Abstract
We construct examples of C∞ smooth submanifolds in ℂn and ℝn of codimension 2 and 1, which intersect every complex, respectively real, analytic curve in a discrete set. The examples are realized either as compact tori or as properly imbedded Euclidean spaces, and are the graphs of quasianalytic functions. In the complex case, these submanifolds contain real n-dimensional tori or Euclidean spaces that are not pluripolar while the intersection with any complex analytic disk is polar.
Highlights
If a real analytic (2n − 2)-dimensional submanifold R in Cn either intersects every complex analytic disk in a discrete set or contains the disk, R is a complex submanifold
There exists a smooth, compact manifold R in Cn, diffeomorphic to a (2n−2)-dimensional torus, which intersects every analytic disk in a discrete set
In [CLP] we proved that quasianalytic curves are pluripolar
Summary
If a real analytic (2n − 2)-dimensional submanifold R in Cn either intersects every complex analytic disk in a discrete set or contains the disk, R is a complex submanifold. The question whether the polarity of intersections of a set implies the pluripolarity of that set when complex lines are replaced by onedimensional varieties of higher degree was open since that time It was posed as an open problem by E. Using the same ideas as in the proof of Theorem 1.1, we construct in Theorem 3.1 a quasianalytic function on Rn whose graph intersects any real analytic curve in a discrete set and, does not contain any analytic curve. It serves as an example of an “extremely” smooth function which is not arc-analytic anywhere (see [BM1]). We would like to thank Al Taylor for introducing us to the notion of quasianalytic functions
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