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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c1302601.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c1302602.png" />-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c1302603.png" />-manifold with countable basis (cf. also [[Differentiable manifold|Differentiable manifold]]) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c1302604.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c1302605.png" /> denotes the [[Vector space|vector space]] of compactly supported differential forms of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c1302606.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c1302607.png" /> (cf. also [[Differential form|Differential form]]). Endow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c1302608.png" /> with the usual structure of a [[Fréchet space|Fréchet space]] by declaring that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c1302609.png" /> tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c13026010.png" /> if there exists a compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c13026011.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c13026012.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c13026013.png" /> and the coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c13026014.png" /> and all their derivatives tend uniformly to those of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c13026015.png" />.
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A current on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c13026016.png" /> is an element of the dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c13026017.png" />. The idea of currents was introduced by G. de Rham in [[#References|[a6]]], to obtain a homology theory including both forms and chains, but a precise definition, see [[#References|[a7]]], [[#References|[a8]]], became only possible after distributions (cf. also [[Generalized function|Generalized function]]) had been introduced by L. Schwartz. See also (the editorial comments to) [[Differential form|Differential form]], whose notation is used here too.
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While exterior products of currents are in general undefined, exterior differentiation can be defined by duality. If the action of a current <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c13026018.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c13026019.png" /> on a form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c13026020.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c13026021.png" />, then one defines the exterior differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c13026022.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c13026023.png" />. In particular, the notions of closed and exact currents are defined.
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Let $V$ be an $n$-dimensional $C ^ { \infty }$-manifold with countable basis (cf. also [[Differentiable manifold|Differentiable manifold]]) and let $\mathcal{D} = \oplus _ { j = 0 } ^ { n } \mathcal{D} ^ { j }$, where $\mathcal{D} ^ { j }$ denotes the [[Vector space|vector space]] of compactly supported differential forms of degree $j$ on $V$ (cf. also [[Differential form|Differential form]]). Endow $\mathcal{D}$ with the usual structure of a [[Fréchet space|Fréchet space]] by declaring that $\{ \phi _ { j } \in \mathcal{D} \}$ tends to $\phi$ if there exists a compact set $K \subset V$ such that $\operatorname{supp} \phi_{j} \subset K$ for all $j$ and the coefficients of $\phi_j$ and all their derivatives tend uniformly to those of $\phi$.
  
Now, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c13026024.png" /> be a [[Complex manifold|complex manifold]]. One has the splitting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c13026025.png" /> for currents just as for forms.
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A current on $V$ is an element of the dual space ${\cal D} ^ { \prime }$. The idea of currents was introduced by G. de Rham in [[#References|[a6]]], to obtain a homology theory including both forms and chains, but a precise definition, see [[#References|[a7]]], [[#References|[a8]]], became only possible after distributions (cf. also [[Generalized function|Generalized function]]) had been introduced by L. Schwartz. See also (the editorial comments to) [[Differential form|Differential form]], whose notation is used here too.
  
A theorem of P. Lelong [[#References|[a4]]] states that any pure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c13026026.png" />-dimensional analytic subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c13026027.png" /> of a Hermitian complex manifold has locally finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c13026028.png" />-volume. As a consequence one can define the current of integration over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c13026029.png" /> by
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While exterior products of currents are in general undefined, exterior differentiation can be defined by duality. If the action of a current $T$ of degree $p$ on a form $\phi$ is denoted by $\langle T , \phi \rangle$, then one defines the exterior differential $d T$ by $\langle d T , \phi \rangle = ( - 1 ) ^ { p + 1 } \langle T , d \phi \rangle$. In particular, the notions of closed and exact currents are defined.
  
<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/c/c130/c130260/c13026030.png" /></td> </tr></table>
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Now, let $V$ be a [[Complex manifold|complex manifold]]. One has the splitting $d = \partial + \overline { \partial }$ for currents just as for forms.
  
