Let \(\gamma\) be a linear combination with integral coefficients of integration currents on oriented closed real curves of class \(C^ 2\) of \(\mathbb{C} \mathbb{P}^ 2(w_ 0:w_ 1:w_ 2)\). The main purpose of this note is to establish the following
Theorem: The following conditions are equivalent:
i) \(\gamma\) is the boundary of a holomorphic 1-chain of finite mass of \(\mathbb{C} \mathbb{P}^ 2 \backslash \text{supp} (\gamma)\)
ii) in a neighborhood of a point \((x_ 0,y_ 0) \in \mathbb{C}^ 2\), the function \(1/2 \pi i \int_ \gamma z_ 2(z_ 2-yz_ 2-x)^{-1}d(z_ 1-yz_ 2-x)\) is equal to \(\sum^ A_{k=1}f_ k(x,y)-\sum^ B_{k=1}g_ k(x,y)\) where \(f_ k\) and \(g_ k\) are holomorphic and satisfy a shock wave equation, \(z_ i=w_ i/w_ 0\).
In the special case where \(A=B=0\), one obtains a result of J. Wermer a proof of which by Harvey and Lawson is extended by the authors of this note in the course of the proof of their main result.