 
Summary: A REMARK ABOUT DONALDSON'S CONSTRUCTION OF
SYMPLECTIC SUBMANIFOLDS
D. AUROUX
Abstract. We describe a simplification of Donaldson's arguments for the
construction of symplectic hypersurfaces [4] or Lefschetz pencils [5] that makes
it possible to avoid any reference to Yomdin's work on the complexity of real
algebraic sets.
1. Introduction
Donaldson's construction of symplectic submanifolds [4] is unquestionably one
of the major results obtained in the past ten years in symplectic topology. What
sets it apart from many of the results obtained during the same period is that
it appeals neither to SeibergWitten theory, nor to pseudoholomorphic curves;
in fact, most of Donaldson's argument is a remarkable succession of elementary
observations, combined in a particularly clever way. One ingredient of the proof that
does not qualify as elementary, though, is an effective version of Sard's theorem for
approximately holomorphic complexvalued functions over a ball in Cn
(Theorem 20
in [4]). The proof of this result, which occupies a significant portion of Donaldson's
paper (§4 and §5 of [4]), appeals to very subtle considerations about the complexity
of real algebraic sets, following ideas of Yomdin [6].
