L’enveloppe local des perturbations d’une union de plans totalement réels qui se coupent en une droite réelle

Abstract

In this note by using techniques similar to that of [2] and [3], we study the local polynomial convexity of perturbation of union of two totally real planes meeting along a real line.

Local polynomial convexity of the union of two totally-real surfaces at their intersection

“…2) It turns out that when T 0 S 1 and T 0 S 2 contain a line, then T 0 S 1 ∪T 0 S 2 is always locally polynomially convex at the origin. There are some partial answers to ( * ) when dim R (T 0 S 1 ∩ T 0 S 2 ) = 1; see, for instance, [3]. However, many of the results that we are aware of require S 1 and S 2 to be real-analytic surfaces (and one of these results contains an error; see Remark 1.6).…”

“…Let S 1 and S 2 be two C 2 -smooth surfaces in C 2 that contain the origin. Assume that [3]). Let ϕ be a real-valued function defined in a neighbourhood of 0 ∈ C and of class C 1 .…”

“…Remark 1.6. Unbeknownst to me, Dieu had announced the following result in [3]: [3]). Let ϕ be a real-valued function defined in a neighbourhood of 0 ∈ C and of class C 1 .…”

“…First, we examine the image of S 1 ∩ U under P . Let us designateφ 1 (z) := Az 2 + Az 2 + C 1 zz + O(|z| 3 ), z ∈ D(0; δ).Thus we have,P (z, z + φ 1 (z)) = z + z + Az 2 + Az 2 + C 1 |z| 2 + αz 2 + αz 2 + O(|z| 3 ), ℑmP (z, z + φ 1 (z)) = ℑmC 1 |z| 2 + O(|z|3 ) ∀z ∈ D(0; δ). (7.1)…”