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arXiv · 1202.2435

Smooth solutions to the complex Hessian equation

Lu Hoang Chinh
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Abstract

Let $(X,ω)$ be a compact Kähler manifold of dimension $n$, and fix $1\leq m\leq n.$ We prove that the complex Hessian equation $(ω+dd^cφ)^m\wedge ω^{n-m}=fω^n$, with $0<f\in \mathcal{C}^{\infty}(X)$ has a smooth admissible solution $ φ\in \mathcal{C}^{\infty}(X)$. This was previously known to hold when $(X,ω)$ has non negative holomorphic bisectional curvature.

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Lu Hoang Chinh. 2012-03-27. Smooth solutions to the complex Hessian equation. https://arxiv.org/abs/1202.2435

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