  We initiate the study of an analogue of the Yamabe problem for complex
manifolds. More precisely, fixed a conformal Hermitian structure on a compact
complex manifold, we are concerned in the existence of metrics with constant
Chern scalar curvature. In this note, we set the problem and we provide a
positive answer when the expected constant Chern scalar curvature is
non-positive. In particular, this includes the case when the Kodaira dimension
of the manifold is non-negative. Finally, we give some remarks on the positive
curvature case, showing existence in some special cases and the failure, in
general, of uniqueness of the solution.
