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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2310.12400 |
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| _version_ | 1866909074611765248 |
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| author | Lin, Genglong |
| author_facet | Lin, Genglong |
| contents | Let $(X,ω)$ be a compact Kähler manifold of dimension $n$ and fix an integer $m$ such that $1\leq m\leq n$. We reformulate most relative pluripotential results of Darvas-DiNezza-Lu's survey \cite{DNL23} to the Hessian setting. As an application, we use a slightly different method and give an characterization of finite energy range of the Hessian operator, which cannot be directly reformulated by \cite{DNL23}.
Given a model potential $ϕ$, we also study degenerate complex Hessian equations of the form $(ω+dd^c φ)^m\wedgeω^{n-m}=F(x,φ)ω^n$. Under some natrual conditions on $F$, we prove that the solution of this type equation has the same singularity type as $ϕ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_12400 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Solution to Hessian type equations with prescribed singularity on compact Kahler manifold Lin, Genglong Differential Geometry Let $(X,ω)$ be a compact Kähler manifold of dimension $n$ and fix an integer $m$ such that $1\leq m\leq n$. We reformulate most relative pluripotential results of Darvas-DiNezza-Lu's survey \cite{DNL23} to the Hessian setting. As an application, we use a slightly different method and give an characterization of finite energy range of the Hessian operator, which cannot be directly reformulated by \cite{DNL23}. Given a model potential $ϕ$, we also study degenerate complex Hessian equations of the form $(ω+dd^c φ)^m\wedgeω^{n-m}=F(x,φ)ω^n$. Under some natrual conditions on $F$, we prove that the solution of this type equation has the same singularity type as $ϕ$. |
| title | Solution to Hessian type equations with prescribed singularity on compact Kahler manifold |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2310.12400 |