Analyticity theorems for parameter-dependent plurisubharmonic functions
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910444426362880 |
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| author | He, Bojie |
| author_facet | He, Bojie |
| contents | In this paper, we first show that a union of upper-level sets associated to fibrewise Lelong numbers of plurisubharmonic functions is in general a pluripolar subset. Then we obtain analyticity theorems for a union of sub-level sets associated to fibrewise complex singularity exponents of some special (quasi-)plurisubharmonic functions. As a corollary, we confirm that, under certain conditions, the logarithmic poles of relative Bergman kernels form an analytic subset when the (quasi-)plurisubharmonic weight function has analytic singularities. In the end, we give counterexamples to show that the aforementioned sets are in general non-analytic even if the plurisubharmonic function is supposed to be continuous. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_07786 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Analyticity theorems for parameter-dependent plurisubharmonic functions He, Bojie Complex Variables Algebraic Geometry 32A60, 32U05, 32U25, 14E15, 14F18 In this paper, we first show that a union of upper-level sets associated to fibrewise Lelong numbers of plurisubharmonic functions is in general a pluripolar subset. Then we obtain analyticity theorems for a union of sub-level sets associated to fibrewise complex singularity exponents of some special (quasi-)plurisubharmonic functions. As a corollary, we confirm that, under certain conditions, the logarithmic poles of relative Bergman kernels form an analytic subset when the (quasi-)plurisubharmonic weight function has analytic singularities. In the end, we give counterexamples to show that the aforementioned sets are in general non-analytic even if the plurisubharmonic function is supposed to be continuous. |
| title | Analyticity theorems for parameter-dependent plurisubharmonic functions |
| topic | Complex Variables Algebraic Geometry 32A60, 32U05, 32U25, 14E15, 14F18 |
| url | https://arxiv.org/abs/2405.07786 |