On 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 relationship between this dimension and the CR dimension of R. 1. Main Results Given a real analytic set R in C (we may identify C with R), we consider the germ Rp at a point p ∈ R, and define Xp to be the unique smallest complex analytic germ at p that contains Rp. We will call dimC Xp the holomorphic closure dimension of R at p (denoted by dimHC Rp). It is natural to ask how this dimension varies with p ∈ R. In [11, Thm. 1.1] the second author showed that, if R is irreducible of pure dimension, then dimHC Rp is constant on a dense open subset of R, which allows us to speak of the generic holomorphic closure dimension of R in this case. Upper semicontinuity implies that the generic value of dimHC Rp is the smallest value of the holomorphic closure dimension on R. It remained an open problem whether the jump in the holomorphic closure dimension actually occurs. In the present paper we construct real analytic sets with non-constant holomorphic closure dimension, and study the structure of the locus of points where this dimension is not generic. Theorem 1.1. Let R be an irreducible real analytic set in C, of dimension d > 0. Then there exists a closed real analytic subset S ⊂ R of dimension less than d, such that the holomorphic closure dimension of R is constant on R \ S. We note that the holomorphic closure dimension is well defined on all of R, including its singular locus. If R is not of pure dimension, then (see, e.g., [4]) the locus of smaller dimension is contained in a proper real analytic subset of dimension less than d, which can be included into S. In particular, S may have non-empty interior in R (see Example 6.4). We also note that unless additional conditions are imposed on R, the germs Xp realizing the holomorphic closure dimension of R at p cannot be glued together to form a global complex analytic set containing R. It should be stressed that the discontinuity in the holomorphic closure dimension is by no means related to (real) singularities of R. As shown in Section 6, the exceptional set S may be nonempty even in the case when R is a smooth real analytic submanifold of C. Instead, the holomorphic closure dimension should be viewed as a measure of how badly a real analytic set is locally twisted with respect to the complex structure of its ambient space. Thus, one can interpret this dimension as a generalization of the concept of a uniqueness set. Indeed, a real analytic set R in C is a local uniqueness set, at a point p ∈ R, for holomorphic functions in C if and only if dimHC Rp = n. The notion of the holomorphic closure dimension can be extended to the class of semianalytic sets. Recall that a set R ⊂ R is semianalytic if, for every point p ∈ R, there exist a neighbourhood U and a