On Holomorphic Curves Tangent to Real Hypersurfaces of Infinite Type

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On Holomorphic Curves Tangent to Real Hypersurfaces of Infinite Type Joe Kamimoto1 Received: 13 August 2020 / Accepted: 12 November 2020 © The Author(s) 2020

Abstract The purpose of this paper is to investigate the geometric properties of real hypersurfaces of D’Angelo infinite type in Cn . In order to understand the situation of flatness of these hypersurfaces, it is natural to ask whether there exists a nonconstant holomorphic curve tangent to a given hypersurface to infinite order. A sufficient condition for this existence is given by using Newton polyhedra, which is an important concept in singularity theory. More precisely, equivalence conditions are given in the case of some model hypersurfaces. Keywords Holomorphic curve · Real hypersurface · D’Angelo type · Bloom–Graham type · Infinite type Mathematics Subject Classification 32F18 (32T25)

1 Introduction Let M be a (C ∞ smooth) real hypersurface in Cn and let p lie on M. Let r be a local defining function for M near p (∇r = 0 when r = 0). In [6,7], the following invariant is introduced: 1 (M, p) := sup

γ ∈

ord(r ◦ γ ) , ord(γ − p)

(1.1)

where  denotes the set of (germs of) nonconstant holomorphic mappings γ : (C, 0) → (Cn , p). (For a C ∞ mapping h : C → C or Cn such that h(0) = 0, let ord(h) denote the order of vanishing of h at 0.) The invariant 1 (M, p) is called the D’Angelo type of M at p. We say that M is of finite type at p if 1 (M, p) < ∞

B 1

Joe Kamimoto [email protected] Faculty of Mathematics, Kyushu University, Motooka 744, Nishi-ku, Fukuoka 819-0395, Japan

123

J. Kamimoto

and of infinite type at p otherwise (the latter case will be denoted by 1 (M, p) = ∞). ¯ The class of finite type plays crucial roles in the study of the local regularity in the ∂Neumann problem over pseudoconvex domains  with smooth boundary ∂. Indeed, it was shown by Catlin [4,5] that M = ∂ is of finite type at p if and only if a local subelliptic estimate at p holds. From its importance, real hypersurfaces of finite type have been deeply investigated from various points of view. On the other hand, to understand the geometric properties of real hypersurfaces of infinite type is also an interesting subject in the study of several complex variables. These hypersurfaces contain some kind of strong flatness. In order to describe the geometric structure of this flatness, the situation of contact of holomorphic curves with the respective hypersurface must be carefully observed. In this paper, we mainly consider the following question: Question 1 When does there exist a nonconstant holomorphic curve γ∞ tangent to M at p to infinite order? Since the condition of the desired curve γ∞ in Question 1 can be written as (r ◦ γ∞ )(t) = O(t N )

as t ∈ C → 0, for every N ∈ N,

(1.2)

the condition 1 (M, p) = ∞ is necessary for the existence of the curve γ∞ . It has been shown in [7,11,19] that when M is real analytic, the above two conditions are equivalent (in this case, the curve γ∞ is contained in M). Moreover, in the case of smooth M, this equivalence is also