A Riemann-Roch theorem in Bott-Chern cohomology

If \(M\) is a complex manifold, the Bott-Chern cohomology \(H_{\mathrm{BC}}^{(\cdot,\cdot)}\left(M,\mathbf{C}\right)\) of \(M\) is a refinement of de Rham cohomology, that takes into account the \((p,q)\) grading of smooth differential forms. By results of Bott and Chern, vector bundles have characteristic classes in Bott-Chern cohomology, which will be denoted with the subscript \(\mathrm{BC}\). Let \(p:M\to S\) be a proper holomorphic submersion of complex manifolds. Let \(F\) be a holomorphic vector bundle on \(M\) and let \(Rp_{*}F\) be its direct image. We will prove that if \(Rp_{*}F\) is locally free, then \[\mathrm{ch}_{\mathrm{BC}}\left(Rp_{*}F\right)=p_{*}\left[ \mathrm{Td}_{\mathrm{BC}}\left(TM/S\right) \mathrm{ch}_{\mathrm{BC}}\left(F\right)\right].\] If \(p\) is projective, this result is a consequence of the theorem of Riemann-Roch-Grothendieck. If \(M\) is Kaehler, it follows from a families version of classical Hodge theory. In the general case, none of these tools is available. We will show how a suitable hypoelliptic deformation of Hodge theory can be used to prove the above result.