Twisted Adiabatic Limit for Complex Structures

Given a complex manifold X and a smooth positive function η thereon, we perturb the standard differential operator d=∂ + ∂ acting on differential forms to a first-order differential operator D_η whose principal part is η∂ + ∂. The role of the zero-th order part is to force the integrability property D_η²=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_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 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 η. This study continues and generalises the one of the operators dₕ=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∈. The operators dₕ 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 η creates positivity in the curvature-like operator that stands one in good stead for various kinds of applications.