IUMJ

Title: Degenerate complex Hessian equations on compact Kahler manifolds

Authors: Chinh Lu and Dong Van Nguyen

Issue: Volume 64 (2015), Issue 6, 1721-1745

Abstract:

Let $(X,\omega)$ be a compact K\"ahler manifold of dimension $n$, and fix $m\in\mathbb{N}$ such that $1\leq m\leq n$. We prove that any $(\omega,m)$-subharmonic function can be approximatedfrom above by smooth $(omega,m)$-subharmonic functions. A potential theory for the complex Hessian equation is also developed that generalizes the classical pluripotential theory on compact K\"ahler manifolds. We then use novel variational tools due to Berman, Boucksom, Guedj, and Zeriahi to solve degenerate complex Hessian equations.