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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.17991 |
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| _version_ | 1866912245344108544 |
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| author | Svensson, Ludvig |
| author_facet | Svensson, Ludvig |
| contents | We consider divergent integrals $\int_X ω$ of certain forms $ω$ on a reduced pure-dimensional complex space $X$. The forms $ω$ are singular along a subvariety defined by the zero set of a holomorphic section $s$ of some holomorphic vector bundle $E$. Equipping $E$ with a smooth Hermitian metric allows us to define a finite part $\mathrm{fp}\,\int_X ω$ of the divergent integral as the action of a certain current extension of $ω$. We introduce a current calculus to compute finite parts for a special class of $ω$. Our main result is a formula that decomposes the finite part of such an $ω$ into sums of products of explicit currents. Lastly, we show that, in principle, it is possible to reduce the computation of $\mathrm{fp}\,\int_X ω$ for a general $ω$ to this class. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_17991 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Calculus for Finite Parts and Residues of some Divergent Complex Geometric Integrals Svensson, Ludvig Complex Variables Mathematical Physics We consider divergent integrals $\int_X ω$ of certain forms $ω$ on a reduced pure-dimensional complex space $X$. The forms $ω$ are singular along a subvariety defined by the zero set of a holomorphic section $s$ of some holomorphic vector bundle $E$. Equipping $E$ with a smooth Hermitian metric allows us to define a finite part $\mathrm{fp}\,\int_X ω$ of the divergent integral as the action of a certain current extension of $ω$. We introduce a current calculus to compute finite parts for a special class of $ω$. Our main result is a formula that decomposes the finite part of such an $ω$ into sums of products of explicit currents. Lastly, we show that, in principle, it is possible to reduce the computation of $\mathrm{fp}\,\int_X ω$ for a general $ω$ to this class. |
| title | A Calculus for Finite Parts and Residues of some Divergent Complex Geometric Integrals |
| topic | Complex Variables Mathematical Physics |
| url | https://arxiv.org/abs/2502.17991 |