On a compact connected 2 m -dimensional K?hler manifold with K?hler form ω, given a smooth function f: M → ? 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.
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