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 on smooth forms with compact support.
is defined on currents, in a natural way, the (p,q)-currents being functionals on
A positive current is defined as a real current of Hodge type (p,p), taking non-negative values on all positive (p,p)-forms.
Using the Hahn–Banach theorem, Harvey and Lawson proved the following criterion of existence of Kähler metrics.
[1] Theorem: Let M be a compact complex manifold.
Note that the de Rham differential maps 3-currents to 2-currents, hence
to a Kähler manifold with 1-dimensional fibers, this theorem leads to the following result of complex algebraic geometry.
Corollary: In this situation, M is non-Kähler if and only if the homology class of a generic fiber of