Very weak solutions of quadratic Hessian equations

We extend the methods of Lewicka - Pakzad, Székelyhidi - Cao and Li - Qiu to study the notion of very weak solutions to the complex ${\sigma }_2$ equation in domains in ${\mathbb{C}}^n,n\ge 2$. As a by-product we sharpen the regularity threshold of the counterexamples obtained by Li and Qiu in the real case.