Indices of holomorphic foliations and the bifurcation conjecture
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912537678708736 |
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| author | Luza, Maycol Falla Fernández-Pérez, Arturo Marín, David Rosas, Rudy |
| author_facet | Luza, Maycol Falla Fernández-Pérez, Arturo Marín, David Rosas, Rudy |
| contents | In this paper, we revisit local invariants (Gómez-Mont-Seade-Verjovsky, variation, Camacho-Sad and Baum-Bott indices) associated with singular holomorphic foliations on $(\mathbb{C}^2 , 0)$ and we provide semi-global formulas for them in terms of the reduction of singularities of the foliation. A key technical ingredient is the Cholesky-type factorization of the intersection matrix of the exceptional divisor, which allows for an explicit control of multiplicities and indices along the resolution process. Using this factorization, we express the Milnor number and other indices as quadratic forms in intersection vectors associated to balanced divisors introduced by Y. Genzmer. As a main application, we address a conjecture posed by A. Szawlowski concerning pencils of plane holomorphic germs. We prove that the excess of Milnor numbers along the pencil is precisely captured by the invariants derived from our formulas, thereby confirming the conjecture in full generality. This also yields a new expression for the dimension of the parameter space of universal unfoldings of meromorphic functions in the sense of T. Suwa. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_10708 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Indices of holomorphic foliations and the bifurcation conjecture Luza, Maycol Falla Fernández-Pérez, Arturo Marín, David Rosas, Rudy Algebraic Geometry Complex Variables 32M25, 14H10 In this paper, we revisit local invariants (Gómez-Mont-Seade-Verjovsky, variation, Camacho-Sad and Baum-Bott indices) associated with singular holomorphic foliations on $(\mathbb{C}^2 , 0)$ and we provide semi-global formulas for them in terms of the reduction of singularities of the foliation. A key technical ingredient is the Cholesky-type factorization of the intersection matrix of the exceptional divisor, which allows for an explicit control of multiplicities and indices along the resolution process. Using this factorization, we express the Milnor number and other indices as quadratic forms in intersection vectors associated to balanced divisors introduced by Y. Genzmer. As a main application, we address a conjecture posed by A. Szawlowski concerning pencils of plane holomorphic germs. We prove that the excess of Milnor numbers along the pencil is precisely captured by the invariants derived from our formulas, thereby confirming the conjecture in full generality. This also yields a new expression for the dimension of the parameter space of universal unfoldings of meromorphic functions in the sense of T. Suwa. |
| title | Indices of holomorphic foliations and the bifurcation conjecture |
| topic | Algebraic Geometry Complex Variables 32M25, 14H10 |
| url | https://arxiv.org/abs/2508.10708 |