I wish to know if this PDE can be solved (for a real smooth function $\rho$) on a compact complex surface X : $\bar{\partial}\partial \rho \wedge \bar{\partial}\partial \rho + \bar{\partial}\partial \rho \wedge \bar{\partial}\partial \sigma + \bar{\partial}\partial \sigma \wedge \bar{\partial}\partial \sigma = 0$ where $\sigma$ is a given smooth real function on X. (Note that, one has solutions to this in $\mathbb{C}^2$ if $\sigma$ is real analytic). The problem is that the linearisation of this equation is not elliptic. The motivation for solving this is to produce a trivial line bundle with a hermitian metric so that its Chern character forms (not cohomology classes) are zero.
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