Smooth solutions to the complex Hessian equation

Abstract

Let $(X,\omega )$ be a compact Kähler manifold of dimension $n$, and fix $1\le m\le n.$ We prove that the complex Hessian equation $(\omega +d{d}^c\phi {)}^m\wedge {\omega }^{n-m}=f{\omega }^n$, with $0<f\in {\mathcal{C}}^{\infty }(X)$ has a smooth admissible solution $\phi \in {\mathcal{C}}^{\infty }(X)$. This was previously known to hold when $(X,\omega )$ has non negative holomorphic bisectional curvature.

This paper has been withdrawn by the author due to a crucial error in the proof of Theorem 3.1