Triangular resolutions and effectiveness for holomorphic subelliptic multipliers

Abstract

© 2021 Elsevier Inc.A solution to the effectiveness problem in Kohn's algorithm for generating holomorphic subelliptic multipliers is provided for general classes of domains of finite type in Cn, that include the so-called special domains given by finite and infinite sums of squares of absolute values of holomorphic functions. Also included is a more general class of domains recently discovered by M. Fassina [23]. More generally, for any smoothly bounded pseudoconvex domain we introduce an invariantly defined associated sheaf S of C-subalgebras of holomorphic function germs, that combined with a result of Fassina, reduces the existence of effective subelliptic estimates at p to a purely algebraic geometric question of controlling the multiplicity of S. Our main new tool, a triangular resolution, is the construction of subelliptic multipliers decomposable as Q∘Γ, where Γ is constructed from pre-multipliers and Q is part of a triangular system. The effectiveness is proved via a sequence of newly proposed procedures, called here meta-procedures, built on top of the holomorphic Kohn's procedures, where the order of subellipticity can be effectively tracked. Important sources of inspiration are algebraic geometric techniques by Y.-T. Siu [54,55] and procedures for triangular systems by D.W. Catlin and J.P. D'Angelo [16,8]. The proposed procedures are purely algebraic and as such can be of wider interest for geometric and computational problems involving Jacobian determinants, such as resolving singularities of holomorphic maps.