Yamabe-type problems on compact Hermitian manifolds

We introduce and study a one-parameter Hermitian deformation of the Yamabe problem on compact complex manifolds. The deformation is defined by adding natural torsion terms to the Riemannian scalar curvature, and includes both the classical Yamabe equation and a scalar-curvature equation arising in locally conformally K\"ahler geometry as a momentum map. We analyze criteria for the existence of solutions, and discuss several examples.