Tameness of Complex Dimension in a Real Analytic Set
نویسنده
چکیده
Given a real analytic set X in a complex manifold and a positive integer d, denote by Ad the set of points p in X at which there exists a germ of a complex analytic set of dimension d contained in X. It is proved that Ad is a closed semianalytic subset of X.
منابع مشابه
Tameness of Holomorphic Closure Dimension in a Semialgebraic Set
Given a semianalytic set S in Cn and a point p ∈ S, there is a unique smallest complex-analytic germ Xp which contains Sp, called the holomorphic closure of Sp. We show that if S is semialgebraic then Xp 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 semial...
متن کاملOn the Holomorphic Closure Dimension of Real Analytic Sets
Given a real analytic (or, more generally, semianalytic) set R in C (viewed as R), there is, for every p ∈ R̄, a unique smallest complex analytic germ Xp that contains the germ Rp. We call dimC Xp the holomorphic closure dimension of R at p. We show that the holomorphic closure dimension of an irreducible R is constant on the complement of a closed proper analytic subset of R, and discuss the re...
متن کاملMetric Dimensions and Tameness in Expansions of the Real Field
For first-order expansions of the field of real numbers, nondefinability of the set of natural numbers is equivalent to equality of topological and Assouad dimension on images of closed definable sets under definable continuous maps.
متن کاملGradient-like Vector Fields on a Complex Analytic Variety Cheol-hyun Cho and Giovanni Marelli
Given any Morse function f on a Whitney stratified complex analytic variety of complex dimension n, we prove the existence of a stratified gradient-like vector field for f such that the unstable set of a critical point p on a stratum S of complex dimension s has real dimension m(p) + n − s as was conjectured by Goresky and MacPherson.
متن کاملTopology of Nonarchimedean Analytic Spaces and Relations to Complex Algebraic Geometry
This note surveys basic topological properties of nonarchimedean analytic spaces, in the sense of Berkovich, including the recent tameness results of Hrushovski and Loeser. We also discuss interactions between the topology of nonarchimedean analytic spaces and classical algebraic geometry.
متن کامل