Positive current

In mathematics, more particularly in complex geometry, algebraic geometry and complex analysis, a positive current is a positive (n-p,n-p)-form over an n-dimensional complex manifold, taking values in distributions.

For a formal definition, consider a manifold M. Currents on M are (by definition) differential forms with coefficients in distributions; integrating over M, we may consider currents as "currents of integration", that is, functionals

${\displaystyle \eta \mapsto \int _{M}\eta \wedge \rho }$

on smooth forms with compact support. This way, currents are considered as elements in the dual space to the space ${\displaystyle \Lambda _{c}^{*}(M)}$ of forms with compact support.

Now, let M be a complex manifold. The Hodge decomposition ${\displaystyle \Lambda ^{i}(M)=\bigoplus _{p+q=i}\Lambda ^{p,q}(M)}$ is defined on currents, in a natural way, the (p,q)-currents being functionals on ${\displaystyle \Lambda _{c}^{p,q}(M)}$.

A positive current is defined as a real current of Hodge type (p,p), taking non-negative values on all positive (p,p)-forms.