A geometric approach to Catlin's boundary systems

TL;DR

This paper introduces new geometric invariants for real hypersurfaces in complex space, linking third and fourth order tensors to boundary systems, Levi form properties, and finite type classifications.

Contribution

It defines invariant cubic and quartic tensors that characterize boundary systems and Levi form degeneracies, connecting these to established invariants like D'Angelo's finite type.

Findings

Invariant cubic tensor $ au^3_p$ vanishes on pseudoconvex hypersurfaces.

Introduction of invariant quartic tensor $ au^4_p$ for Levi rank stratification.

Explicit algebraic description of tangent spaces using these tensors.

Abstract

For a point $p$ in a smooth real hypersurface $M\subset\C^n$, where the Levi form has the nontrivial kernel $K^{10}_p$, we introduce an invariant cubic tensor $\tau^3_p \colon \C T_p \times K^{10}_p \times \overline{K^{10}_p} \to \C\otimes (T_p/H_p)$, which together with Ebenfelt's tensor $\psi_3$, constitutes the full set of $3$rd order invariants of $M$ at $p$. Next, in addition, assume $M\subset\C^n$ to be {\em (weakly) pseudoconvex}. Then $\tau^3_p$ must identically vanish. In this case we further define an invariant quartic tensor $\tau^4_p \colon \C T_p \times \C T_p \times K^{10}_p\times \overline{K^{10}_p} \to \C\otimes (T_p/H_p)$, and for every $q=0, \ldots, n-1$, an invariant submodule sheaf of $(1,0)$ vector fields in terms of the Levi form, and an invariant ideal sheaf of complex functions generated by certain derivatives of the Levi form, such that the set of points of Levi rank $q$ is locally contained in certain real submanifolds defined by real parts of the functions in the ideal sheaf, whose tangent spaces have explicit algebraic description in terms of the quartic tensor $\tau^4$. Finally, we relate the introduced invariants with D'Angelo's finite type, Catlin's mutlitype and Catlin's boundary systems.