Smooth solutions to the complex Hessian equation

Abstract

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