielles" , Hermann (1967) MR0231303 Zbl 0184.12701
[9] P. Deligne, P. Griffiths, J. Morgan, D. Sullivan, "The real homotopy of Kaehler manifolds" Invent. Math. , 29 (1975) pp. 245–274
[10] R. Bott, L.W. Tu, "Differential forms in algebraic topology" , Springer (1982) MR0658304 Zbl 0496.55001
[11] D. Sullivan, "Infinitesimal computations in topology" Publ. Math. IHES , 47 (1977) pp. 269–331 MR0646078 Zbl 0374.57002

Comments

In the Western literature, by a differential $ p $- form on a differentiable manifold $ M $ one always means a smooth section of the $ p $- th exterior power of the tangent bundle $ T M $, i.e. a smooth section of $ \Lambda ^ {p} ( T M ) $. (These are called exterior differential forms in the main article.)

The value of the operator $ L _ {X} $ on a differential form $ x $ is also called the pullback of $ x $ under the infinitesimal transformation generated by the vector field $ X $.

Let $ X $ be a $ C ^ \infty $- manifold of dimension $ n $. A current on $ X $ is a linear functional defined on the space $ \Omega _ {c} ( X) $ of smooth (anti-symmetric) differential forms of compact support on $ X $. The current $ T $ is said to be homogeneous of dimension $ p $ if $ T ( \phi ) = 0 $ for all $ \phi \in \Omega _ {c} ^ {q} ( X) $ with $ q \neq p $.

The degree of a current $ T $ is $ n - p $, if $ p $ is its dimension (if $ T $ is homogeneous).

Define a chain element in $ X $( of dimension $ p $) as a smooth mapping $ s : \Delta _ {p} \rightarrow X $ of the standard cube of dimension $ p $ into $ X $. If $ \omega $ is a $ p $- form on $ X $, then $ \int _ {s} \omega $ is defined as $ \int _ {\Delta _ {p} } f $ if $ s {} ^ \star ( \omega ) = f d x _ {1} \wedge \dots \wedge d x _ {p} $( cf. the main article above). If $ s $ is smooth and orientation-preserving, the integral $ \int _ {s} \omega $ only depends on the image $ s ( \Delta _ {p} ) \subset X $( and on $ \omega $ over $ s ( \Delta _ {p} ) $). More generally, a $ p $- chain on a manifold $ X $ is a formal linear combination $ c = \sum a _ {i} s _ {i} $ of chain elements. The corresponding integral is defined as $ \int _ {c} \omega = \sum a _ {i} \int _ {s _ {i} } \omega $. Each $ p $- chain thus defines a current on $ X $ of dimension $ p $ and degree $ n - p $. If $ \alpha $ is a $ q $- form, it defines a homogeneous $ n - q $ current by the formula $ T _ \alpha ( \omega ) = \int _ {V} \alpha \wedge \omega $. Let $ b $ be a contravariant $ p $- vector with local coordinates $ b ^ {i _ {1} \dots i _ {p} } $. Let the local coordinates of the $ p $- form $ \omega $ be $ \omega _ {i _ {1} \dots i _ {p} } $; then

$$ T _ {b} ( \omega ) = \sum _ {i _ {1} < \dots < i _ {p} } \omega _ {i _ {1} \dots i _ {p} } b ^ {i _ {1} \dots i _ {p} } $$

defines a current of dimension $ p $. Thus, currents generalize both forms (i.e. covariant vectors) and contravariant vectors. They are also a global generalization of the idea of distributions (as a generalization of functions). The role of a space of test functions is played by the smooth forms of compact support. The name "current" comes from the fact that in $ \mathbf R ^ {3} $ the currents of dimension 1 can be interpreted as electric currents.

In the theory of several complex variables one defines the currents of type $ ( p , q ) $ over a domain $ D $ as the (complex-valued) linear functionals on the space $ \Omega _ {c} ^ {n - p , n - q } ( D) $ of $ ( n - p , n - q ) $( complex-valued) forms with compact support. In this area currents and their applications are a very active field of research at the moment [a2].

