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A generalization of the classical calculus of differential forms and differential operators to analytic spaces. For the calculus of differential forms on complex manifolds see [[Differential form|Differential form]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d0318601.png" /> be an [[Analytic space|analytic space]] over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d0318602.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d0318603.png" /> be the diagonal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d0318604.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d0318605.png" /> be the sheaf of ideals defining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d0318606.png" /> and generated by all germs of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d0318607.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d0318608.png" /> is an arbitrary germ from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d0318609.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186010.png" /> be projection on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186011.png" />-th factor.
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The analytic sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186012.png" /> is known as the sheaf of analytic differential forms of the first order on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186013.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186014.png" /> is the germ of an analytic function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186015.png" />, then the germ <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186016.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186017.png" /> and defines the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186019.png" /> known as the differential of the germ <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186020.png" />. This defines a sheaf homomorphism of vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186021.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186022.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186023.png" /> is the free sheaf generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186024.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186025.png" /> are the coordinates in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186026.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186027.png" /> is an analytic subspace in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186028.png" />, defined by a sheaf of ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186029.png" />, then
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<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/d031/d031860/d03186030.png" /></td> </tr></table>
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A generalization of the classical calculus of differential forms and differential operators to analytic spaces. For the calculus of differential forms on complex manifolds see [[Differential form|Differential form]]. Let  $  ( X, {\mathcal O} _ {X} ) $
 +
be an [[Analytic space|analytic space]] over a field  $  k $,
 +
let  $  \Delta $
 +
be the diagonal in  $  X \times X $,
 +
let  $  J $
 +
be the sheaf of ideals defining  $  \Delta $
 +
and generated by all germs of the form  $  \pi _ {1}  ^ {*} f - \pi _ {2}  ^ {*} f $,
 +
where  $  f $
 +
is an arbitrary germ from  $  {\mathcal O} _ {X} $,
 +
and let  $  \pi _ {i} : X \times X \rightarrow X $
 +
be projection on the  $  i $-
 +
th factor.
  
Each analytic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186031.png" /> may be related to a sheaf of relative differentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186032.png" />. This is the analytic sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186033.png" /> inducing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186034.png" /> on each fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186035.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186036.png" />) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186037.png" />; it is defined from the exact sequence
+
The analytic sheaf  $  \pi _ {1} ( J / J  ^ {2} ) = \Omega _ {X}  ^ {1} $
 +
is known as the sheaf of analytic differential forms of the first order on  $  X $.  
 +
If  $  f $
 +
is the germ of an analytic function on $  X $,
 +
then the germ  $  \pi _ {1}  ^ {*} f - \pi _ {2}  ^ {*} f $
 +
belongs to  $  J $
 +
and defines the element  $  df $
 +
of  $  \Omega _ {X}  ^ {1} $
 +
known as the differential of the germ  $  f $.  
 +
This defines a sheaf homomorphism of vector spaces  $  d : {\mathcal O} _ {X} \rightarrow \Omega _ {X}  ^ {1} $.  
 +
If  $  X = k  ^ {n} $,
 +
then  $  \Omega _ {X}  ^ {1} $
 +
is the free sheaf generated by  $  dx _ {1} \dots dx _ {n} $,
 +
where  $  x _ {1} \dots x _ {n} $
 +
are the coordinates in  $  k  ^ {n} $.  
 +
If  $  X $
 +
is an analytic subspace in  $  k  ^ {n} $,
 +
defined by a sheaf of ideals  $  J $,
 +
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/d031/d031860/d03186038.png" /></td> </tr></table>
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$$
 +
\Omega _ {X}  ^ {1}  \cong  \Omega _ {k  ^ {n}  }  ^ {1} / ( J \Omega _ {k  ^ {n}  }  ^ {1} + dJ) \mid  _ {X} .
 +
$$
  
The sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186039.png" /> is called the sheaf of germs of analytic vector fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186040.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186041.png" /> is a manifold, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186043.png" /> are locally free sheaves, which are naturally isomorphic to the sheaf of analytic sections of the cotangent and the tangent bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186044.png" />, respectively.
+
Each analytic mapping  $  f : X \rightarrow Y $
 +
may be related to a sheaf of relative differentials  $  \Omega _ {X/Y}  ^ {1} $.  
 +
This is the analytic sheaf $  \Omega _ {X/Y}  ^ {1} $
 +
inducing  $  \Omega _ {X _ {s}  }  ^ {1} $
 +
on each fibre  $  X _ {s} $(
 +
$  s \in Y $)
 +
of $  f $;
 +
it is defined from the exact sequence
  
