Abstract Given a complex manifold X and a smooth positive function $$\eta $$ η thereon, we perturb the standard differential operator $$d=\partial + \bar{\partial }$$ d = ∂ + ∂ ¯ acting on differential forms to a first-order differential operator $$D_\eta $$ D η whose principal part is $$\eta \partial + \bar{\partial }$$ η ∂ + ∂ ¯ . The role of the zero-th order part is to force the integrability property $$D_\eta ^2=0$$ D η 2 = 0 that leads to a cohomology isomorphic to the de Rham cohomology of X, while the components of types $$(0,\,1)$$ ( 0 , 1 ) and $$(1,\,0)$$ ( 1 , 0 ) of $$D_\eta $$ D η 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 $$\eta $$ η . 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 $$\eta $$ η . This study continues and generalises the one of the operators $$d_h=h\partial + \bar{\partial }$$ d h = h ∂ + ∂ ¯ that we introduced and investigated recently for a positive constant h that was then let to converge to 0 and, more generally, for constants $$h\in \mathbb {C}$$ h ∈ C . The operators $$d_h$$ 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 $$\eta $$ η creates positivity in the curvature-like operator that stands one in good stead for various kinds of applications.