Logrithmic Versions of Ginzburg's Sharp Operation for Free Divisors
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915518433198080 |
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| author | Liao, Xia Zhang, Xiping |
| author_facet | Liao, Xia Zhang, Xiping |
| contents | Let $M$ be a complex manifold, $D\subset M$ a free divisor and $U=M\setminus D$ its complement. In this paper we study the characteristic cycle $\textup{CC}(γ\cdot \ind_U)$ of the restriction of a constructible function $γ$ on $U$. We globalise Ginzburg's local sharp construction and introduce the log transversality condition, which is a new transversality condition about the relative position of $γ$ and $D$. We prove that the log transversality condition is satisfied if either $D$ is normal crossing and $γ$ is arbitrary, or $D$ is holonomic strongly Euler homogheneous and $γ$ is non-characteristic. Under the log transversality assumption we establish a logarithmic pullback formula for $\textup{CC}(γ\cdot \ind_U)$. Mixing Ginzburg's sharp construction with the logarithmic pullback, we obtain a double restriction formula for the Chern-Schwartz-MacPherson class $c_*(γ\cdot \ind_{D\cup V})$ where $V$ is any reduced hypersurface in $M$. Applications of our results include the non-negativity of Euler characteristics of effective constructible functions, and CSM classes of hypersurfaces in the open manifold $\mathbb{P}^n\setminus D$ when $D$ is a linear free divisor or a free hyperplane arrangement. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_24236 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Logrithmic Versions of Ginzburg's Sharp Operation for Free Divisors Liao, Xia Zhang, Xiping Algebraic Geometry 14B05, 14C17, 32S60, 32S05 Let $M$ be a complex manifold, $D\subset M$ a free divisor and $U=M\setminus D$ its complement. In this paper we study the characteristic cycle $\textup{CC}(γ\cdot \ind_U)$ of the restriction of a constructible function $γ$ on $U$. We globalise Ginzburg's local sharp construction and introduce the log transversality condition, which is a new transversality condition about the relative position of $γ$ and $D$. We prove that the log transversality condition is satisfied if either $D$ is normal crossing and $γ$ is arbitrary, or $D$ is holonomic strongly Euler homogheneous and $γ$ is non-characteristic. Under the log transversality assumption we establish a logarithmic pullback formula for $\textup{CC}(γ\cdot \ind_U)$. Mixing Ginzburg's sharp construction with the logarithmic pullback, we obtain a double restriction formula for the Chern-Schwartz-MacPherson class $c_*(γ\cdot \ind_{D\cup V})$ where $V$ is any reduced hypersurface in $M$. Applications of our results include the non-negativity of Euler characteristics of effective constructible functions, and CSM classes of hypersurfaces in the open manifold $\mathbb{P}^n\setminus D$ when $D$ is a linear free divisor or a free hyperplane arrangement. |
| title | Logrithmic Versions of Ginzburg's Sharp Operation for Free Divisors |
| topic | Algebraic Geometry 14B05, 14C17, 32S60, 32S05 |
| url | https://arxiv.org/abs/2505.24236 |