Pfaffian Intersections and Multiplicity Cycles

TL;DR

This paper establishes a method to bound intersection multiplicities between algebraic varieties and Pfaffian foliations using algebraic cycles, with applications to Milnor fibers and classical multiplicity estimates.

Contribution

It introduces a new approach to estimate intersection multiplicities via algebraic cycles, extending classical results and providing uniform bounds in various geometric contexts.

Findings

Bounded intersection multiplicity by algebraic cycle multiplicity

Derived uniform estimates for Milnor fiber complexity

Provided an alternative proof for classical multiplicity estimates

Abstract

We consider the problem of estimating the intersection multiplicity between an algebraic variety and a Pfaffian foliation, at every point of the variety. We show that this multiplicity can be majorized at every point $p$ by the local algebraic multiplicity at $p$ of a suitably constructed algebraic cycle. The construction is based on Gabrielov's complex analog of the Rolle-Khovanskii lemma. We illustrate the main result by deriving similar uniform estimates for the complexity of the Milnor fiber of a deformation (under a smoothness assumption) and for the order of contact between an algebraic hypersurface and an arbitrary non-singular one-dimensional foliation. We also use the main result to give an alternative geometric proof for a classical multiplicity estimate in the context of commutative group varieties.