Degenerate complex Hessian equations on compact Kahler manifolds
Chin-Pi LuDong Nguyen
32W2032U0532Q15complex Hessianpotential theoryvariational methodregularization
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.
Indiana University Mathematics Journal
2015
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10.1512/iumj.2015.64.5680
10.1512/iumj.2015.64.5680
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Indiana Univ. Math. J. 64 (2015) 1721 - 1745
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