Smooth solutions to the complex Hessian equation

TL;DR

This paper proves the existence of smooth solutions to the complex Hessian equation on compact Kähler manifolds, extending previous results to more general curvature conditions.

Contribution

It establishes the existence of smooth admissible solutions to the complex Hessian equation without the non-negative curvature assumption.

Findings

Existence of smooth solutions for the complex Hessian equation.

Extension of previous results beyond non-negative holomorphic bisectional curvature.

Solution regularity established on compact Kähler manifolds.

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.