Monodromie

Monodromie, or monodromy, is a concept in mathematics that describes how objects behave when they are analytically continued along paths that loop around singularities or other special points. It captures the idea that transporting a object along different loops can yield different results, even if the loops start and end at the same base point.

In topology and geometry, monodromy arises from the study of fibrations and covering spaces. If a space E maps to a base B with fiber F, then the fundamental group of B acts on F by transporting points along loops. This action, called the monodromy representation, encodes how the fibers are permuted or transformed when one travels around nontrivial loops in B.

In complex analysis and the theory of differential equations, monodromy describes the change of analytic continuations of multivalued functions or of solutions to linear differential equations around singular points. For a linear system with singularities, continuation around a loop yields a linear transformation on the space of solutions, represented by a monodromy matrix. The set of such matrices forms the monodromy group, an important invariant of the equation.

In algebraic geometry and Hodge theory, monodromy appears in families of varieties and in the study of period maps. The Gauss–Manin connection provides a flat, or locally constant, structure on cohomology along the base, generating a monodromy representation of the base’s fundamental group on cohomology groups.

Monodromy provides a unifying framework for understanding how local data extend globally in settings ranging from analytic functions to fiber bundles, and it plays a central role in many areas of mathematics.