18 research outputs found
Tameness of holomorphic closure dimension in a semialgebraic set
Given a semianalytic set S in a complex space and a point p in S, there is a
unique smallest complex-analytic germ at p which contains the germ of S, called
the holomorphic closure of S at p. We show that if S is semialgebraic then its
holomorphic closure is a Nash germ, for every p, and S admits a semialgebraic
filtration by the holomorphic closure dimension. As a consequence, every
semialgebraic subset of a complex vector space admits a semialgebraic
stratification into CR manifolds satisfying a strong version of the condition
of the frontier.Comment: Published versio
A descending chain condition for groups definable in o-minimal structures
We prove that if G is a group definable in a saturated o-minimal structure, then G has no infinite descending chain of type-definable subgroups of bounded index. Equivalently, G has a smallest (necessarily normal) type-definable subgroup G 00 of bounded index and G/G 00 equipped with the “logic topology” is a compact Lie group. These results give partial answers to some conjectures of the fourth author