I will describe a geometric problem on families of elliptic operators, which is solved via a deformation to a family of non self-adjoint Fredholm operators. Let $p:M\to S$ be a proper holomorphic projection of complex manifolds. Let $F$ be a holomorphic vector bundle on $M$. We assume that the $H^{(0,p)}(Xs, F|X_s)$ have locally constant dimension. They are the fibers of a holomorphic vector bundle on $S$. The problem we will address is the computation of characteristic classes associated with the above vector bundle in a refinement of the ordinary de Rham cohomology of $S$, its Bott-Chern cohomology, and the proof of a corresponding theorem of Riemann-Roch-Grothendieck. When $M$ is not K\"ahler, none of the existing techniques to prove such a result using the fiberwise Dolbeault Laplacians can be used. The solution is obtained via a proper deformation of the corresponding Dolbeault Laplacians to a family of hypoelliptic Laplacians, for which the corresponding result can be proved. This deformation is made to destroy the geometric obstructions which exist in the elliptic theory, like the fact the metric should be K\"ahler, or the K\"ahler form to be $\bar\partial\partial$-closed.