GENERIC BEHAVIOR OF ASYMPTOTICALLY HOLOMORPHIC
JAUME AMOR ´OS, VICENTE MU~NOZ, AND FRANCISCO PRESAS
Abstract. We prove that the vanishing spheres of the Lefschetz pencils constructed by
Donaldson on symplectic manifolds of any dimension are conjugated under the action of the
symplectomorphism group of the fiber. More precisely, a number of generalized Dehn twists
may be used to conjugate those spheres. This implies the non-existence of homologically
trivial vanishing spheres in these pencils. To develop the proof, we discuss some basic
topological properties of the space of asymptotically holomorphic transverse sections.
In this article we analyze the generic behavior of vanishing spheres in the symplectic pencils
introduced by Donaldson in , henceforth referred to as Donaldson's -transverse Lefschetz
pencils (see Section 2 for precise definitions), and show it to be similar to the case of Lefschetz
pencils for complex projective varieties. Using the pencils as a tool, we start the study of
the symplectic analogue of the dual variety in algebraic geometry, which we believe will be
of interest in symplectic topology.
The property of Lefschetz pencils on projective algebraic varieties that we seek to extend
Theorem 1.1 (cf. SGA 7 XVIII, 6.6.2). Let M be a complex projective manifold, and L M