Given a complex manifold X and a smooth positive function \(\eta \) thereon, we perturb the standard differential operator \(d=\partial + \bar{\partial }\) acting on differential forms to a first-order differential operator \(D_\eta \) whose principal part is \(\eta \partial + \bar{\partial }\) . The role of the zero-th order part is to force the integrability property \(D_\eta ^2=0\) that leads to a cohomology isomorphic to the de Rham cohomology of X, while the components of types \((0,\,1)\) and \((1,\,0)\) of \(D_\eta \) induce cohomologies isomorphic to the Dolbeault and conjugate-Dolbeault cohomologies. We compute Bochner-Kodaira-Nakano-type formulae for the Laplacians induced by these operators and a given Hermitian metric on X. The computations throw up curvature-like operators of order one that can be made (semi-)positive under appropriate assumptions on the function \(\eta \) . As applications, we obtain vanishing results for certain harmonic spaces on complete, non-compact, manifolds and for the Dolbeault cohomology of compact complex manifolds that carry certain types of functions \(\eta \) . This study continues and generalises the one of the operators \(d_h=h\partial + \bar{\partial }\) that we introduced and investigated recently for a positive constant h that was then let to converge to 0 and, more generally, for constants \(h\in \mathbb {C}\) . The operators \(d_h\) had, in turn, been adapted to complex structures from the well-known adiabatic limit construction for Riemannian foliations. Allowing now for possibly non-constant functions \(\eta \) creates positivity in the curvature-like operator that stands one in good stead for various kinds of applications.
