Positive current

Summary

A positive current is a current of bidegree $(p, p)$ on a complex manifold with a positivity property. Its wedge product with products of the forms $i \alpha \wedge \bar\alpha$, for $(1, 0)$-forms $\alpha$, is a nonnegative measure. Pierre Lelong introduced closed positive currents, and they are now a central tool of complex geometry. The two basic examples are the current of integration over a complex analytic subvariety and the current $dd^c u$ of a plurisubharmonic function $u$. The coefficients of a positive current are complex measures, and closed positive currents carry Lelong numbers that measure their singularities.