tailieunhanh - A note on infinite type germs of a real hypersurface in C2

The purpose of this article is to show that there exists a smooth real hypersurface germ (M p), of D'Angelo infinite type in C2 such that it does not admit any (singular) holomorphic curve that has infinite order contact with M at p . | VNU Journal of Science Mathematics - Physics Vol. 35 No. 2 2019 82-87 Original Article A Note on Infinite Type Germs of a Real Hypersurface in c2 Nguyen Thi Kim Son1 Chu Van Tiep2 1Department of Mathematics Hanoi University of Mining and Geology 18 Pho Vien Bac Tu Liem Hanoi Department of Mathematics Da Nang University of Education at Da Nang 459 Ton Duc Thang Lien Chieu Da Nang Received 02 April 2019 Revised 10 April 2019 Accepted 10 April 2019 Abstract The purpose of this article is to show that there exists a smooth real hypersurface germ M p of D Angelo infinite type in c2 such that it does not admit any singular holomorphic curve that has infinite order contact with M at p . 2010 Mathematics Subject Classification. Primary 32T25 Secondary 32C25. Key words and phrases Holomorphic vector field automorphism group real hypersurface infinite type point. 1. Introduction Let M p be a germ at p of a real smooth hypersurface in C and let r be a local defining function for M near p . The normalized order of contact of the curve Y with M at p is defined by v r v r M Y p Vv Where Y 0 p and V y is the vanishing order of Y t Y 0 at t 0 v k is the vanishing order of r y í at t 0. The D Angelo type of M at p is defined by Corresponding author. Email address https 2588-1124 82 . Son C. V. Tiep VNU Journal of Science Mathematics - Physics Vol. 35 No. 2 2019 82-87 83 t M p sup t M r p sup r r v roỵ v r where the supremum is taken over all germs r A C of non-constant holomorphic curves with r 0 p . Here and in what follows A z e c z 0 and A A . We say that p is of D Angelo finite type if T M p and of D Angelo infinite type if otherwise. Throughout the paper we assume that M p is of D Angelo infinite type. Then there exists a v roỵ sequence of non-constant holomorphic curves rn such that A 00 as n . It is natural v rn to ask whether there exists a variety that has infinite order contact with M p . This question pertains to