Copyright © 2012 Asma Jbilou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

On a compact connected -dimensional Kähler manifold with Kähler form , given a smooth function and an integer , we want to solve uniquely in the equation , relying on the notion of -positivity for (the extreme cases are solved: by (Yau in 1978), and trivially). We solve by the continuity method the corresponding complex elliptic th Hessian equation, more difficult to solve than the Calabi-Yau equation (), under the assumption that the holomorphic bisectional curvature of the manifold is nonnegative, required here only to derive an a priori eigenvalues pinching.