## Lifts of Analytic Discs from $X$ to $T^*X$

J. Math. Sci. Univ. Tokyo
Vol. 5 (1998), No. 4, Page 713--725.

Baracco, Luca ; Zampieri, Giuseppe
Lifts of Analytic Discs from $X$ to $T^*X$
We state a general criterion for existence of analytic discs attached to conormal bundles of CR manifolds. In particular let $S$ be a CR (non--generic) submanifold of $X=\C^n$ and $E^*$ a CR subbundle of the complex conormal bundle $T^*_SX\cap\im T^*_SX$ such that $E^*+\sqrt{-1}E^*=T^*_SX\cap\sqrt{-1}T^*_SX$ (where sum and multiplication by $\sqrt{-1}$ are understood in the sense of the fibers). We then show that for any small disc $A$ attached to $S$ through $z_o$ , and for any point $p_o\in (E^*)_{z_o}$, there is an analytic lift $A^*$ attached to $E^*$ through $p_o$. In particular we regain the theorem by Trepreau and Tumanov \cite{T 3} on existence of lifts for discs attached to non--minimal manifolds. Our criterion also applies to discs attached to manifolds with a constant number of negative Levi--eigenvalues. We finally state the uniqueness of small discs attached to (non--necessarily CR) manifolds $M$ through a given point $z_o$ and with prescribed components in $T^\C_{z_o}M$. This is a slight, but perhaps interesting, generalization of the classical result (often used all through this paper), on uniqueness of lifts of small discs attached to generic manifolds.