Indices of holomorphic foliations and the bifurcation conjecture

TL;DR

This paper develops formulas for local invariants of singular holomorphic foliations on complex surfaces, linking them to the resolution of singularities, and confirms a conjecture relating Milnor number variations in holomorphic pencil families.

Contribution

It introduces semi-global formulas for foliation invariants using Cholesky-type factorizations and verifies Szawlowski's conjecture on Milnor number excess in holomorphic pencils.

Findings

Formulas express indices as quadratic forms in intersection vectors.

Confirmed Szawlowski's conjecture for all cases.

Provided new insights into the parameter space of unfoldings.

Abstract

In this paper, we revisit local invariants (G\'omez-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.