Poincaré-Lelong type formulas and Segre numbers
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866914939241758720 |
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| author | Andersson, Mats |
| author_facet | Andersson, Mats |
| contents | Let $E$and $F$ be Hermitian vector bundles over a complex manifold $X$ and let $g\colon E\to F$ be a holomorphic morphism. We prove a Poincaré-Lelong type formula with a residue term $M^g$. The currents $M^g$ so obtained have an expected functorial property. We discuss various applications: If $F$ has a trivial holomorphic subbundle of rank $r$ outside the analytic set $Z$, then we get currents with support on $Z$ that represent the Bott-Chern classes $\hat c_k(E)$ for $k >\rank E-r$. We also consider Segre and Chern forms associated with certain singular metrics on $F$. The multiplicities (Lelong numbers) of the various components of $M^g$ only depend on the cokernel of the adjoint sheaf morphism $g^*$. This leads to a notion of distinguished varieties and Segre numbers of an arbitrary coherent sheaf, generalizing these notions, in particular the Hilbert-Samuel multiplicity, in case of an ideal sheaf. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_01905 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Poincaré-Lelong type formulas and Segre numbers Andersson, Mats Complex Variables Algebraic Geometry 32Q99 Let $E$and $F$ be Hermitian vector bundles over a complex manifold $X$ and let $g\colon E\to F$ be a holomorphic morphism. We prove a Poincaré-Lelong type formula with a residue term $M^g$. The currents $M^g$ so obtained have an expected functorial property. We discuss various applications: If $F$ has a trivial holomorphic subbundle of rank $r$ outside the analytic set $Z$, then we get currents with support on $Z$ that represent the Bott-Chern classes $\hat c_k(E)$ for $k >\rank E-r$. We also consider Segre and Chern forms associated with certain singular metrics on $F$. The multiplicities (Lelong numbers) of the various components of $M^g$ only depend on the cokernel of the adjoint sheaf morphism $g^*$. This leads to a notion of distinguished varieties and Segre numbers of an arbitrary coherent sheaf, generalizing these notions, in particular the Hilbert-Samuel multiplicity, in case of an ideal sheaf. |
| title | Poincaré-Lelong type formulas and Segre numbers |
| topic | Complex Variables Algebraic Geometry 32Q99 |
| url | https://arxiv.org/abs/2312.01905 |