Pfaffian intersections and multiplicity cycles

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–Khovanskiĭ 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.