Degenerate complex Hessian equations on compact Kähler manifolds
Abstract
Let $(X,\omega)$ be a compact Kähler manifold of dimension $n$ and fix $m\in \mathbb{N}$ such that $1\leq m \leq n$. We prove that any $(\omega,m)$sh function can be approximated from above by smooth $(\omega,m)$sh functions. A potential theory for the complex Hessian equation is also developed which generalizes the classical pluripotential theory on compact Kähler manifolds. We then use novel variational tools due to Berman, Boucksom, Guedj and Zeriahi to study degenerate complex Hessian equations.
 Publication:

arXiv eprints
 Pub Date:
 February 2014
 arXiv:
 arXiv:1402.5147
 Bibcode:
 2014arXiv1402.5147L
 Keywords:

 Mathematics  Complex Variables;
 Mathematics  Analysis of PDEs;
 Mathematics  Differential Geometry