A note on pseudoconvex hypersurfaces of infinite type in $\mathbb C^n$

Abstract

The purpose of this article is to prove that there exists a real smooth pseudoconvex hypersurface germ (M, p) of D'Angelo infinite type in C n+1 such that it does not admit any (singular) holomorphic curve in C n+1 tangent to M at p to infinite order.2010 Mathematics Subject Classification. Primary 32T25; Secondary 32C25.

On Holomorphic Curves Tangent to Real Hypersurfaces of Infinite Type

“…(6) ⇒ (7) is easy to see from Lemma 4.2. (7) ⇒ (2) is shown in [9]. (Lemma 5 in [9] states (7) ⇒ (1), but its proof actually implies the above stronger implication.)…”

“…It is easy to check that the examples of hypersurface constructed in [3,9,16,22] do not admit any N -canonical coordinates. More exactly, we will give equivalence conditions in more restricted cases in Sects.…”

“…Here f ∈ C ∞ 0 (C) is a subharmonic function admitting a divergent Taylor series at the origin. It was constructed in [9] (see Remark 5.8).…”

“…Furthermore, it is shown in [9] that when {c j } j∈N is an increasing sequence of positive real numbers, the above function f can be selected to be a subharmonic function on C. Therefore, there are many pseudoconvex real hypersurfaces satisfying the condition (i) in the corollary.…”

“…But, in general, the sufficient direction is not true. Indeed, the nonexistence of the curve γ ∞ in (1.2) is shown in the case of some real hypersurfaces constructed in [3,9,16,22]. Understanding flatness on hypersurfaces of infinite type has been recognized to be a delicate issue.…”