References

[a1] E.M. Chirka, "Complex analytic sets" , Kluwer (1989) (Translated from Russian) MR1111477 Zbl 0683.32002
[a2] P. Lelong, L. Gruman, "Entire functions of several complex variables" , Springer (1986) MR0837659 Zbl 0583.32001

A differential form on an algebraic variety is the analogue of the concept of a differential form on a differentiable manifold. Let $ X $ be an irreducible algebraic variety of dimension $ d $ over an algebraically closed field $ k $( cf. Irreducible variety) and let $ K $ be its field of rational functions. A differential form of degree $ r $ on $ X $ is an element of the $ K $- space

$$ \Omega ^ {r} ( X) = \wedge ^ { r } \Omega _ {K / k } ^ {1} , $$

where $ \Omega _ {K/k} $ is the module of derivations (cf. Derivations, module of) of the field $ K $ over the field $ k $. If $ x _ {1} \dots x _ {d} $ is a separable basis of transcendency of the extension $ K/k $, any differential form $ \omega \in \Omega ^ {r} ( X) $ can be written as

$$ \omega = \sum a _ {i _ {1} \dots i _ {r} } dx _ {i} \wedge \dots \wedge dx _ {i _ {r} } , $$

where $ a _ {i _ {1} \dots i _ {r} } \in K $. A differential form $ \omega $ is called regular on an open set $ U \subset X $ if $ \omega $ belongs to the submodule $ \Omega _ {k [ U] / k } ^ {r} $ of $ \Omega ^ {r} ( X) $, regarded as a module over the ring $ k [ U] $ of regular functions on the subset $ U $. A differential form $ \omega $ is called regular if any point $ x \in X $ has a neighbourhood $ U $ such that $ \omega $ is regular on $ U $. The regular differential forms on $ X $ form a module over $ k [ X] $, denoted by $ \Omega ^ {r} [ X] $. Its elements are identified with the sections of the sheaf $ \Omega _ {X/k} ^ {r} $ over the variety $ X $. In a neighbourhood of each point $ x \in X $ a regular differential form $ \omega \subset \Omega ^ {r} [ X] $ is written as

$$ \omega = \sum \alpha _ {i _ {1} \dots i _ {r} } df _ {i _ {1} } \wedge \dots \wedge df _ {i _ {r} } , $$

where the functions $ \alpha _ {i _ {1} \dots i _ {r} } $, $ f _ {i _ {1} } \dots f _ {i _ {r} } $ are regular at the point $ x $. If $ X $ is a complete algebraic variety, the spaces $ \Omega ^ {r} [ X] $ are finite-dimensional, and if $ X $ is non-singular, the dimension $ p _ {g} ( X) = \mathop{\rm dim} _ {k} \Omega ^ {d} [ X] $ is known as the geometric genus of the variety $ X $. If $ X $ is a complete variety over the field of complex numbers, the space $ \Omega ^ {r} [ X] $ is identical with the space of holomorphic differential forms of degree $ r $ on the corresponding analytic space $ X ^ {\textrm{ an } } $.

Let $ X $ be a normal variety and let $ \omega \in \Omega ^ {d} [ X] $; for any point $ x \in X ^ {(} 1) $ of codimension one the differential form $ \omega $ may be written as

$$ \tag{* } \omega = a dt \wedge dt _ {1} \wedge \dots \wedge dt _ {d- 1 } , $$

where $ a $ belongs to the field of fractions $ K _ {x} $ of the local ring $ {\mathcal O} _ {X,x } $, $ t $ is the generator of its maximal ideal, and $ t _ {1} \dots t _ {d-} 1 $ is a separable basis of transcendency over $ k $ of the residue field of the ring $ {\mathcal O} _ {X,x } $. The value of the valuation at the element $ a $, as defined by the ring $ {\mathcal O} _ {X,x } $, does not depend on the choice of the representation of $ \omega $ in the form (*) and is denoted by $ \nu _ {x} ( \omega ) $. The divisor