The analytic sheaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186045.png" /> are called sheaves of analytic exterior differential forms of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186047.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186048.png" /> (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186049.png" />, they are also called holomorphic forms). For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186050.png" /> one may define a sheaf homomorphism of vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186051.png" />, which for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186052.png" /> coincides with the one introduced above, and which satisfies the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186053.png" />. The complex of sheaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186054.png" /> is called the de Rham complex of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186055.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186056.png" /> is a manifold and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186057.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186058.png" />, the de Rham complex is an exact complex of sheaves. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186059.png" /> is a Stein manifold or a real-analytic manifold, the cohomology groups of the complex of sections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186060.png" />, which is also often referred to as the de Rham complex, are isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186061.png" />.
+
$$
 +
f ^ { * } \Omega _ {Y}  ^ {1}  \rightarrow  \Omega _ {X}  ^ {1}  \rightarrow  \Omega _ {X/Y}  ^ {1}  \rightarrow  0 .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186062.png" /> has singular points, the de Rham complex need not be exact. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186063.png" />, a sufficient condition for the exactness of the de Rham complex at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186064.png" /> is the presence of a complex-analytic contractible neighbourhood at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186065.png" />. The hyperhomology groups of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186066.png" /> contain, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186067.png" />, the cohomology groups of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186068.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186069.png" /> as direct summands, and are identical with them if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186070.png" /> is smooth. The sections of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186071.png" /> are called analytic (and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186072.png" />, also holomorphic) vector fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186073.png" />. For any open <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186074.png" /> the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186075.png" /> defines a derivation in the algebra of analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186076.png" />, acting according to the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186077.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186078.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186079.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186080.png" /> defines a local one-parameter group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186081.png" /> of automorphisms of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186082.png" />. If, in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186083.png" /> is compact, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186084.png" /> is globally definable.
+
The sheaf  $  \Theta _ {X} = \mathop{\rm Hom} _ { {\mathcal O} _ {X}  } ( \Omega _ {X}  ^ {1} , {\mathcal O} _ {X} ) $
 +
is called the sheaf of germs of analytic vector fields on $  X $.  
 +
If  $  X $
 +
is a manifold,  $  \Omega _ {X}  ^ {1} $
 +
and  $  \Theta _ {X} $
 +
are locally free sheaves, which are naturally isomorphic to the sheaf of analytic sections of the cotangent and the tangent bundle over  $  X $,  
 +
respectively.
  
The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186085.png" /> provided with the Lie bracket is a Lie algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186086.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186087.png" /> is a compact complex space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186088.png" /> is the Lie algebra of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186089.png" />.
+
The analytic sheaves  $  \Omega _ {X}  ^ {p} = \wedge _ { {\mathcal O} _ {X}  }  ^ {p} \Omega _ {X}  ^ {1} $
 +
are called sheaves of analytic exterior differential forms of degree  $  p $
 +
on  $  X $(
 +
if  $  k = \mathbf C $,
 +
they are also called holomorphic forms). For any  $  p\geq  0 $
 +
one may define a sheaf homomorphism of vector spaces  $  d  ^ {p} :  \Omega _ {X}  ^ {p} \rightarrow \Omega _ {X}  ^ {p+} 1 $,
 +
which for  $  p= 0 $
 +
coincides with the one introduced above, and which satisfies the condition  $  d  ^ {p+} 1 d  ^ {p} = 0 $.  
 +
The complex of sheaves  $  ( \Omega _ {X}  ^ {*} , d) $
 +
is called the de Rham complex of the space  $  X $.  
 +
If $  X $
 +
is a manifold and  $  k = \mathbf C $
 +
or  $  \mathbf R $,
 +
the de Rham complex is an exact complex of sheaves. If  $  X $
 +
is a Stein manifold or a real-analytic manifold, the cohomology groups of the complex of sections  $  \Gamma ( \Omega _ {X}  ^ {*} ) $,  
 +
which is also often referred to as the de Rham complex, are isomorphic to  $  H  ^ {p} ( X , k) $.
  
