Compactness of the \(\overline{\partial}\)-Neumann problem on convex domains (Q1281651)

For a bounded convex domain \(\Omega\) in \(\mathbb{C}^n\) and \(1\leq q\leq n\), the following assertions are proved to be equivalent: (1) there exists a compact solution operator for \(\overline\partial\) on \((0,q)\)-forms; (2) \(\partial\Omega\) does not contain any affine varieties of dimension \(\geq q\); (3) \(\partial\Omega\) does not contain any analytic variety of dimension \(\geq q\); (4) the \(\overline\partial\)-Neumann operator (inverse of \(\overline \partial\overline\partial^*+ \overline\partial^* \overline\partial)\) is compact.