One-parameter Families of Homeomorphisms, Topological Monodromies, and Foliations

Abstract

The homological monodromy of a degeneration whose singular fiber has at most normal crossings was described by C. H. Clemens. In his work, local monodromies were described in detail. It is actually a classical result that the local monodromy around a node is a Dehn twist. For higher-dimensional case, we describe local monodromies alternatively: On a local smooth fiber of dimension $n \geq 2$, we construct $n+1$ singular foliations and then describe the action of the local monodromy on each leaf. Here the $i$th singular foliation is used for describing its action on the $i$th face of the boundary of a local smooth fiber.