Complex Hessian Equations on Some Compact Kähler Manifolds

Source abstract

On a compact connected 2 m -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 ω ̃ k ∧ ω m - k = e f ω m , relying on the notion of k -positivity for ω ̃ ∈ [ 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 k th 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.