 
Summary: SEMIGLOBAL EXTENSION OF MAXIMALLY
COMPLEX SUBMANIFOLDS
GIUSEPPE DELLA SALA,
ALBERTO SARACCO
Abstract. Let A be a domain of the boundary of a strictly pseu
doconvex domain of Cn
and M a smooth, closed, maximally
complex submanifold of A. We find a subdomain A of , depend
ing only on and A, and a complex variety W A such that
bW A = M. Moreover, a generalization to analytic sets of depth
at least 4 is given.
1. Introduction
In the last fifty years, the boundary problem, i.e. the problem of
characterizing real submanifolds which are boundaries of "something"
analytic, has been widely treaten.
The first result of this kind is due to Wermer [19]: compact real
curves in Cn
are boundaries of complex varieties if and only if they
satisfy a global integral condition, the moments condition. For greater
dimension the problem was solved, by Harvey and Lawson [8], prov