Differential operators on an analytic space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186090.png" /> are defined in analogy to the differential operators on a module (cf. [[Differential operator on a module|Differential operator on a module]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186091.png" /> are analytic sheaves on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186092.png" />, then a linear differential operator of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186094.png" />, acting from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186095.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186096.png" />, is a sheaf homomorphism of vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186097.png" /> which extends to an analytic homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186098.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d03186099.png" /> is smooth and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d031860100.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d031860101.png" /> are locally free, this definition gives the usual concept of a differential operator on a vector bundle , [[#References|[4]]].
+
If  $  X $
 +
has singular points, the de Rham complex need not be exact. If  $  k = \mathbf C $,
 +
a sufficient condition for the exactness of the de Rham complex at a point  $  x \in X $
 +
is the presence of a complex-analytic contractible neighbourhood at  $  x $.
 +
The hyperhomology groups of the complex  $  \Gamma ( \Omega _ {X}  ^ {*} ) $
 +
contain, for  $  k= \mathbf C $,
 +
the cohomology groups of the space  $  X $
 +
with coefficients in  $  \mathbf C $
 +
as direct summands, and are identical with them if  $  X $
 +
is smooth. The sections of the sheaf  $  \Theta _ {X} $
 +
are called analytic (and if  $  k= \mathbf C $,
 +
also holomorphic) vector fields on $  X $.  
 +
For any open  $  U \subset  X $
 +
the field  $  Z \in \Gamma ( X , \Theta _ {X} ) $
 +
defines a derivation in the algebra of analytic functions  $  \Gamma ( U , {\mathcal O} _ {X} ) $,  
 +
acting according to the formula  $  \phi \rightarrow Z _  \phi  = Z ( d \phi ) $.  
 +
If  $  k= \mathbf C $
 +
or  $  \mathbf R $,  
 +
then  $  Z $
 +
defines a local one-parameter group  $  \mathop{\rm exp}  Z $
 +
of automorphisms of the space  $  X $.  
 +
If, in addition,  $  X $
 +
is compact, the group  $  \mathop{\rm exp}  Z $
 +
is globally definable.
  
The germs of the linear differential operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d031860102.png" /> form an analytic sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d031860103.png" /> with filtration
+
The space  $  \Gamma ( X, \Theta _ {X} ) $
 +
provided with the Lie bracket is a Lie algebra over  $  k $.
 +
If  $  X $
 +
is a compact complex space,  $  \Gamma ( X, \Theta _ {X} ) $
 +
is the Lie algebra of the group  $  \mathop{\rm Aut}  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/d031/d031860/d031860104.png" /></td> </tr></table>
+
Differential operators on an analytic space  $  ( X, {\mathcal O} _ {X} ) $
 +
are defined in analogy to the differential operators on a module (cf. [[Differential operator on a module|Differential operator on a module]]). If  $  F, G $
 +
are analytic sheaves on  $  X $,
 +
then a linear differential operator of order  $  \leq  l $,
 +
acting from  $  F $
 +
into  $  G $,
 +
is a sheaf homomorphism of vector spaces  $  F \rightarrow G $
 +
which extends to an analytic homomorphism  $  F \otimes \pi _ {1} ( {\mathcal O} _ {X \times X }  / I  ^ {l+} 1 ) \rightarrow G $.
 +
If  $  X $
 +
is smooth and  $  F $
 +
and  $  G $
 +
are locally free, this definition gives the usual concept of a differential operator on a vector bundle , [[#References|[4]]].
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d031860105.png" /> is the sheaf of germs of operators of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d031860106.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d031860107.png" /> is a filtered sheaf of associative algebras over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d031860108.png" /> under composition of mappings. One has
+
The germs of the linear differential operators $  F \rightarrow G $
 +
form an analytic sheaf  $  \mathop{\rm Diff} ( F, G) $
 +
with filtration
  
<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/d031/d031860/d031860109.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Diff}  ^ {0} ( F, G)  \subset  \dots \subset    \mathop{\rm Diff}  ^ {l} ( F, G)  \subset  \dots ,
 +
$$
  
<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/d031/d031860/d031860110.png" /></td> </tr></table>
+
where  $  \mathop{\rm Diff}  ^ {l} ( F, G) $
 +
is the sheaf of germs of operators of order  $  < l $.  
 +
In particular,  $  \mathop{\rm Diff} ( {\mathcal O} , {\mathcal O} ) $
 +
is a filtered sheaf of associative algebras over  $  k $
 +
under composition of mappings. One has
  
The sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d031860111.png" /> was studied (for the non-smooth case) only for certain special types of singular points. In particular, it was proved in the case of an irreducible one-dimensional [[Complex space|complex space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d031860112.png" /> that the sheaf of algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031860/d031860113.png" /> and the corresponding sheaf of graded algebras have finite systems of generators [[#References|[5]]].
+
$$
 +
\mathop{\rm Diff}  ^ {0} ( F, G)  \cong  \mathop{\rm Hom} _  {\mathcal O}  ( F, G) ,
 +
$$
 +
 
 +
$$
 +
\mathop{\rm Diff}  ^ {1} ( {\mathcal O} , {\mathcal O} ) /
 +
\mathop{\rm Diff}  ^ {0} ( {\mathcal O} , {\mathcal O} )  \cong  \Theta _ {X} .
 +
$$
 +
 
 +
The sheaf  $  \mathop{\rm Diff} ( {\mathcal O} , {\mathcal O} ) $
 +
was studied (for the non-smooth case) only for certain special types of singular points. In particular, it was proved in the case of an irreducible one-dimensional [[Complex space|complex space]] $  X $
 +
that the sheaf of algebras $  \mathop{\rm Diff} ( {\mathcal O} , {\mathcal O} ) $
 +
and the corresponding sheaf of graded algebras have finite systems of generators [[#References|[5]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B. Malgrange,  "Analytic spaces"  ''Enseign. Math. Ser. 2'' , '''14''' :  1  (1968)  pp. 1–28</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Kaup,  "Infinitesimal Transformationsgruppen komplexer Räume"  ''Math. Ann.'' , '''160''' :  1  (1965)  pp. 72–92</TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top">  L. Schwartz,  "Variedades analiticas complejas elipticas" , Univ. Nac. Colombia  (1956)</TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top">  L. Schwartz,  "Ecuaciones differenciales parciales" , Univ. Nac. Colombia  (1956)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.O. Wells jr.,  "Differential analysis on complex manifolds" , Springer  (1980)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  Th. Bloom,  "Differential operators on curves"  ''Rice Univ. Stud.'' , '''59''' :  2  (1973)  pp. 13–19</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  R. Berger,  R. Kiehl,  E. Kunz,  H.-J. Nastold,  "Differentialrechnung in der analytischen Geometrie" , Springer  (1967)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  G. Fischer,  "Complex analytic geometry" , Springer  (1976)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B. Malgrange,  "Analytic spaces"  ''Enseign. Math. Ser. 2'' , '''14''' :  1  (1968)  pp. 1–28</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Kaup,  "Infinitesimal Transformationsgruppen komplexer Räume"  ''Math. Ann.'' , '''160''' :  1  (1965)  pp. 72–92</TD></TR><TR><TD valign="top">[3a]</TD> <TD valign="top">  L. Schwartz,  "Variedades analiticas complejas elipticas" , Univ. Nac. Colombia  (1956)</TD></TR><TR><TD valign="top">[3b]</TD> <TD valign="top">  L. Schwartz,  "Ecuaciones differenciales parciales" , Univ. Nac. Colombia  (1956)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.O. Wells jr.,  "Differential analysis on complex manifolds" , Springer  (1980)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  Th. Bloom,  "Differential operators on curves"  ''Rice Univ. Stud.'' , '''59''' :  2  (1973)  pp. 13–19</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  R. Berger,  R. Kiehl,  E. Kunz,  H.-J. Nastold,  "Differentialrechnung in der analytischen Geometrie" , Springer  (1967)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  G. Fischer,  "Complex analytic geometry" , Springer  (1976)</TD></TR></table>

Revision as of 17:33, 5 June 2020


A generalization of the classical calculus of differential forms and differential operators to analytic spaces. For the calculus of differential forms on complex manifolds see Differential form. Let $ ( X, {\mathcal O} _ {X} ) $ be an analytic space over a field $ k $, let $ \Delta $ be the diagonal in $ X \times X $, let $ J $ be the sheaf of ideals defining $ \Delta $ and generated by all germs of the form $ \pi _ {1} ^ {*} f - \pi _ {2} ^ {*} f $, where $ f $ is an arbitrary germ from $ {\mathcal O} _ {X} $, and let $ \pi _ {i} : X \times X \rightarrow X $ be projection on the $ i $- th factor.