$$ D = \sum _ {x \in X ^ {(} 1) } \nu _ {x} ( \omega ) \{ \overline{x}\; \} $$

is defined and is known as divisor of the differential form $ \omega $. A differential form $ \omega $ is regular if and only if its divisor $ D \geq 0 $, i.e. $ \nu _ {x} ( \omega ) \geq 0 $ for all $ x \in X ^ {(} 1) $. The divisors of any two differential forms are equivalent and, moreover, the divisors of all differential forms on a given algebraic variety form a divisor class with respect to linear equivalence. This class is known as the canonical class of the variety $ X $ and is denoted by $ K _ {X} $. For a non-singular variety $ X $ the class $ K _ {X} $ is identical with the first Chern class of the invertible sheaf $ \Omega _ {X/k } ^ {d} $; in particular,

$$ \Omega _ {X / k } ^ {d} \simeq {\mathcal O} _ {X} ( D ) $$

for any $ D \in K _ {X} $.

Any dominant rational mapping between algebraic varieties $ f : X ^ { \prime } \rightarrow X $ induces a canonical homomorphism

$$ f ^ { * } : \Omega ^ {r} ( X) \rightarrow \Omega ^ {r} ( X ^ { \prime } ) . $$

If $ X $ and $ X ^ { \prime } $ are non-singular and $ X $ is complete, $ f ^ { * } $ maps regular differential forms to regular ones. In particular, if two non-singular complete varieties $ X $ and $ X ^ { \prime } $ are birationally isomorphic, the vector spaces $ \Omega ^ {r} [ X] $ and $ \Omega ^ {r} [ X ^ { \prime } ] $ are isomorphic over the field $ k $.

For any $ i > 1 $ the elements of the $ i $- th symmetric power $ S ^ {i} ( \Omega ^ {r} ( X) ) $ of the $ K $- space $ \Omega ^ {r} ( X) $ are known as $ i $- tuple differential forms of degree $ r $ on $ X $. Each such differential form may be considered as a rational section of the sheaf $ S ^ {i} ( \Omega _ {X/k } ^ {r} ) $. The regular sections

$$ \omega \in \Gamma ( X , S ^ {i} ( \Omega _ {X/k } ^ {r} ) ) $$

are known as regular $ i $- tuple differential forms of degree $ r $ on $ X $. For a non-singular complete variety $ X $ the dimension

$$ P _ {i} ( X) = \mathop{\rm dim} _ {k} \Gamma ( X , S ^ {i} ( \Omega _ {X/k } ^ {d} )) $$

is known as the $ i $ genus of the variety $ X $. The $ i $ genera of birationally isomorphic varieties are identical.

References

[1] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
[2] M. Baldassarri, "Algebraic varieties" , Springer (1956) MR0082172 Zbl 0995.14003 Zbl 0075.15902
[3] R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001

I.V. Dolgachev

Comments

In terms of the description of (Cartier) divisors by local functions (cf. Divisor), the divisor associated to a differential form $ \omega $ of degree $ n $, $ n = \mathop{\rm dim} X $, $ X $ smooth, can be described as follows. For each $ x \in X $ there is an open affine $ U $ such that $ \omega $ can be represented in $ U $ as $ \omega = g d u _ {1} \wedge \dots \wedge d u _ {n} $. Now cover $ X $ by open affine $ U _ {i} $. Let the representation of $ \omega $ in $ U _ {i} $ be $ \omega = g ^ {(} i) d u _ {1} ^ {(} i) \wedge \dots \wedge d u _ {n} ^ {(} i) $. Then on $ U _ {i} \cap U _ {j} $, $ g ^ {(} j) $ is equal to $ g ^ {(} i) $ times the Jacobian of the $ u _ {1} ^ {(} i) \dots u _ {n} ^ {(} i) $ with respect to the $ u _ {1} ^ {(} j) \dots u _ {n} ^ {(} j) $. Thus the local functions $ g ^ {(} i) $ on $ U _ {i} $ define a divisor on $ X $, denoted by $ ( \omega ) $. One has $ ( f \omega ) = ( f ) + ( \omega ) $ for all $ f \in k ( X) $ and thus, because $ \Omega ^ {n} ( X) $ is one-dimensional over $ k ( X) $, all $ 0 \neq \omega \in \Omega ^ {n} ( X) $ define the same divisor class, the canonical class (canonical divisor class) of $ X $.

How to Cite This Entry:
Differential form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Differential_form&oldid=46684
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article