Indices of holomorphic foliations and the bifurcation conjecture

Falla Luza M.; Fernández-Pérez A.; Marín D.; Rosas R. · International Mathematics Research Notices (2026) · DOI: 10.1093/imrn/rnag234

Resumen

In this paper, we revisit local invariants (Gómez-Mont–Seade–Verjovsky, variation, Camacho–Sad and Baum–Bott indices) associated with singular holomorphic foliations on (Formula presented) 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 self-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 Genzmer. As a main application, we address a conjecture posed by 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 Suwa. © The Author(s) 2026. Published by Oxford University Press. All rights reserved. For commercial re-use, please contact reprints@oup.com for reprints and translation rights for reprints. All other permissions can be obtained through our RightsLink service via the Permissions link on the article page on our site—for further information please contact journals.permissions@oup.com. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/pages/standard-publication-reuse-rights)

Instituciones: Pontificia Universidad Católica del Perú