The $\bar\partial$-problem on $Z(q)$-domains

Debraj Chakrabarti, Phillip S. Harrington and Andrew Raich

Vol. 342 (2026), No. 1, 1–35

DOI: 10.2140/pjm.2026.342.1

Abstract

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.

Keywords

Hermitian manifolds, $Z(q)$, $q$-complete, $q$-convex, $q$-plurisubharmonic, $\bar\partial$-problem, $L^2$-theory

Mathematical Subject Classification

Primary: 32F10, 32F32, 32W05

Milestones

Received: 14 March 2025

Revised: 19 September 2025

Accepted: 21 January 2026

Published: 23 March 2026