Solutions to degenerate complex Hessian equations

Details Solutions to degenerate complex Hessian equations Solutions to degenerate complex Hessian equations

2012-02-11

arxiv.org/abs/1202.2436v2

Mathematics

Complex Variables

Scientific paper

Let $(X,\omega)$ be a $n$-dimensional compact K\"{a}hler manifold. In this paper we study degenerate complex Hessian equations of the form $(\omega+dd^c\varphi)^m\wedge \omega^{n-m}=F(x,\varphi)\omega^n.$ We develop the first steps of a potential theory for the associated complex Hessian operator. In particular we define a notion of bounded weak solution for this equation. We then prove that under some natural conditions on $F$ and a curvature assumption on $\omega$, this equation has a unique continuous solution.

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