Local Metric Properties of Zero Sets and Positive Closed Currents

Abstract

A biholomorphic mapping F: ℂⁿ→ℂⁿ induces a mapping of the underlying real coordinates ℝ²ⁿ whose determinant is just ∣J(F)∣², where J(F) is the Jacobian of F, and this determinant is positive. Thus, if one chooses once and for all a volume form an ℂⁿ this choice determines an orientation in ℂⁿ which is invariant with respect to holomorphic isomorphisms, and its restriction to subspaces L^p, p<n, or to complex submanifolds, determines a volume element and hence an orientation. This led to the introduction of a positive differential form in the exterior differential algebra E_2n(dz) with involution (given by dz→d$dz \to d\bar z$) and its generalization, the positive closed current.