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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2511.22373 |
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| _version_ | 1866918333197058048 |
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| author | Hong, Nguyen Xuan |
| author_facet | Hong, Nguyen Xuan |
| contents | Let $φ$ be a plurisubharmonic function defined in a neighborhood of the origin in $\mathbb C^n$. For each real number $t>-n$, we associate to $φ$ the weighted log canonical threshold \[ c_t(φ):=\sup\Bigl\{c\geq 0:\|z\|^{2t}e^{-2cφ}\in L^1_{\mathrm{loc}} \text{ near }0\Bigr\}. \] In this paper, we prove a sharp slope inequality showing that all difference quotients of the function $t\mapsto c_t(φ)$ are uniformly controlled by the Lelong number $ν_φ(0)$. Moreover, we derive explicit lower bounds for the growth of $c_t(φ)$ in terms of the complex Monge-Ampère mass of $φ$ at the origin. Our arguments combine weighted integrability estimates, restrictions to complex lines, and techniques from pluripotential theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_22373 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some inequalities for the weighted log canonical thresholds Hong, Nguyen Xuan Complex Variables Differential Geometry Let $φ$ be a plurisubharmonic function defined in a neighborhood of the origin in $\mathbb C^n$. For each real number $t>-n$, we associate to $φ$ the weighted log canonical threshold \[ c_t(φ):=\sup\Bigl\{c\geq 0:\|z\|^{2t}e^{-2cφ}\in L^1_{\mathrm{loc}} \text{ near }0\Bigr\}. \] In this paper, we prove a sharp slope inequality showing that all difference quotients of the function $t\mapsto c_t(φ)$ are uniformly controlled by the Lelong number $ν_φ(0)$. Moreover, we derive explicit lower bounds for the growth of $c_t(φ)$ in terms of the complex Monge-Ampère mass of $φ$ at the origin. Our arguments combine weighted integrability estimates, restrictions to complex lines, and techniques from pluripotential theory. |
| title | Some inequalities for the weighted log canonical thresholds |
| topic | Complex Variables Differential Geometry |
| url | https://arxiv.org/abs/2511.22373 |