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The jet transcendence degree of a real hypersurface and Huang-Ji-Yau Conjecture

Jan Gregorovic, Ilya Kossovskiy

Abstract

We investigate the problem of holomorphic algebraizibility for real hypersurfaces in complex space. We introduce a new invariant of a (real-analytic) Levi-nondegenerate hypersurface called {\em the jet transcendence degree}. Using this invariant, we solve in the negative the Conjecture of Huang, Ji and Yau on the algabraizability of real hypersurfaces with algebraic syzygies.

The jet transcendence degree of a real hypersurface and Huang-Ji-Yau Conjecture

Abstract

We investigate the problem of holomorphic algebraizibility for real hypersurfaces in complex space. We introduce a new invariant of a (real-analytic) Levi-nondegenerate hypersurface called {\em the jet transcendence degree}. Using this invariant, we solve in the negative the Conjecture of Huang, Ji and Yau on the algabraizability of real hypersurfaces with algebraic syzygies.
Paper Structure (13 sections, 3 theorems, 56 equations)

This paper contains 13 sections, 3 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The method of associated differential equations
  4. Invariants of Segre families and the obstruction of Huang, Ji and Yau
  5. Cartan connections
  6. The jet transcendence degree of a Levi-nondegenerate hypersurface
  7. Explicit construction of invariants of Segre families
  8. Construction of the Cartan connection
  9. Construction of the invariants
  10. Algebraicity of invariants in the Main Theorem
  11. Proof of the Main Theorem
  12. The Cartan connection $\omega$ of type $(PSU(1,2),B)$
  13. Coframe for general hypersurfaces in $\mathbb{C}^2$

Key Result

Theorem 1

Suppose a rigid real-analytic hypersurface in $\mathbb{C}^2$ satisfies that $H_{z \bar{z}}$ is algebraic and everywhere nonzero. Then all its Cartan-Chern-Moser syzygies are algebraic and thus the obstructions of Huang, Ji and Yau do not occur. However, if the rigid hypersurface in addition satisfies that $H_{z}(z, \bar{z})$ is non-algebraic

Theorems & Definitions (7)

  • Theorem 1: Main Theorem
  • Definition 1
  • Definition 2
  • Proposition 3
  • Proposition 4
  • proof
  • proof : Proof of Main Theorem