Local study of ordinary double points

Analogously to real Morse theory, every holomorphic function around an ordinary double points is of the form

where are suitable coordinates. Let us consider the fiber around a critical point for some .


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 and . 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 .