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Pfaffian intersections and multiplicity cycles

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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 Gabrièlov’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.

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Notes

  1. This restriction can be relaxed significantly by a perturbation argument which we omit for simplicity.

  2. In fact, a more general stratified version holds also without the assumption on p, see [12].

  3. The same property holds, with a minor technical modification, for arbitrary invariant vector fields on commutative group varieties.

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Authors and Affiliations

  1. University of Toronto, Toronto, ON, Canada

    Gal Binyamini

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  1. Gal Binyamini
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Correspondence to Gal Binyamini.

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The author was supported by the Banting Postdoctoral Fellowship and the Rothschild Fellowship.

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Binyamini, G. Pfaffian intersections and multiplicity cycles. Sel. Math. New Ser. 22, 297–318 (2016). https://doi.org/10.1007/s00029-015-0191-0

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