A non-elliptic PDE - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T09:26:14Z http://mathoverflow.net/feeds/question/40670 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/40670/a-non-elliptic-pde A non-elliptic PDE Vamsi 2010-09-30T20:18:31Z 2010-09-30T20:18:31Z <p>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.</p>