Calculus inequalities derived from holomorphic Morse inequalities (Q914002)

The author proves the following result: Let D be a bounded polynomially convex domain in \({\mathbb{C}}^ n\) and let V be a real-valued function on D such that V is plurisubharmonic near the boundary of D. Let \(i\partial \partial\) be the complex Monge-Ampère operator and let \(D_ q\) be the set of points where the real (1,1)-form \(i\partial {\bar \partial}\) is nondegenerate and has at most q negative eigenvalues, \(0\leq q\leq n\). Then \[ \int_{D}(i\partial {\bar \partial}V)^ n\geq \int_{D_ q}(i\partial {\bar \partial}V)^ n\text{ for } q\quad even\text{ and } \int_{D_ q}(i\partial {\bar \partial}V)^ n\leq \int_{D}(i\partial {\bar \partial}V)^ n\text{ for } q\quad odd. \] Similar results are proven for complex Stein manifolds and for convex functions defined in domains in \({\mathbb{R}}^ n\).