A geometric approach to Catlin’s boundary systems

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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.