The analytic sheaf $ \pi _ {1} ( J / J ^ {2} ) = \Omega _ {X} ^ {1} $ is known as the sheaf of analytic differential forms of the first order on $ X $. If $ f $ is the germ of an analytic function on $ X $, then the germ $ \pi _ {1} ^ {*} f - \pi _ {2} ^ {*} f $ belongs to $ J $ and defines the element $ df $ of $ \Omega _ {X} ^ {1} $ known as the differential of the germ $ f $. This defines a sheaf homomorphism of vector spaces $ d : {\mathcal O} _ {X} \rightarrow \Omega _ {X} ^ {1} $. If $ X = k ^ {n} $, then $ \Omega _ {X} ^ {1} $ is the free sheaf generated by $ dx _ {1} \dots dx _ {n} $, where $ x _ {1} \dots x _ {n} $ are the coordinates in $ k ^ {n} $. If $ X $ is an analytic subspace in $ k ^ {n} $, defined by a sheaf of ideals $ J $, then

$$ \Omega _ {X} ^ {1} \cong \Omega _ {k ^ {n} } ^ {1} / ( J \Omega _ {k ^ {n} } ^ {1} + dJ) \mid _ {X} . $$

Each analytic mapping $ f : X \rightarrow Y $ may be related to a sheaf of relative differentials $ \Omega _ {X/Y} ^ {1} $. This is the analytic sheaf $ \Omega _ {X/Y} ^ {1} $ inducing $ \Omega _ {X _ {s} } ^ {1} $ on each fibre $ X _ {s} $( $ s \in Y $) of $ f $; it is defined from the exact sequence

$$ f ^ { * } \Omega _ {Y} ^ {1} \rightarrow \Omega _ {X} ^ {1} \rightarrow \Omega _ {X/Y} ^ {1} \rightarrow 0 . $$

The sheaf $ \Theta _ {X} = \mathop{\rm Hom} _ { {\mathcal O} _ {X} } ( \Omega _ {X} ^ {1} , {\mathcal O} _ {X} ) $ is called the sheaf of germs of analytic vector fields on $ X $. If $ X $ is a manifold, $ \Omega _ {X} ^ {1} $ and $ \Theta _ {X} $ are locally free sheaves, which are naturally isomorphic to the sheaf of analytic sections of the cotangent and the tangent bundle over $ X $, respectively.

The analytic sheaves $ \Omega _ {X} ^ {p} = \wedge _ { {\mathcal O} _ {X} } ^ {p} \Omega _ {X} ^ {1} $ are called sheaves of analytic exterior differential forms of degree $ p $ on $ X $( if $ k = \mathbf C $, they are also called holomorphic forms). For any $ p\geq 0 $ one may define a sheaf homomorphism of vector spaces $ d ^ {p} : \Omega _ {X} ^ {p} \rightarrow \Omega _ {X} ^ {p+} 1 $, which for $ p= 0 $ coincides with the one introduced above, and which satisfies the condition $ d ^ {p+} 1 d ^ {p} = 0 $. The complex of sheaves $ ( \Omega _ {X} ^ {*} , d) $ is called the de Rham complex of the space $ X $. If $ X $ is a manifold and $ k = \mathbf C $ or $ \mathbf R $, the de Rham complex is an exact complex of sheaves. If $ X $ is a Stein manifold or a real-analytic manifold, the cohomology groups of the complex of sections $ \Gamma ( \Omega _ {X} ^ {*} ) $, which is also often referred to as the de Rham complex, are isomorphic to $ H ^ {p} ( X , k) $.

