Stein fillability and the realization of contact manifolds

This paper proves that two possible notions of Stein fillability for a contact structure are the same. A contact geometer typically says a contact manifold $(M,\xi)$(M,ξ) is Stein fillable if $M$M is the noncritical level set of a proper plurisubharmonic function $f\colon X\to [0,\infty)$f:X→[0,∞) and $\xi$ξ is the set of complex tangencies to the level set $M.$M. This is a very useful notion in contact geometry and implies strong geometric properties for the contact structure if $M$M is 3-dimensional. (In particular, the contact structure will be tight. Moreover, many tight contact structures are constructed by constructing a Stein filling of a 3-manifold.) There is another, "more intrinsic'', notion of Stein fillability. Suppose $X$X is a Stein manifold and $X$X is the interior of a closed manifold $\overline{X}$X−− such that $\overline{X}=M\cup X$X−−=M∪X and the complex structure on $X$X extends to one on $\overline{X}. $X−−. Then we can say the contact manifold $(M,\xi)$(M,ξ) is filled by $X$X, where $\xi$ξ is the set of complex tangencies to $M.$M. In this case the authors say that $M$M is the "intrinsic'' boundary of $X.$X. The main result of this paper shows the equivalence of these two notions. The first notion obviously implies the second, but the converse is non-obvious. Along the way the authors prove several results. In particular, they show that if $(M,\xi)$(M,ξ) is a 3-dimensional Stein fillable contact manifold then $M$M embeds in complex 4-space such that $\xi$ξ is the set of complex tangencies to the embedding. They also show that if $M$M is an intrinsic boundary of $X$X then $\overline{X}$X−− embeds as a domain in a larger open complex manifold with strictly pseudoconvex boundary $M.$M. However, the germ of this extension of $\overline{X}$X−− is not unique.