Twisted Adiabatic Limit for Complex Structures

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