Bi-Lipschitz triviality of function-germs on singular varieties
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917917522657280 |
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| author | Sinha, Raúl Oset Ruas, Maria Aparecida Soares |
| author_facet | Sinha, Raúl Oset Ruas, Maria Aparecida Soares |
| contents | In this paper we study the bi-Lipschitz triviality of deformations of an analytic function germ $f$ defined on a germ of an analytic variety $(X, 0)$ in $\mathbb C^n$. We introduce the notion of strongly rational $\mathscr R_X$-bi-Lipschitz trivial families and give an infinitesimal criterion which is a sufficient condition for the bi-Lipschitz triviality of deformations of $f$ on $(X,0).$ As a corollary it follows that when $X$ and $f$ are homogeneous of the same degree, all deformation of $f$ of the same or higher degrees are bi-Lipschitz trivial. We then prove a rigidity result for deformations of $f$ on $X$ when both are weighted homogeneous with respect to the same set of weights. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_06436 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bi-Lipschitz triviality of function-germs on singular varieties Sinha, Raúl Oset Ruas, Maria Aparecida Soares Algebraic Geometry Complex Variables Differential Geometry Primary 14B05, Secondary 58K40, 32S25 In this paper we study the bi-Lipschitz triviality of deformations of an analytic function germ $f$ defined on a germ of an analytic variety $(X, 0)$ in $\mathbb C^n$. We introduce the notion of strongly rational $\mathscr R_X$-bi-Lipschitz trivial families and give an infinitesimal criterion which is a sufficient condition for the bi-Lipschitz triviality of deformations of $f$ on $(X,0).$ As a corollary it follows that when $X$ and $f$ are homogeneous of the same degree, all deformation of $f$ of the same or higher degrees are bi-Lipschitz trivial. We then prove a rigidity result for deformations of $f$ on $X$ when both are weighted homogeneous with respect to the same set of weights. |
| title | Bi-Lipschitz triviality of function-germs on singular varieties |
| topic | Algebraic Geometry Complex Variables Differential Geometry Primary 14B05, Secondary 58K40, 32S25 |
| url | https://arxiv.org/abs/2502.06436 |