 
Summary: ESTIMATED TRANSVERSALITY IN SYMPLECTIC
GEOMETRY AND PROJECTIVE MAPS
DENIS AUROUX
1. Introduction
Since Donaldson's original work [7], approximately holomorphic tech
niques have proven themselves most useful in symplectic geometry and topol
ogy, and various classical constructions from algebraic geometry have been
extended to the case of symplectic manifolds [3, 4, 8, 11]. All these results re
ly on an estimated transversality statement for approximately holomorphic
sections of very positive bundles, obtained by Donaldson [7, 8]. However,
the arguments require transversality not only for sections but also for their
covariant derivatives, which makes it necessary to painstakingly imitate the
arguments underlying Thom's classical strong transversality theorem for jets.
It is our aim in this paper to formulate and prove a general result of
estimated transversality with respect to finite stratifications in jet bundles.
The transversality properties obtained in the various abovementioned pa
pers then follow as direct corollaries of this result, thus allowing some of the
arguments to be greatly simplified. The result can be formulated as follows
(see §2 and §3 for definitions) :
Theorem 1.1. Let (Ek)k 0 be an asymptotically very ample sequence of lo
