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.