Positive current


In mathematics, more particularly in complex geometry,
algebraic geometry and complex analysis, a positive current
is a positive -form over an n-dimensional complex manifold,
taking values in distributions.
For a formal definition, consider a manifold M.
Currents on M are
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. This way, currents
are considered as elements in the dual space to the space
of forms with compact support.
Now, let M be a complex manifold.
The Hodge decomposition
is defined on currents, in a natural way, the '-currents being
functionals on.
A positive current is defined as a real current
of Hodge type ', taking non-negative values on all positive
-forms.

Characterization of [Kähler manifold]s

Using the Hahn–Banach theorem, Harvey and Lawson proved the following criterion of existence of Kähler metrics.
Theorem: Let M be a compact complex manifold. Then M does not admit a Kähler structure if and only if M admits a non-zero positive -current which is a -part of an exact 2-current.
Note that the de Rham differential maps 3-currents to 2-currents, hence is a differential of a 3-current; if is a current of integration of a complex curve, this means that this curve is a -part of a boundary.
When M admits a surjective map 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 is a -part of a boundary.