A higher-dimensional Van den Essen type formula for projective foliations and applications

TL;DR

This paper extends Van den Essen's formula to higher dimensions for projective foliations, providing tools to analyze singularities and compute Milnor numbers after blowups, with applications to foliation resolution.

Contribution

It introduces a higher-dimensional Van den Essen type formula for projective foliations and applies it to bound Milnor numbers and resolve singularities.

Findings

Derived a formula for Milnor number variation under blowups in higher dimensions.

Established bounds on the number of blow-ups needed for foliation resolution.

Provided lower bounds for Milnor numbers based on invariants of the singular set.

Abstract

Let $F$ be a one-dimensional holomorphic foliation on $\mathbb{P}^n$ such that $W\subset Sing(F)$, where $W$ is a smooth complete intersection variety. We determine and compute the variation of the Milnor number $ \mu(F, W)$ under blowups, which depends on the vanishing order of the pullback foliation along the exceptional divisor, as well as on numerical and topological invariants of $W$. This represents a higher-dimensional version of Van den Essen's formula for projective foliations of dimension one. As an application, we obtain a lower bound for the Milnor number of the foliation. Also, we use this formula to show that for a foliation on $\mathbb{P}^n$ that is singular along a smooth curve, there exists a finite number of blow-ups with centers on smooth curves such that the induced foliation has multiplicity equal to 1 and that for generic points of the curves in the final stage, the singularities are elementary. Moreover, we obtain a bound on the maximum number of blow-ups needed to resolve the foliation, depending on the numerical and topological invariants of the curve.