Degenerate Complex Hessian Equations on Compact Kahler Manifolds

Magazine article, 2015

Let (X, omega) be a compact Kahler manifold of dimension n, and fix m is an element of N such that 1 <= m <= n. We prove that any (omega, m)-subharmonic function can be approximated from 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 Kahler manifolds. We then use novel variational tools due to Berman, Boucksom, Guedj, and Zeriahi to solve degenerate complex Hessian equations.