Explicit calculation of Siu’s Effective Termination in Kohn’s Algorithm for Special Domains in C^{3}

Mercredi 10 mai 2017 14:00-17:00 - Wei Guo Foo - Orsay

Résumé : 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
[<br class='autobr' />
\Omega :=<br class='autobr' />
\bigg\{(z_{1},z_{2},z_{3})\in\mathbb{C}^{3} :\ <br class='autobr' />
2\text{Re}\ z_{3}+<br class='autobr' />
\sum_{i=1}^{\<span class="caps">NN</span>}|F_{i}(z_{1},z_{2})|^{2}<0<br class='autobr' />
\bigg\}<br class='autobr' />
]<br class='autobr' />
with holomorphic function germs F_1,\dots,F_NN\in\mathcalO_\mathbbC^2,0 of finite intersection multiplicity<br class='autobr' />
[<br class='autobr' />
s :=\dimsmall_{\mathbb{C}}\ <br class='autobr' />
\mathcal{O}_{\mathbb{C}^{2},0}<br class='autobr' />
\big/<br class='autobr' />
\langle <br class='autobr' />
F_{1},\dots, F_{\NN}<br class='autobr' />
\rangle<br class='autobr' />
<<br class='autobr' />
\infty,<br class='autobr' />
]<br class='autobr' />
we show that an \varepsilon-subelliptic regularity for (0,1)-forms holds whenever, just in terms of s,<br class='autobr' />
[<br class='autobr' />
\varepsilon<br class='autobr' />
\geqslant <br class='autobr' />
\frac{1}{<br class='autobr' />
2^{(4s^{2}-1)s+3}<br class='autobr' />
s^{2}(4s^{2}-1)^{4}<br class='autobr' />
\binom{8s+1}{8s-1}}.<br class='autobr' />
]

Lieu : Salle 113-115

Explicit calculation of Siu’s Effective Termination in Kohn’s Algorithm for Special Domains in C^{3}  Version PDF