Here, the integration is over the regular points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c13026031.png" /> (cf. also [[Analytic set|Analytic set]]). <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c13026032.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c13026033.png" />-closed current of bi-dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c13026034.png" />. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c13026035.png" /> is positive, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c13026036.png" /> is positive for forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c13026037.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c13026038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c13026039.png" /> being the volume form on the regular points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c13026040.png" />. See also [[#References|[a2]]], [[#References|[a5]]].
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A theorem of P. Lelong [[#References|[a4]]] states that any pure $p$-dimensional analytic subset $A$ of a Hermitian complex manifold has locally finite $2 p$-volume. As a consequence one can define the current of integration over $A$ by
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\begin{equation*} \langle [ A ] , \phi \rangle = \int _ { \operatorname { reg } A } \phi. \end{equation*}
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Here, the integration is over the regular points of $A$ (cf. also [[Analytic set|Analytic set]]). $[ A ]$ is a $d$-closed current of bi-dimension $( p , p )$. Moreover, $[ A ]$ is positive, that is, $\langle [ A ] , \phi \rangle$ is positive for forms $\phi = \lambda d V _ { A }$, with $\lambda &gt; 0$ and $d V _ { A }$ being the volume form on the regular points of $A$. See also [[#References|[a2]]], [[#References|[a5]]].
  
 
Thus, currents can be viewed as an extension of the notion of [[Analytic manifold|analytic manifold]]. This idea has been very fruitful in complex analysis. See e.g. [[#References|[a1]]], [[#References|[a3]]] and their references.
 
Thus, currents can be viewed as an extension of the notion of [[Analytic manifold|analytic manifold]]. This idea has been very fruitful in complex analysis. See e.g. [[#References|[a1]]], [[#References|[a3]]] and their references.
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Ben Messaoud,  H. El Mir,  "Tranchage et prolongement des courants positifs fermés"  ''Math. Ann.'' , '''307'''  (1997)  pp. 473–487  {{MR|}} {{ZBL|0879.32009}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.M. Chirka,  "Complex analytic sets" , ''MAIA'' , '''46''' , Kluwer Acad. Publ.  (1989)  (In Russian)  {{MR|1111477}} {{ZBL|0683.32002}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Duval,  N. Sibony,  "Hulls and positive closed currents"  ''Duke Math. J.'' , '''95'''  (1998)  pp. 621–633  {{MR|1658760}} {{ZBL|0958.32004}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P. Lelong,  "Integration sur un ensemble analytique complexe"  ''Bull. Soc. Math. France'' , '''85'''  (1957)  pp. 239–262  {{MR|0095967}} {{ZBL|0079.30901}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  P. Lelong,  "Fonctions plurisousharmoniques et formes différentielles positives" , Gordon &amp; Breach  (1968)  {{MR|0243112}} {{ZBL|0195.11603}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  G. De Rham,  "Sur l'analyse situs des varietés a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c130/c130260/c13026041.png" /> dimensions (Thèse)"  ''J. Math. Pures Appl.'' , '''10'''  (1931)  pp. 115–200</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  G. De Rham,  "Differentiable manifolds" , Springer  (1984)  (Translated from French)  (Edition: Third)  {{MR|1859366}} {{ZBL|0534.58003}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  L. Schwartz,  "Théorie des distributions" , Hermann  (1966)  {{MR|0209834}} {{ZBL|0149.09501}} </TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  H. Ben Messaoud,  H. El Mir,  "Tranchage et prolongement des courants positifs fermés"  ''Math. Ann.'' , '''307'''  (1997)  pp. 473–487  {{MR|}} {{ZBL|0879.32009}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top">  E.M. Chirka,  "Complex analytic sets" , ''MAIA'' , '''46''' , Kluwer Acad. Publ.  (1989)  (In Russian)  {{MR|1111477}} {{ZBL|0683.32002}} </td></tr><tr><td valign="top">[a3]</td> <td valign="top">  J. Duval,  N. Sibony,  "Hulls and positive closed currents"  ''Duke Math. J.'' , '''95'''  (1998)  pp. 621–633  {{MR|1658760}} {{ZBL|0958.32004}} </td></tr><tr><td valign="top">[a4]</td> <td valign="top">  P. Lelong,  "Integration sur un ensemble analytique complexe"  ''Bull. Soc. Math. France'' , '''85'''  (1957)  pp. 239–262  {{MR|0095967}} {{ZBL|0079.30901}} </td></tr><tr><td valign="top">[a5]</td> <td valign="top">  P. Lelong,  "Fonctions plurisousharmoniques et formes différentielles positives" , Gordon &amp; Breach  (1968)  {{MR|0243112}} {{ZBL|0195.11603}} </td></tr><tr><td valign="top">[a6]</td> <td valign="top">  G. De Rham,  "Sur l'analyse situs des varietés a $n$ dimensions (Thèse)"  ''J. Math. Pures Appl.'' , '''10'''  (1931)  pp. 115–200</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  G. De Rham,  "Differentiable manifolds" , Springer  (1984)  (Translated from French)  (Edition: Third)  {{MR|1859366}} {{ZBL|0534.58003}} </td></tr><tr><td valign="top">[a8]</td> <td valign="top">  L. Schwartz,  "Théorie des distributions" , Hermann  (1966)  {{MR|0209834}} {{ZBL|0149.09501}} </td></tr></table>

