Current

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.

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