Positive current

In mathematics, more particularly in complex geometry,
algebraic geometry and complex analysis, a positive currentis 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. ; integratingover "M", we may consider currents as "currents of integration",that is, functionals

:$eta\; mapsto\; int\_M\; etawedge\; ho$

on smooth forms with compact support. This way, currentsare considered as elements in the dual space to the space$Lambda\_c^*(M)$ of forms with compact support.

Now, let "M" be a complex manifold.The Hodge decomposition is defined on currents, in a natural way, the "(p,q)"-currents beingfunctionals on $Lambda\_c^\{p,\; q\}(M)$.

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

Characterization of Kahler manifolds

Using the Hahn-Banach theorem, Harvey and Lawson proved the following criterion of existence of Kahler metrics. [R. Harvey and H. B. Lawson, "An intrinsic characterisation of Kahler manifolds," Invent. Math 74 (1983) 169-198.]

Theorem: Let "M" be a compact complex manifold. Then "M" does not admit a Kahler structure if and only if "M" admits a non-zero positive (1,1)-current $Theta$ which is a (1,1)-part of an exact 2-current.

Note that the de Rham differential maps 3-currents to 2-currents, hence $Theta$ is a differential of a 3-current; if $Theta$ is a current of integration of a complex curve, this means that this curve is a (1,1)-part of a boundary.

When "M" admits a surjective map $pi:;\; M\; mapsto\; X$ to a Kahler manifold with 1-dimensional fibers, this theorem leads to the following result of complex algebraic geometry.

Corollary: In this situation, "M" is non-Kahler if and only if the homology class of a generic fiber of $pi$ is a (1,1)-part of a boundary.

Notes

*Phillip Griffiths and Joseph Harris (1978), "Principles of Algebraic Geometry", Wiley. ISBN 0471327921

*J.-P. Demailly, " [http://arxiv.org/abs/alg-geom/9410022 $L^2$ vanishing theorems for positive line bundles and adjunction theory, Lecture Notes of a CIME course on "Transcendental Methods of Algebraic Geometry" (Cetraro, Italy, July 1994)] "

References