Given a complex manifold containing a relatively compact Z(q) domain, we give sufficient geometric conditions on the domain so that its L 2 -cohomology in degree ( p, q) (known to be finite-dimensional) vanishes.The condition consists in the existence of a smooth weight function in a neighborhood of the closure of the domain, where the complex Hessian of the weight has a prescribed number of eigenvalues of a particular sign, along with good interaction at the boundary of the Levi form with the complex Hessian, encoded in a subbundle of common positive directions for the two Hermitian forms.
Chakrabarti et al. (2026) studied this question.