Wednesday 21 October from 11:00–12:00 in Carslaw 535A

A Valuation Criterion of Hironaka

Pierre Milman (Toronto)

Abstract

For ideals in the rings of convergent power series expansions their `integral closures' and, the so called, `arc-closures' coincide. (I'll start by defining both notions.) "Integral closure contains the arc-closure" is the nontrivial inclusion of this criterion (first stated by H. Hironaka after which Lejeune-Teissier announced a proof). To prove the inclusion it suffices to construct an ideal whose product with our ideal contains its product with the arc-closure of our ideal. Jointly with Grant-Melles I proved a stronger version of the latter with ideals being replaced by coherent ideals on manifolds. (I'll explain the notion of coherency.) Our proof (that appeared in 2006 in Ann. Fac. Sci. Toulouse Math. in Appendix to Ch. V) is essentially an application of our algebraic generalization of the well known Chow Theorem, i.e. that closed complex analytic subsets of a complex projective space are algebraic.