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Continuous Solutions for Degenerate Complex Hessian Equation

Hichame Amal , Saïd Asserda , $icon-email$ Manar Bouhssina

Abstract

Let $(X, ω)$ be an $n$ -dimensional compact Kähler manifold and fix an integer $m$ such that $1 \leq m \leq n$ . Let $μ$ be a finite Borel measure on $X$ satisfying the conditions $H_{m} (δ, A, ω)$ . We study degenerate complex Hessian equations of the form $(ω + d d^{c} φ)^{m} \land ω^{n - m} = F (φ, .) d μ$ . Under some natural conditions on $F$ , we prove that if $0 < δ < \frac{m}{n - m}$ , then this equation has a unique continuous solution.

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Continuous Solutions for Degenerate Complex Hessian Equation

Abstract