Summary: SYMPLECTIC HYPERSURFACES IN THE COMPLEMENT
OF AN ISOTROPIC SUBMANIFOLD
DENIS AUROUX, DAMIEN GAYET, AND JEAN-PAUL MOHSEN
Abstract. Using Donaldson's approximately holomorphic techniques,
we construct symplectic hypersurfaces lying in the complement of any
given compact isotropic submanifold of a compact symplectic manifold.
We discuss the connection with rational convexity results in the Kšahler
case and various applications.
It was first observed by Duval (see e.g. [Du]) that, in Kšahler geometry,
the notions of isotropy and rational convexity are tightly related to each
other. Recall that a compact subset N of Cn or more generally of a complex
algebraic manifold is said to be rationally convex if there exists a complex
algebraic hypersurface passing through any given point in the complement of
N and avoiding N. Among the results motivating the interest in this notion,
one can mention the classical theorem of Oka and Weil (further improved
by subsequent work) stating that every holomorphic function over a neigh-
borhood of a rationally convex compact subset N Cn can be uniformly
approximated over N by rational functions.
It was shown in 1995 by Duval and Sibony that, if a smooth compact