Complex Hessian Equations on Some Compact Kähler Manifolds

• Main Author: Asma Jbilou
• Format: Article
• Language: English
• Published: Hindawi Limited 2012-01-01
• Series: International Journal of Mathematics and Mathematical Sciences
• Online Access: http://dx.doi.org/10.1155/2012/350183

On a compact connected 2m-dimensional Kähler manifold with Kähler form ω, given a smooth function f:M→ℝ and an integer 1<k<m, we want to solve uniquely in [ω] the equation ω̃k∧ωm-k=efωm, relying on the notion of k-positivity for ω̃∈[ω] (the extreme cases are solved: k=m by (Yau in 1978), and k=1 trivially). We solve by the continuity method the corresponding complex elliptic kth Hessian equation, more difficult to solve than the Calabi-Yau equation (k=m), under the assumption that the holomorphic bisectional curvature of the manifold is nonnegative, required here only to derive an a priori eigenvalues pinching.