Analogously to real Morse theory, every holomorphic function around an ordinary double points is of the form
are suitable coordinates. Let us consider the fiber around a critical point
We define a "vanishing cycle" for , where , to be the sphere
We define a "Lefschetz thimble"
to be the union of all vanishing spheres at some real path between
. Note that
depends on the choice of the path.
One can check that a vanishing cycle is Lagrangian in .
Assume that and are regular values of . Let be all critical points. Connect with by some real paths, and define together with for to be the Lefschetz pencil and the Lefschetz thimble at , respectively, with respect to the ordinary double point . Then is homotopic to .
Proof (Sketch of the proof)
Let be the path connecting and . Define a skeleton . Then, the standard "run a flow" argument implies that is homotopic to . Similarily, we retract every fiber over to .
In the case when , the proposition above implies the Lefschetz hyperplane theorem. In general, one can easily reprove Lefschetz hyperplane theorem by a careful inductive argument comparing and .