Explicit calculation of Siu's Effective Termination in Kohn's Algorithm for Special Domains in $\mathbb{C}^{3}$

Abstract

In this article, we follow the arguments in a paper of Y-T. Siu to study the effective termination of Kohn's algorithm for special domains in $\mathbb{C}^{3}$. We make explicit the effective constants and generic conditions that appear there, and we obtain an explicit expression for the regularity of the Dolbeault laplacian for the $\overline{\partial}$-Neumann problem. Specifically, on a local peudoconvex domain of the special shape \[ Ω:= \bigg\{(z_{1},z_{2},z_{3})\in\mathbb{C}^{3}:\ 2\text{Re}\ z_{3}+ \sum_{i=1}^{N}|F_{i}(z_{1},z_{2})|^{2}<0 \bigg\} \] with holomorphic function germs $F_{1},\dots,F_{N}\in\mathcal{O}_{\mathbb{C}^{2},0}$ of finite intersection multiplicity \[ s:=\dim_{\mathbb{C}}\ \mathcal{O}_{\mathbb{C}^{2},0} \big/ \langle F_{1},\dots, F_{N} \rangle < \infty, \] we show that an $\varepsilon$-subelliptic regularity for $(0,1)$-forms holds whenever, just in terms of $s$, \[ \varepsilon \geqslant \frac{1}{ 2^{(4s^{2}-1)s+3} s^{2}(4s^{2}-1)^{4} \binom{8s+1}{8s-1}}. \]