Latest revision as of 17:00, 1 July 2020

Let $V$ be an $n$-dimensional $C ^ { \infty }$-manifold with countable basis (cf. also Differentiable manifold) and let $\mathcal{D} = \oplus _ { j = 0 } ^ { n } \mathcal{D} ^ { j }$, where $\mathcal{D} ^ { j }$ denotes the vector space of compactly supported differential forms of degree $j$ on $V$ (cf. also Differential form). Endow $\mathcal{D}$ with the usual structure of a Fréchet space by declaring that $\{ \phi _ { j } \in \mathcal{D} \}$ tends to $\phi$ if there exists a compact set $K \subset V$ such that $\operatorname{supp} \phi_{j} \subset K$ for all $j$ and the coefficients of $\phi_j$ and all their derivatives tend uniformly to those of $\phi$.

A current on $V$ is an element of the dual space ${\cal D} ^ { \prime }$. The idea of currents was introduced by G. de Rham in [a6], to obtain a homology theory including both forms and chains, but a precise definition, see [a7], [a8], became only possible after distributions (cf. also Generalized function) had been introduced by L. Schwartz. See also (the editorial comments to) Differential form, whose notation is used here too.

While exterior products of currents are in general undefined, exterior differentiation can be defined by duality. If the action of a current $T$ of degree $p$ on a form $\phi$ is denoted by $\langle T , \phi \rangle$, then one defines the exterior differential $d T$ by $\langle d T , \phi \rangle = ( - 1 ) ^ { p + 1 } \langle T , d \phi \rangle$. In particular, the notions of closed and exact currents are defined.

Now, let $V$ be a complex manifold. One has the splitting $d = \partial + \overline { \partial }$ for currents just as for forms.

A theorem of P. Lelong [a4] states that any pure $p$-dimensional analytic subset $A$ of a Hermitian complex manifold has locally finite $2 p$-volume. As a consequence one can define the current of integration over $A$ by

\begin{equation*} \langle [ A ] , \phi \rangle = \int _ { \operatorname { reg } A } \phi. \end{equation*}

Here, the integration is over the regular points of $A$ (cf. also Analytic set). $[ A ]$ is a $d$-closed current of bi-dimension $( p , p )$. Moreover, $[ A ]$ is positive, that is, $\langle [ A ] , \phi \rangle$ is positive for forms $\phi = \lambda d V _ { A }$, with $\lambda > 0$ and $d V _ { A }$ being the volume form on the regular points of $A$. See also [a2], [a5].

Thus, currents can be viewed as an extension of the notion of analytic manifold. This idea has been very fruitful in complex analysis. See e.g. [a1], [a3] and their references.

See also Geometric measure theory.

References

[a1] H. Ben Messaoud, H. El Mir, "Tranchage et prolongement des courants positifs fermés" Math. Ann. , 307 (1997) pp. 473–487 Zbl 0879.32009
[a2] E.M. Chirka, "Complex analytic sets" , MAIA , 46 , Kluwer Acad. Publ. (1989) (In Russian) MR1111477 Zbl 0683.32002
[a3] J. Duval, N. Sibony, "Hulls and positive closed currents" Duke Math. J. , 95 (1998) pp. 621–633 MR1658760 Zbl 0958.32004
[a4] P. Lelong, "Integration sur un ensemble analytique complexe" Bull. Soc. Math. France , 85 (1957) pp. 239–262 MR0095967 Zbl 0079.30901
[a5] P. Lelong, "Fonctions plurisousharmoniques et formes différentielles positives" , Gordon & Breach (1968) MR0243112 Zbl 0195.11603
[a6] G. De Rham, "Sur l'analyse situs des varietés a $n$ dimensions (Thèse)" J. Math. Pures Appl. , 10 (1931) pp. 115–200
[a7] G. De Rham, "Differentiable manifolds" , Springer (1984) (Translated from French) (Edition: Third) MR1859366 Zbl 0534.58003
[a8] L. Schwartz, "Théorie des distributions" , Hermann (1966) MR0209834 Zbl 0149.09501
How to Cite This Entry:
Current. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Current&oldid=50355
This article was adapted from an original article by J. Wiegerinck (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article