If $ X $ has singular points, the de Rham complex need not be exact. If $ k = \mathbf C $, a sufficient condition for the exactness of the de Rham complex at a point $ x \in X $ is the presence of a complex-analytic contractible neighbourhood at $ x $. The hyperhomology groups of the complex $ \Gamma ( \Omega _ {X} ^ {*} ) $ contain, for $ k= \mathbf C $, the cohomology groups of the space $ X $ with coefficients in $ \mathbf C $ as direct summands, and are identical with them if $ X $ is smooth. The sections of the sheaf $ \Theta _ {X} $ are called analytic (and if $ k= \mathbf C $, also holomorphic) vector fields on $ X $. For any open $ U \subset X $ the field $ Z \in \Gamma ( X , \Theta _ {X} ) $ defines a derivation in the algebra of analytic functions $ \Gamma ( U , {\mathcal O} _ {X} ) $, acting according to the formula $ \phi \rightarrow Z _ \phi = Z ( d \phi ) $. If $ k= \mathbf C $ or $ \mathbf R $, then $ Z $ defines a local one-parameter group $ \mathop{\rm exp} Z $ of automorphisms of the space $ X $. If, in addition, $ X $ is compact, the group $ \mathop{\rm exp} Z $ is globally definable.

The space $ \Gamma ( X, \Theta _ {X} ) $ provided with the Lie bracket is a Lie algebra over $ k $. If $ X $ is a compact complex space, $ \Gamma ( X, \Theta _ {X} ) $ is the Lie algebra of the group $ \mathop{\rm Aut} X $.

Differential operators on an analytic space $ ( X, {\mathcal O} _ {X} ) $ are defined in analogy to the differential operators on a module (cf. Differential operator on a module). If $ F, G $ are analytic sheaves on $ X $, then a linear differential operator of order $ \leq l $, acting from $ F $ into $ G $, is a sheaf homomorphism of vector spaces $ F \rightarrow G $ which extends to an analytic homomorphism $ F \otimes \pi _ {1} ( {\mathcal O} _ {X \times X } / I ^ {l+} 1 ) \rightarrow G $. If $ X $ is smooth and $ F $ and $ G $ are locally free, this definition gives the usual concept of a differential operator on a vector bundle , [4].

The germs of the linear differential operators $ F \rightarrow G $ form an analytic sheaf $ \mathop{\rm Diff} ( F, G) $ with filtration

$$ \mathop{\rm Diff} ^ {0} ( F, G) \subset \dots \subset \mathop{\rm Diff} ^ {l} ( F, G) \subset \dots , $$

where $ \mathop{\rm Diff} ^ {l} ( F, G) $ is the sheaf of germs of operators of order $ < l $. In particular, $ \mathop{\rm Diff} ( {\mathcal O} , {\mathcal O} ) $ is a filtered sheaf of associative algebras over $ k $ under composition of mappings. One has

$$ \mathop{\rm Diff} ^ {0} ( F, G) \cong \mathop{\rm Hom} _ {\mathcal O} ( F, G) , $$

$$ \mathop{\rm Diff} ^ {1} ( {\mathcal O} , {\mathcal O} ) / \mathop{\rm Diff} ^ {0} ( {\mathcal O} , {\mathcal O} ) \cong \Theta _ {X} . $$

The sheaf $ \mathop{\rm Diff} ( {\mathcal O} , {\mathcal O} ) $ was studied (for the non-smooth case) only for certain special types of singular points. In particular, it was proved in the case of an irreducible one-dimensional complex space $ X $ that the sheaf of algebras $ \mathop{\rm Diff} ( {\mathcal O} , {\mathcal O} ) $ and the corresponding sheaf of graded algebras have finite systems of generators [5].

References

[1] B. Malgrange, "Analytic spaces" Enseign. Math. Ser. 2 , 14 : 1 (1968) pp. 1–28
[2] W. Kaup, "Infinitesimal Transformationsgruppen komplexer Räume" Math. Ann. , 160 : 1 (1965) pp. 72–92
[3a] L. Schwartz, "Variedades analiticas complejas elipticas" , Univ. Nac. Colombia (1956)
[3b] L. Schwartz, "Ecuaciones differenciales parciales" , Univ. Nac. Colombia (1956)
[4] R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980)
[5] Th. Bloom, "Differential operators on curves" Rice Univ. Stud. , 59 : 2 (1973) pp. 13–19
[6] R. Berger, R. Kiehl, E. Kunz, H.-J. Nastold, "Differentialrechnung in der analytischen Geometrie" , Springer (1967)
[7] G. Fischer, "Complex analytic geometry" , Springer (1976)
How to Cite This Entry:
Differential calculus (on analytic spaces). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Differential_calculus_(on_analytic_spaces)&oldid=14009
This article was adapted from an original article by D.A. Ponomarev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article