On \(L^2\)-estimates for \(\overline{\partial}\) on pseudoconvex domains in a complete Kähler manifold with positive holomorphic bisectional curvature (Q447878)

This note studies \(L^{2}\)-estimates for \(\overline{\partial}\) on relatively compact pseudoconvex domains in a Kähler manifold which posess a plurisubharmonic defining function. Such estimates are established, in particular, under a condition on an expression involving the complex Hessian (with respect to the Kähler metric) of the boundary distance. The expression is one that occurs when estimating the complex Hessian of powers of the boundary distance. The \(L^{2}\)-estimates are then obtained via the method of \textit{B. Berndtsson} and \textit{P. Charpentier} [Math. Z. 235, No. 1, 1--10 (2000; Zbl 0969.32015)].